QUANTUM PHASE TRANSITIONS OF MAGNETIC IMPURITIES IN ...ufdcimages.uflib.ufl.edu/UF/E0/04/23/86/00001/cheng_m.pdf · I dedicate this dissertation to my family, particularly, to myself

QUANTUM PHASE TRANSITIONS OF MAGNETIC IMPURITIES IN DISSIPATIVEENVIRONMENTS

By

MENGXING CHENG

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010

c⃝ 2010 Mengxing Cheng

2

I dedicate this dissertation to my family, particularly, to myself for working on this Ph. D.program for more than five years; to my parents for raising and supporting me; to my

wife for her patience and understanding; to my son for bringing me enormous pleasureeveryday; to my parents-in-law for their help; and to my grandparents, uncles, and aunts

for encouraging me.

3

ACKNOWLEDGMENTS

My greatest appreciation goes to Kevin Ingersent for supervising me as adviser

throughout the time it took me to complete this research and write the dissertation.

Kevin has been a role model for inquisitiveness, healthy skepticism, clear writing, and

honesty in the scientific community. His patience and encouragement has helped me

through several periods of doubt. His physical insight and knowledge have guided my

Ph.D. studies. The members of my dissertation committee, Gregory Stewart, Selman

Hershfield, Sergei Obukhov, and Simon Phillpot, have generously given their time and

expertise to improve my work. I thank them for their contributions and their good-natured

support.

I am grateful to persons who shared their memories and experiences, especially

Yinan Yu and family, Bo Liu and family, Yuning Wu, Xingyuan Pan, Chungwei Wang,

Yun-Wen Chen, Vivek Mishra, Lili Deng, Tomoyuki Nakayama, and many other friends at

the Department of Physics. I must also acknowledge former group colleagues, Matthew

Glossop and Brian Lane, who generously shared their understanding of physics and

computing skills during my first two years at UF as a junior graduate student. I have to

thank Luis G. G. V. Dias da Silva for valuable discussions.

I thank as well the many teachers, friends, colleagues, and staff who assisted,

advised, and supported my research and writing efforts over the years. Especially, I

need to express deep appreciation to Kristin Nichola for her hospitality and assistance.

My thanks must go also to David Hansen and Brent Nelson for their technical support.

I also need to express my gratitude to the staff of the UF High-Performance Computing

Center at which most of the computational work included in this dissertation was

performed. Financial support has been provided by the Division of Materials Research

of the U.S. National Science Foundation.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 The Kondo Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Overview of Quantum Phase Transitions . . . . . . . . . . . . . . . . . . . 161.4 Quantum Criticality of Heavy-Fermion Materials . . . . . . . . . . . . . . . 181.5 Quantum Phase Transitions in Quantum-Dot and Single-Molecule Devices 23

2 NUMERICAL RENORMALIZATION-GROUP METHOD FOR QUANTUM IMPURITYMODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 Overview of Numerical Renormalization-Group Method . . . . . . . . . . 252.2 Numerical Renormalization Group with a Fermionic Band . . . . . . . . . 27

2.2.1 Logarithmic Discretization . . . . . . . . . . . . . . . . . . . . . . . 282.2.2 Mapping onto a Semi-Infinite Chain . . . . . . . . . . . . . . . . . . 302.2.3 Iterative Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Numerical Renormalization Group with a Bosonic Bath . . . . . . . . . . . 34

3 RESULTS FOR CHARGE-COUPLED BOSE-FERMI ANDERSON MODEL . . 38

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Model and Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.1 Charge-Coupled Bose-Fermi Anderson Hamiltonian and RelatedModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.2 Numerical Renormalization-Group Treatment . . . . . . . . . . . . 423.3 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1 Zero Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3.2 Zero Electron-Boson Coupling . . . . . . . . . . . . . . . . . . . . . 503.3.3 Expectations for the Full Model . . . . . . . . . . . . . . . . . . . . 53

3.4 Results: Symmetric Model with Sub-Ohmic Dissipation . . . . . . . . . . 573.4.1 NRG Flows and Fixed Points . . . . . . . . . . . . . . . . . . . . . 583.4.2 Critical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.3 Crossover Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4.4 Thermodynamic Susceptibilities . . . . . . . . . . . . . . . . . . . . 653.4.5 Local Charge Response . . . . . . . . . . . . . . . . . . . . . . . . 68

5

3.4.5.1 Static local charge response . . . . . . . . . . . . . . . . 683.4.5.2 Dynamical local charge susceptibility . . . . . . . . . . . 70

3.4.6 Impurity Spectral Function . . . . . . . . . . . . . . . . . . . . . . . 723.4.7 Spin-Kondo to Charge-Kondo Crossover . . . . . . . . . . . . . . . 75

3.5 Results: Symmetric Model with Ohmic Dissipation . . . . . . . . . . . . . 763.5.1 Fixed Points and Thermodynamic Susceptibilities . . . . . . . . . . 763.5.2 Static Local Charge Susceptibility and Crossover Scale . . . . . . 773.5.3 Impurity Spectral Function . . . . . . . . . . . . . . . . . . . . . . . 78

3.6 Results: Asymmetric Model . . . . . . . . . . . . . . . . . . . . . . . . . . 783.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4 RESULTS FOR PSEUDOGAP ANDERSON-HOLSTEIN MODEL . . . . . . . . 117

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.2 Model Hamiltonian and Preliminary Analysis . . . . . . . . . . . . . . . . 118

4.2.1 Pseudogap Anderson-Holstein Model . . . . . . . . . . . . . . . . 1184.2.2 Review of Related Models . . . . . . . . . . . . . . . . . . . . . . . 120

4.2.2.1 Pseudogap Anderson model. . . . . . . . . . . . . . . . . 1204.2.2.2 Anderson-Holstein model . . . . . . . . . . . . . . . . . . 121

4.3 Results: Symmetric PAH Model with Band Exponents 0 < r < 12

. . . . . . 1224.3.1 NRG Spectrum and Fixed Points . . . . . . . . . . . . . . . . . . . 1234.3.2 Phase Boundaries and Crossover Scales . . . . . . . . . . . . . . 1264.3.3 Thermodynamic Properties . . . . . . . . . . . . . . . . . . . . . . 1274.3.4 Local Response and Universality Class . . . . . . . . . . . . . . . 128

4.4 Results: General PAH Model with Band Exponents 0 < r < 1 . . . . . . . 1314.5 Results: Double Quantum Dots with U2 = 0 . . . . . . . . . . . . . . . . . 132

4.5.1 Effective Pseudogap Model for Double Quantum Dots . . . . . . . 1324.5.2 Impurity Thermodynamic Properties . . . . . . . . . . . . . . . . . 1344.5.3 Phase Diagrams and Critical Couplings . . . . . . . . . . . . . . . 1354.5.4 Linear Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5 CONCLUSIONS AND FUTURE DIRECTIONS . . . . . . . . . . . . . . . . . . 157

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6

LIST OF TABLES

Table page

3-1 Crossover e-b coupling λc0 for the zero-hybridization model . . . . . . . . . . . 50

3-2 Correlation-length critical exponent ν vs bath exponent s . . . . . . . . . . . . . 65

3-3 Static critical exponents β, 1/δ, x , and γ . . . . . . . . . . . . . . . . . . . . . . 71

4-1 Correlation-length critical exponents ν1 and ν2 vs band exponent r . . . . . . . 127

4-2 Exponents describing the local spin response at the critical point Cs of theparticle-hole-symmetric PAH model . . . . . . . . . . . . . . . . . . . . . . . . 129

4-3 Exponents describing the local spin response of the particle-hole-asymmetricPAH model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7

LIST OF FIGURES

Figure page

3-1 Level crossing of the zero-hybridization model . . . . . . . . . . . . . . . . . . . 84

3-2 Dependence of the level-crossing coupling λc0 on the discretization Λ for thezero-hybridization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3-3 Schematic phase diagram of the symmetric charge-coupled BFA model forbath exponents 0 < s < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3-4 Schematic renormalization-group flows on the λ-∆ plane for 0 < s < 1 . . . . . 87

3-5 Low-lying many-body energies EN vs even iteration number N for s = 0.2 . . . 88

3-6 CouplingWd vs λ− λc in the localized phase near the phase boundary . . . . 89

3-7 Dependence of the energy of the first bosonic excitation at the critical pointon the NRG truncation parameters Nb and Ns . . . . . . . . . . . . . . . . . . . 90

3-8 Dependence of the energy of the first bosonic excitation in the localized phaseon the NRG truncation parameters Nb and Ns . . . . . . . . . . . . . . . . . . . 91

3-9 Dependence of ⟨B20⟩ on the NRG truncation parameters Nb for s = 0.2 ands = 0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3-10 Critical coupling λc vs hybridization width Γ for four bath exponents s . . . . . . 93

3-11 Variation with bath exponent s of the critical couplings λc , λc0, and gc/2 . . . . 94

3-12 Crossover scale T∗ vs λc − λ on the Kondo side of the critical point for fourbath exponents s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3-13 Temperature dependence of the impurity contribution to the static spin andcharge susceptibilities for s = 0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3-14 Impurity charge Qloc(λ,ϕ → 0;T = 0) vs e-b coupling λ − λc for four bathexponents s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3-15 Impurity charge Qloc(ϕ;λ=λc ,T =0) vs local electric potential |ϕ| for four bathexponents s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3-16 Static local charge susceptibility χc,loc(T ;ω = 0) vs temperature T for s = 0.4 . 99

3-17 Imaginary part of the dynamical local charge susceptibility χ′′c,loc(ω;T = 0) vs

frequency ω for s = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3-18 Critical static and dynamical response: χc,loc(T ;λ = λc ,ω = 0) vs T andχ′′c,loc(ω;λ = λc ,T = 0) vs ω for two bath exponents s = 0.2 and s = 0.8 . . . . 101

8

3-19 Scaling with ω/T of the imaginary part of the dynamical local charge susceptibilityχ′′c,loc(ω,T ) at the critical e-b coupling λc for s = 0.2 . . . . . . . . . . . . . . . 102

3-20 Impurity spectral function A(ω;T = 0) vs frequency ω for s = 0.8 . . . . . . . . 103

3-21 Variation with e-b coupling λ of two characteristic energy scales ωH and ΓKfor s = 0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3-22 Detail of the impurity spectral function A(ω;T = 0) around frequency ω = 0for s = 0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3-23 Variation with e-b coupling λ < λc of the Kondo resonance width 2ΓK, theinverse static local spin susceptibility 1/χs,loc(ω = 0,T = 0), and the inversestatic local charge susceptibility 4/χc,loc(ω = 0,T = 0) . . . . . . . . . . . . . . 106

3-24 Phase boundary λc(Γ) and crossover boundary λX(Γ) for s = 0.8 . . . . . . . . 107

3-25 Schematic renormalization-group flows on the λ-∆ plane for the symmetricmodel with bath exponent s = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3-26 Static local charge susceptibility χc,loc(T ;ω = 0) vs temperature T for s = 1 . . 109

3-27 Variation with e-b coupling λ of the local charge susceptibility χc,loc(ω = T =0) in the Kondo phase and of the order parameter Qloc(ϕ → 0−;T = 0) in thelocalized phase for s = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3-28 Impurity spectral function A(ω;T = 0) vs ω for s = 1 . . . . . . . . . . . . . . . 111

3-29 Variation with e-b coupling λ of two characteristic energy scales ωH and ΓKfor s = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3-30 Impurity spectral function A(ω;T = 0) vs frequency ω on a logarithmic scalefor s = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3-31 Variation in the magnitude ⟨1 − nd⟩0 of the ground-state impurity charge withe-b coupling λ for s = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3-32 Variation in the bosonic localization temperature TL with coupling λ for s = 0.4 115

3-33 Static local charge susceptibility χc,loc(T ;ω = 0) vs temperature T for s = 0.4 . 116

4-1 Schematic Γ-δd phase diagrams of the pseudogap Anderson model . . . . . . 140

4-2 NRG energy EN vs even iteration number N for symmetric PAH model at weakbosonic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4-3 NRG energy EN vs even iteration number N for symmetric PAH model at strongbosonic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4-4 Schematic renormalization-group flows on the Γ-Ueff plane for the symmetricPAH model with a band exponent 0 < r < 1

2. . . . . . . . . . . . . . . . . . . . 143

9

4-5 Phase boundaries of the symmetric PAH model: Variation with bosonic couplingλ of the critical hybridization widths Γc1 and Γc2 . . . . . . . . . . . . . . . . . . 144

4-6 Symmetric PAH model at weak bosonic coupling: Temperature dependenceof the impurity contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4-7 Symmetric PAH model at strong bosonic coupling: Temperature dependenceof the impurity contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4-8 Local spin response for symmetric PAH model near the spin-sector quantumphase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4-9 Local charge response for symmetric PAH model near the charge-sector quantumphase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4-10 Behaviors of local properties for symmetric PAH model at both weak and strongcouplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4-11 Phase boundaries of the PAH model on the δd -Γ plane for three weak bosoniccouplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4-12 Phase boundaries of the symmetric PAH model on the h-Γ plane for three strongbosonic couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4-13 Temperature dependence of impurity contributions for U2 = 0 double-quantum-dotdevice at weak bosonic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4-14 Temperature dependence of impurity contributions for U2 = 0 double-quantum-dotdevice at strong bosonic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4-15 Phase diagrams of U2 = 0 double-quantum-dot device . . . . . . . . . . . . . . 154

4-16 Linear conductance g for U2 = 0 double-quantum-dot device near the spin-sectorquantum phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4-17 Linear conductance g for U2 = 0 double-quantum-dot device near the charge-sectorquantum phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

QUANTUM PHASE TRANSITIONS OF MAGNETIC IMPURITIES IN DISSIPATIVEENVIRONMENTS

By

Mengxing Cheng

December 2010

Chair: Kevin IngersentMajor: Physics

This dissertation presents results of theoretical research on quantum phase

transitions in systems where a magnetic impurity hybridizes with a fermionic host and

is also coupled, via the impurity charge, to one or more bosonic modes representing

dissipative environments. Two such dissipative quantum impurity models are studied

using the numerical renormalization-group technique.

The charge-coupled Bose-Fermi Anderson model describes a magnetic impurity

that hybridizes with conduction electrons in a metal and is also coupled to a bath of

dispersive bosons. The metallic host is described by a constant density of states, while

the bath is described by a spectral density proportional to ωs , where the value of the

exponent s depends on the particular realization of the model. The following properties

of the model are established: (i) As the impurity-bath coupling increases from zero

at fixed impurity-band hybridization, the effective Coulomb interaction between two

electrons in the impurity level is progressively renormalized from its repulsive bare value

until it eventually becomes attractive. For weak hybridization, this renormalization in turn

produces a crossover from a conventional, spin-sector Kondo effect to a charge Kondo

effect. (ii) At particle-hole symmetry, and for sub-Ohmic bath exponents 0 < s < 1,

further increase in the impurity-bath coupling results in a continuous, zero-temperature

quantum phase transition to a broken-symmetry phase in which the ground-state

impurity occupancy nd acquires an expectation value ⟨nd⟩0 = 1. The response of the

11

impurity occupancy to a locally applied electric potential features the hyperscaling of

critical exponents and ω/T scaling that are expected at an interacting critical point. For

the Ohmic case s = 1, the transition is instead of Kosterlitz-Thouless type. (iii) Away

from particle-hole symmetry, the quantum phase transition is replaced by a smooth

crossover, but signatures of the symmetric quantum critical point remain in the physical

properties at elevated temperatures and/or frequencies.

In the pseudogap Anderson-Holstein model, a magnetic impurity level hybridizes

with a fermionic host whose density of states vanishes as |ϵ|r at the Fermi energy

(ϵ = 0) and is also coupled, via the impurity charge, to a local boson mode. We

find that the pseudogap Anderson-Holstein model shows distinctive low-temperature

quantum fluctuations in two regimes, depending on the strength of the impurity-boson

coupling. We study two cases of band exponents: 0 < r < 1 and r = 2. (i) For

0 < r < 1, the pseudogap Anderson-Holstein model exhibits continuous quantum phase

transitions with anomalous critical exponents. At fixed weak impurity-boson couplings,

as the impurity-band hybridization increases from zero, transitions occur between a

local-moment phase and two strong-coupling (Kondo) phases. However, at fixed strong

impurity-boson couplings, increase in the impurity-band hybridization instead leads

to continuous quantum phase transitions from a local-charge phase to another two

strong-coupling phases. Particle-hole asymmetry in the model with weak impurity-boson

couplings acts in a manner analogous to a local magnetic field applied to the model with

strong impurity-boson couplings. (ii) For r = 2, the pseudogap Anderson-Holstein model

can effectively describe a particular boson-coupled two-quantum-dot setup. In this case,

quantum phase transitions between local spin (charge) and strong-coupling phases are

manifested by peak-and-valley features in the gate-voltage (magnetic-field) dependence

of the linear electrical conductance through the device.

12

CHAPTER 1INTRODUCTION

1.1 Outline

The dissertation is organized as follows. This first chapter provides background

knowledge and motivation for the research. We first review the history and models

of Kondo physics. After a brief overview of the general concept of a quantum phase

transition, we provide a discussion of relevant experimental results and review related

theory, motivating the study of quantum phase transitions of a magnetic impurity

in a dissipative environment. Chapter 2 describes the formalism of the numerical

renormalization group, the technique we use to investigate this area of physics. Chapter

3 presents results for a charge-coupled Bose-Fermi Anderson model describing a

magnetic impurity that hybridizes with conduction electrons in a host metal and is also

coupled to a dispersive bosonic bath. Chapter 4 presents results for a pseudogap

Anderson-Holstein model of a magnetic impurity that hybridizes with a pseudogapped

fermionic host and is also coupled, via its charge, to a local boson mode. Chapter 5

summarizes our results and indicates future directions.

1.2 The Kondo Effect

The behavior of magnetic impurities in metals has been a challenging and

stimulating field since the 1930s [1]. The experimental observation of a low-temperature

minimum in the resistance of nominally pure metals such as silver and gold was not

explained until 1964, when Kondo [2] calculated the resistivity in the single-impurity s-d

model, now more usually referred to as the Kondo model, described by the Hamiltonian

HK =∑kσ

ϵkc†kσckσ +

J

Nk

∑k,k′,σ,σ′

c†kσ1

2σσσ′ck′σ′ · S. (1–1)

Here, ckσ is the annihilation operator for a conduction electron of spin z component

σ = ±12

and energy ϵk. J is the strength of the exchange interaction between the impurity

spin S and the conduction electron spin at the impurity site, written in the terms of the

13

set of Pauli matrices σσσ′. Nk is the number of unit cells in the host metal and, hence,

the number of inequivalent wavevector k values.

Kondo’s perturbative calculation to the third order in J correctly describes the

observed upturn of the resistance at low temperatures [2]. However, it also produces an

unphysical logarithmic divergence of the resistance at T = 0. Later, a more advanced

perturbative approach using many-body diagrammatic techniques [3] to carry out

infinite-order summation of leading diagrams worked well for ferromagnetic exchange

(J < 0) but for antiferromagnetic exchange (J > 0) moved the divergence from

T = 0 to T = TK ≃ (kB)−1D exp[−1/(ρ0J)], where kB is Boltzmann’s constant,

D is half of the bandwidth, and ρ0 is the Fermi-energy density of states. In the late

1960s, Anderson and his collaborators developed a new scaling approach [4–7] which

shed light on the “Kondo problem”. By adjusting J to compensate for progressively

eliminating high-energy excitations, they found a scaling trajectory suggesting that the

antiferromagnetic coupling J approaches infinity at very low temperatures. This means

that the impurity is strongly coupled to the conduction electrons to form a spin singlet,

leaving a completely compensated ground state with non-magnetic behavior. In 1975,

Wilson combined the idea of scaling with that of renormalization from high-energy theory

to develop the numerical renormalization-group (NRG) method, a non-perturbative tool

which he initially applied to the Kondo problem [8], successfully giving the entire picture

of crossover from an asymptotically free magnetic impurity at high temperatures to a

strongly bound Kondo singlet at low temperatures.

In order to understand the formation of the local moment in a host metal, Anderson

had previously proposed a model considering scattering of conduction electrons off

transition metal or rare earth impurities whose d levels mix weakly with the host metal’s

conduction band and experience a Coulomb interaction for double occupancy [9]. The

14

Anderson model is

HA =∑kσ

ϵkc†kσckσ + ϵd nd +

1√Nk

∑kσ

(Vk c†kσdσ + V

∗k d

†σckσ) + Und↑nd↓. (1–2)

Here, dσ annihilates an electron of spin z component σ = ±12

(or σ = ↑, ↓) and energy

ϵd < 0 in the impurity level, ndσ = d †σdσ, nd = nd↑ + nd↓, and U > 0 is the Coulomb

repulsion between two electrons in the impurity level. Nk has the same definition as in

Eq. (1–1). Vk is the hybridization between the impurity and a conduction-band state of

energy ϵk annihilated by fermionic operator ckσ. Henceforth, we follow common practice

and assume a local hybridization Vk = V .

In the atomic limit V = 0, the possible impurity configurations are (i) an empty

state |0⟩ with energy 0; (ii) singly occupied states |↑⟩ and |↓⟩ with energy ϵd ; and (iii) a

doubly occupied state |↑↓⟩ with energy 2ϵd + U. If −ϵd ≫ kBT and ϵd + U ≫ kBT ,

the singly occupied states will be energetically favored, leaving a net spin σ =↑ or ↓

on the impurity. In the noninteracting limit U = 0, the effect of the hybridization V is to

broaden the impurity d level to a Lorentzian resonance of width Γ = πρ0|V |2, where ρ0

is the conduction band density of states. For a general case with U > 0 and V = 0,

the conditions for local-moment formation are modified to be −ϵd ≫ max(Γ, kBT ) and

ϵd + U ≫ max(Γ, kBT ). In this limit, the Anderson Hamiltonian Eq. (1–2) can be mapped

onto the Kondo Hamiltonian Eq. (1–1) through the Schrieffer-Wolff transformation [10],

giving the exchange interaction J as

J = 2|V |2(1

−ϵd+

1

ϵd + U

). (1–3)

Thus, the Kondo model describes a limiting situation of the more general Anderson

model.

Since the end of 1990s, the Kondo and Anderson models of magnetic impurities

have been realized both in semiconductor quantum dots [11, 12] and in single-molecule

transistors [13, 14]. At low temperatures, interaction between a localized electron with

15

an unpaired spin in the dot or molecule and conduction electrons in the electrical leads

produces a Kondo (or Abrikosov-Suhl) resonance in the impurity density of states at the

Fermi energy, which greatly enhances the zero-bias conductance even up to the unitary

limit [15]. Zero-bias conductance data collapse to a universal function of a normalized

ratio T/TK , indicating that the Kondo temperature TK is the characteristic energy scale.

1.3 Overview of Quantum Phase Transitions

Phase transitions are ubiquitous and familiar. In our daily life, we all have observed

water boiling and snow melting. Among classic examples in physics textbooks are

magnetic and superconducting phase transitions reached upon lowering temperature.

These classical phase transitions result from a delicate balance of energy and entropy

reached by tuning the temperature [16].

During recent years, enormous attention has been attracted to another kind of

phase transition, namely, quantum phase transitions (QPTs) occurring at the absolute

zero of temperature (T = 0) [17, 18]. QPTs are closely associated with many

fundamental problems in condensed matter physics, including the rich behaviors

of heavy-fermion metals [19], the high-temperature superconductivity in cuprate

compounds [20, 21], the metal-insulator transition in disordered electronic systems [22],

and the quantum Hall effect in two-dimensional electron systems [23].

At T = 0, thermal fluctuations are completely suppressed. However, Heisenberg’s

uncertainty principle tells us that it is impossible to simultaneously determine both

the position and momentum of a particle. Thus the basic law of quantum mechanics

predicts non-thermal zero-point fluctuations that may destroy long-range macroscopic

order just as thermal fluctuations do at classical phase transitions. At first glance,

QPTs seem to be of little practical interest because the absolute zero of temperature

is not accessible in any real-world experiment. However, as found over the last two

decades, the presence of a QPT at T = 0 has great influence on the measurable

16

physical properties at T > 0, producing fascinating phase diagrams in a variety of

materials [19, 24, 25].

To understand the physics of QPTs, it is useful to draw analogies with the theory of

classical phase transitions [17]. A classical phase transition takes place when a system

is cooled down to a transition temperature, whereas a quantum phase transition occurs

at the absolute zero of temperature when some external non-thermal parameter K

such as pressure or magnetic field is tuned to its critical value Kc . Traditionally, phase

transitions are classified a first-order or second-order (continuous). At a first-order

classical phase transition (for examples, ice melts at 0 Celsius), a latent heat is involved

and the order parameter jumps discontinuously at the transition temperature, while

at a first-order quantum phase transition, there is a level-crossing of the ground state

at K = Kc . More interesting is the second-order phase transition where long-range

ordering, arising from spontaneous symmetry breaking, vanishes continuously as

approaching the transition. Classically, as T → Tc , the spatial correlation length ξ

diverges as ξ ∝ |(T − Tc)/Tc |−ν . Here ν is the correlation-length critical exponent.

As a consequence of the diverging ξ, many physical properties show scaling forms

controlled by a set of critical exponents characterizing the universality class of the phase

transition. In contrast, continuous quantum phase transitions are more complicated

because space and time are intertwined in the critical region (as explained in the next

paragraph). Indeed, as K → Kc , not only the spatial correlation length ξ diverges as

ξ ∝ |(K − Kc)/Kc |−ν , but the correlation length ξτ in the time direction also diverges as

ξτ ∝ ξz , where z is the dynamical critical exponent.

A quantum phase transition can be understood in the language of quantum

statistical mechanics as follows [18]. The weight function e−βH (β = 1/kBT , where kB is

the Boltzmann constant) in the partition function looks exactly like the quantum-mechanical

time-evolution operator e−iHT /~ for an imaginary time T = −i~β. Within Feynman’s

path-integral formalism, the imaginary time acts like an additional dimension. In

17

this sense, the expression for the partition function for a real quantum system in d

dimensions looks like the partition function for a classical system in d + 1 dimensions,

except that the time dimension has a finite length ~β. At T = 0, the extra time dimension

is extended to infinity and we have a true d + 1 classical system. At a non-zero

temperature, a finite-size crossover in the time direction occurs when the correlation

time ξτ becomes larger than ~β.

An important point missing from the intuitive picture presented in the previous

paragraph is that space and time need not to enter the path integral on the same

footing [18]. In general, time scales as the power z of a spatial dimension, where

z is the aforementioned dynamical exponent. This suggests that a quantum phase

transition in d dimensions corresponds more closely to a classical phase transition in

D = d + z dimensions. Whether this long-believed correspondence between quantum

and classical transitions generically holds true has recently been the subject of heated

debate [26, 27], as described in more detail in the following section 1.4.

1.4 Quantum Criticality of Heavy-Fermion Materials

The study of low-temperature excitations of interacting fermions is among

the most important aspects of condensed-matter physics. Historically, Landau

proposed a Fermi-liquid theory postulating that the low-energy excitations of interacting

fermions can be described by long-lived quasiparticles whose quantum numbers

(such as charge and spin) are the same as those of the fermions in the absence of

interactions [28]. To lowest order in kBT/ϵF , where ϵF is the Fermi energy, Fermi-liquid

theory predicts a quadratic temperature dependence of the resistivity ρ(T ) =

ρ0 + AT2, a linear temperature dependence of the specific heat C(T ) = γT , and a

temperature-independent magnetic susceptibility χ = M/B ≈ const. Fermi-liquid theory

has been successfully applied to a variety of interacting systems ranging from helium-3

liquid to normal metals like copper or even complex compounds like CeCu6 in which the

interactions between localized f -electrons are very strong.

18

Since the 1990s, a series of experiments in heavy-fermion materials [29–32], a

group of d− and f−electron compounds famous for their gigantic effective masses

(determined experimentally by measuring the ratio of specific heat to temperature C/T ),

have challenged the validity of Landau Fermi-liquid theory. For instance, CeCu6−xAux

exhibits marked non-Fermi-liquid behaviors at x = 0.1, i.e., C/T = aln(T0/T ),

χ = M/B = χ0 − α√T , and ρ(T ) = ρ0 + A

′T [33]. Similar non-Fermi-liquid behaviors

were also observed in YbRh2Si2 and its variants by tuning an external magnetic field;

in fact there is an upturn C/T ∝ T−0.3 below a very low temperature at which the lnT

behavior of C/T is cut off [34, 35]. These non-Fermi-liquid behaviors in heavy-fermion

materials seem always to be associated with quantum phase transitions involving

magnetic ordering.

Hertz first put forward a theory to describe magnetic quantum phase transitions in

metals [36] and the work was later re-examined and corrected by Millis [37]. Assuming

that low-energy fermionic excitations can be completely integrated out and the order

parameter is the only significant fluctuation mode near the transition, a QPT in d

dimensions can be connected to a classical ϕ4-field theory in d + z dimensions, where

z is the aforementioned dynamical exponent. Specifically, it is found that z = 3 for a

ferromagnetic QPT and z = 2 for an antiferromagnetic QPT.

Hertz-Millis theory predicts C/T ∝√T and ∆ρ ∝ T 3/2 for antiferromagnetic

(z = 2) systems in d = 3 dimensions. These temperature dependences are

consistent with experimental observations in a number of heavy-fermion materials

such as CeNi2Ge2 and CeCu2Si2 [30, 32]. However, they contradict those behaviors

mentioned above in CeCu6−xAux and YbRh2Si2. Rosch in 1997 pointed out that the

lnT behavior of C/T and the linear dependence of resistivity in CeCu6−xAux can be

explained within Hertz-Millis theory by assuming that spin fluctuations exist only in

d = 2 dimensions [38]. However, subsequent experimental results have made this

explanation untenable. First, neutron-scattering data for CeCu6−xAux exhibit E/T

19

scaling with an anomalous fractional exponent over essentially the entire Brillouin

zone [39], suggesting an interacting type of critical point only expected below the upper

critical dimension du. However, Hertz-Millis theory predicts du = 4−2 = 2, so fluctuations

in d = 2 or 3 dimensions would be at or above the upper critical dimension, leading to a

Gaussian (mean-field) critical point. Second, experiments on the Hall effect for YbRh2Si2

have observed a sharp change of the Fermi surface at the critical magnetic field [40],

which contradicts the smooth evolution of the Fermi surface implied by Hertz-Millis

theory and indicates a sudden disappearance of a large number of charge carriers

at the critical point. In order to explain the experimental phenomena contradicting the

conventional theory of QPT, Si and collaborators proposed a theory of locally critical

quantum phase transitions [26, 27]. The main idea of the theory is that local degrees

of freedom associated with the Kondo effect and the long-wavelength fluctuations of

the order parameter become critical simultaneously at the QPT. This is different from

the Hertz-Millis picture where only critical long-range order-parameter fluctuations are

important. In the following of this section, we briefly review the theory of locally critical

quantum phase transitions and motivate our study on a related model.

It is generally accepted that QPTs in heavy-fermion materials stem from competition

between Kondo screening of local moments and the Ruderman-Kittel-Kasuya-Yosida

(RKKY) interaction attempting to align those local moments [41]. A microscopic model

capturing this essential physics is the Kondo lattice model,

HKL =∑ij ,σ

tijc†iσcjσ +

∑i

JKSi · sc,i +∑ij

IijSi · Sj . (1–4)

Here, tij is the tight-binding parameter that determines the conduction-band dispersion

εk and hence the density of states ρ(ϵ) = (Nk)−1∑

k δ(ϵ − εk). c†iσ creates an electron

of spin z component σ = ±12

at site i . JK is the Kondo coupling between a localized

spin Si and the conduction-electron spin sc,i at the same lattice site. The tendency

toward magnetism enters through the RKKY interaction Iij between the localized spins

20

on different sites i and j . The Fourier transform of Iij determines the RKKY density of

states ρl(ϵ) = (Nk)−1∑

k δ(ϵ− Ik).

Si et al. have studied the Kondo lattice model within the framework of extended

dynamical mean-field theory (EDMFT), where the lattice model (1–4) is mapped onto a

dissipative quantum impurity model – the Bose-Fermi Kondo model (BFKM), described

by the Hamiltonian [26, 27]

HBFKM =∑k,σ

ϵkc†kσckσ +

∑q

ωqϕ†q · ϕq + JKS · sc +

∑q

gqS · (ϕq + ϕ†−q). (1–5)

Here, a single localized spin S is coupled not only to the on-site conduction-band spin

sc but also to three dissipative bosonic baths of harmonic oscillators described by ϕq,

which represent magnetic fluctuations generated by spins at all other lattice sites. In the

dissipative impurity model, the coupling of the impurity to the conduction band is fully

specified by the exchange function J(ϵ) = (Nk)−1JK

∑k δ(ϵ − ϵk), while the interaction

between the impurity and the bosonic baths is completely described by a bath spectral

function B(ω) ≡ (Nq)−1π

∑q g2qδ(ω − ωq). To reproduce the effect of other lattice sites

on the representative site, the exchange function and the bath spectral function entering

the impurity model are determined by a pair of self-consistency equations involving the

aforementioned conduction-band density of states ρ(ϵ) and the RKKY density of states

ρl(ϵ) of the Kondo lattice model.

It is believed that the RKKY interaction in CeCu6−xAux is highly anisotropic and can

be approximated as∑ij IijS

zi Szj [42, 43]. Within EDMFT, this situation is described by a

self-consistent Ising-anisotropic Bose-Fermi Kondo model, which has been solved using

various methods including ϵ-expansion [27], quantum Monte Carlo [42, 43], and the

numerical renormalization group [44, 45]. All the methods agree that: (a) For magnetic

fluctuations in 3D where the RKKY density of states ρl(ϵ) has a square-root onset at its

lower edge, the lattice static susceptibility at the ordering momentum diverges at the

critical point whereas the local static susceptibility stays finite. (b) By contrast, for 2D

21

magnetic fluctuations where the RKKY density of states undergoes a jump at its lower

edge, both the lattice static susceptibility at the ordering momentum and the local static

susceptibility become singular simultaneously at the critical point. Furthermore, the

critical local static susceptibility shows a power-law dependence on temperature with

a fractional exponent α ≃ 0.75 as observed in experiments in CeCu6−xAux [39], while

the critical local dynamical susceptibility exhibits E/T scaling with the same fractional

exponent.

Although the locally critical theory based on the Kondo lattice model reproduces

the features of several key experiments, the original model (1–5) takes into account

local spin fluctuations but ignores local charge fluctuations. It is natural is to ask what

happens if charge fluctuations are included in a two-band extended Hubbard model [46]

HTBHM =∑i

[ϵd ndi + Undi↑ndi↓ +

∑σ

t(d †iσciσ + H.c.) + V ndi nci + JKSdi · sci

](1–6)

+∑ij

(tij∑σ

c†iσcjσ + Vij(ndi − 1)(ndj − 1) + IijSdi · Sdj

).

The model involves two kinds of electrons, namely, the strongly correlated d electrons

and the band c electrons. Here, ϵd is the energy of the d level and U is the Hubbard

interaction for d-electron double occupancy at the same site. ndi =∑

σ ndiσ. t is the

hybridization between d electrons and c electrons at the same site. V is the local

charge-screening interaction and JK is the Kondo coupling. tij is the intersite hopping for

c electrons. Vij and Iij are the non-local density-density interactions and spin-exchange

interactions between d electrons. Within the framework of EDMFT, the two-band

extended Hubbard model is mapped onto a dissipative quantum impurity model, which

has been found in its mixed-valence regime (and at infinite U) to exhibit a novel phase

associated with standard Fermi-liquid spin excitations and unusual non-Fermi-liquid

charge excitations [46, 47]. This impurity model is closely related to a charge-coupled

Bose-Fermi Anderson model that was first developed by Haldane to address the

22

mixed-valence phenomenon in rare-earth materials [48]. Haldane solved the model

with a mean-field approximation [49] that leaves many open questions about the

strong-correlation regime. In Chapter 3, we report essentially exact numerical results for

the charge-coupled Bose-Fermi Anderson model.

1.5 Quantum Phase Transitions in Quantum-Dot and Single-Molecule Devices

Studying gate-tunable artificial nanoscale devices, such as semiconductor quantum

dots and single-molecule transistors, is one of the most active areas of current scientific

research driven by potential applications in nanoelectronics and quantum computing.

These devices’ outstanding features, such as precise experimental control by means of

gate electrostatics and large spacings between discrete energy levels due to nanoscale

spatial confinement of a small number of electrons, make it feasible to observe quantum

phenomena at relatively high temperatures. For instance, the Kondo (or Abrikosov-Suhl)

resonance has been observed in both semiconductor quantum dots [11, 12] and

single-molecule transistors [13, 14].

While heavy-fermion compounds are prototypical materials for studying bulk

quantum phase transitions as we have seen in the last section, semiconductor quantum

dots and single-molecule transistors provide new means to investigate boundary

quantum phase transitions in which only a zero-measure subset of system degrees of

freedom become critical [50]. Experimentally, boundary QPTs have been realized in

quantum-dot setups where two conduction electron reservoirs are competing to localize

a magnetic dot [51] as well as in single-C60 transistors showing a magnetic field and

gate-induced singlet-triplet transition [52]. Theoretically, certain quantum-dot systems

have been proposed to realize an XY-anisotropic Bose-Fermi Anderson model exhibiting

an interesting boundary QPT [53].

One boundary QPT of theoretical interest is that occurring in the Kondo and

Anderson impurity models when the conduction-band density of states vanishes as

|ϵ|r at the Fermi energy (ϵ = 0) [54], a situation that can be realized in a number of

23

systems including unconventional d-wave superconductors (r = 1) [55] and certain

double-quantum-dot setups (r = 2) [56, 57]. The pseudogap in the density of states

inhibits Kondo screening and incurs a local-moment phase for weak impurity-band

hybridization, whereas the magnetic impurity is still screened by the conduction band

through the Kondo effect for strong impurity-band hybridization. These two stable

phases are separated by a QPT, at which there is a divergent response to a magnetic

field applied locally to the impurity spin [58].

Furthermore, in quantum-dot or single-molecule devices, it is natural that vibrational

modes (local phonons) play significant role. A number of recent experiments have

investigated phonon-assisted electron transport through a quantum dot [59] or

molecule [60, 61] in the Kondo regime. The essential physics of these experiments

seems to be captured by the Anderson-Holstein model [62]

HAH =∑σ

εd ndσ + Und↑nd↓ +∑k,σ

εkc†kσckσ +

1√Nk

∑k,σ

(Vk c†kσdσ + V

∗k d

†σckσ)

+ λ(nd − 1)(a + a†) + ω0a†a, (1–7)

which augments the Anderson impurity model Eq. (1–2) with a Holstein coupling [63] of

the impurity occupancy to a local boson (phonon) mode of energy ω0 (setting ~ = 1).

In Chapter 4, we report our study of a pseudogap Anderson-Holstein model

incorporating the effects both of a pseudogapped conduction band and coupling to a

local boson mode. The model turns out to exhibit interesting QPTs that have remarkable

signatures in the finite-temperature properties.

24

CHAPTER 2NUMERICAL RENORMALIZATION-GROUP METHOD FOR QUANTUM IMPURITY

MODELS

In this chapter, a brief introduction to the numerical renormalization-group (NRG)

method for solving quantum impurity models in Section 2.1 is followed by a more

technical description of the method. The conventional NRG method for pure-fermionic

problems is summarized in Section 2.2. Section 2.3 describes the extension of the NRG

to treat problems including bosonic baths.

2.1 Overview of Numerical Renormalization-Group Method

A quantum impurity model describes a small subsystem with a few degrees

of freedom coupled to a large subsystem with a continuum of states [64]. A typical

example is the Kondo problem (as discussed in Sec. 1.2) where a magnetic impurity

interacts with a metallic conduction band. Study of quantum impurity problems is

challenging because one has to deal with a wide range of energies from an ultra-violet

cutoff (often of order electron volts) down to the smallest scale (typically set by kBT ,

which may be as low as 1µeV in quantum-dot experiments). Perturbation theory usually

fails for such problems due to the appearance of infrared divergences.

In the 1970s, K. G. Wilson developed a non-perturbative method - the numerical

renormalization group - to solve the Kondo problem. The NRG first successfully gave

the entire crossover from a free-magnetic-moment regime at high temperatures to a

strongly coupled Kondo-singlet regime at low temperatures [8]. The NRG involves three

key steps: discretization of the conduction band, tridiagonalization of the Hamiltonian,

and iterative solution.

Discretization: The full range of conduction-band energies −D ≤ ϵ ≤ D is divided

into a set of logarithmic intervals bounded by ϵ = ±DΛ−k for k = 0, 1, 2, ..., where Λ > 1

is the Wilson discretization parameter. The continuum of states within each interval is

replaced by a single state, namely, the particular linear combination of band states within

the interval that directly couples to the impurity site.

25

Tridiagonalization: The Lanczos procedure [65] is used to convert the conduction-band

part of the Hamiltonian [the first term in Eqs. (1–1) and (1–2)] into a tight-banding form

Hband = D∑

σ

∑∞n=0

[ϵnf

†nσfnσ+τn

(f †nσfn−1,σ+ f

†n−1,σfnσ

)]. The operator f0σ annihilates a

spin-σ electron in the linear combination of conduction-band states that couple to the

impurity. The operator fnσ for n > 0 annihilates a linear combination of electrons of

energy |ϵ| ≤ DΛ−n.

Iterative solution: The full quantum-impurity problem can be solved by iteratively

diagonalizing a sequence of Hamiltonians HN (N = 0, 1, 2, ...) satisfying a recursive

relation HN+1 = Λ1/2HN +∑

σ εN+1f†N+1,σfN+1,σ +

∑σ τN+1

(f †N+1,σfNσ+f

†NσfN+1,σ

), which

equivalently describe tight-binding chains of increasing length. Many-body energies and

matrix elements from the diagonalization are then used to calculate static and dynamic

properties. Practically, it is not feasible to keep track of all the eigenstates beyond the

first few iterations because the dimension of the Fock space increases exponentially as

we add sites to the chain. Instead, the Ns lowest lying many-particle states are retained

after each iteration. It is necessary to check that Ns has been chosen sufficiently large to

ensure convergence of physical properties.

After its first application to the Kondo model [8], the NRG was then generalized

to many situations: the Anderson model including charge fluctuations at the impurity

site [66, 67]; the two-channel Kondo model where the magnetic impurity is coupled

to two independent conduction bands [68]; the two-impurity Kondo model treating

competition between Kondo screening and correlation between magnetic impurities [69];

the more complex two-impurity, two-channel Kondo model [70]; and the Kondo/Anderson

model with a pseudogap in the density of states [71]. Another generalization has been

the development of a bosonic NRG, first applied to the spin-boson model [72] and then

to the Bose-Fermi Kondo model [73, 74]. Many of these applications are described in a

recent review by Bulla et al. [64].

26

Sections 2.2 and 2.3 describe the technical formalism of the NRG in more depth.

The reader who is not interested in these details may skip the rest of this chapter and

move directly to the presentation of results in chapter 3.

2.2 Numerical Renormalization Group with a Fermionic Band

This section describes the NRG treatment [66, 67] of the single-impurity Anderson

model (1–2), which can be written conveniently as

HA = Himp + Hband + Himp−band, (2–1)

where

Himp = ϵd nd + Und↑nd↓, (2–2)

Hband =∑k,σ

ϵk c†kσckσ, (2–3)

Himp-band =1√Nk

∑k,σ

(Vk c

†kσdσ + V

∗k d

†σckσ

). (2–4)




repulsion between two electrons in the impurity level. Vk is the hybridization between the

impurity and a conduction-band state of energy ϵk annihilated by fermionic operator ckσ.

Nk is the number of unit cells in the host metal and, hence, the number of inequivalent k

values. Hereafter in this chapter, summation over repeated spin indices σ is assumed.

For simplicity, we consider only spatially isotropic problems, which means that ϵk and

Vk depend only on |k|, so that the impurity interacts only with s-wave states about the

impurity site. Then, it is convenient to replace summation over k by integration over

ϵ ≡ ϵk and replace Vk by V (ϵ).

The interaction between the impurity and the conduction band is completely

determined by the hybridization-width function Γ(ϵ) = πρ(ϵ)|V (ϵ)|2, where ρ(ϵ) is the

27

density of states of the conduction band. In this work, we consider functions of the form

Γ(ϵ) =

Γ0|ϵ/D|r for |ϵ| ≤ D,

0 otherwise.(2–5)

The case r = 0 corresponds to a regular metallic band with a constant density of states,

whereas r > 0 describes a pseudogapped density of states.

Introducing a dimensionless scale ε = ϵ/D for convenience, the Hamiltonian Eq.

(2–1) can be transformed into the one-dimensional form:

Himp = ϵd nd + Und↑nd↓, (2–6)

Hband = D

∫ 1−1dε εc†εσcεσ, (2–7)

Himp−band =

√Γ0D

πF (f †0σdσ + d

†σf0σ), (2–8)

where

f0σ = F−1∫ 1−1dεw(ε)cεσ, (2–9)

which includes a weighting function

w(ε) =

√Γ(εD)

Γ0, (2–10)

and a normalization factor

F 2 =

∫ 1−1dε [w(ε)]2. (2–11)

2.2.1 Logarithmic Discretization

The continuous band spectrum −1 ≤ ε ≤ 1 (−D ≤ ϵ ≤ D) is divided into a sequence

of intervals. The nth positive (negative) interval extends over energy ϵ (-ϵ) from Λ−(n+1) to

Λ−n (n = 0, 1, 2, ...). In his original treatment of the Kondo and Anderson models in which

the conduction-band density of states is constant, namely w(ε) = 1, Wilson defined a

28

complete set of orthonormal functions [8]

ψ±np(ε) =

1√dne±iωnpε for Λ−(n+1) ≤ ± ε ≤ Λ−n,

0 outside this interval,(2–12)

where dn = Λ−n(1 − Λ−1) is the width of the nth interval, p is the Fourier harmonic index

running from −∞ to +∞, ωn = 2π/dn is the fundamental frequency for the nth interval,

and the superscript + (−) stands for positive (negative) ε. The conduction electron

operator cεσ can be expanded in this basis as

cεσ =∑np

[anpσψ

+np(ε) + bnpσψ

−np(ε)

], (2–13)

in terms of operators

anpσ =

∫ 1−1dε [ψ+np(ε)]

∗cεσ, (2–14)

bnpσ =

∫ 1−1dε [ψ−

np(ε)]∗cεσ, (2–15)

which satisfy the usual fermionic anticommutation relations. With this expansion, the

operator f0σ defined in Eq. (2–9) is expanded as

f0σ = F−1∑np

[anpσ

∫ Λ−nΛ−(n+1)

dε ψ+np(ε) + bnpσ

∫ −Λ−(n+1)

−Λ−ndε ψ−

np(ε)]. (2–16)

The integrals in Eq. (2–16) are zero unless p = 0. This means that the p = 0 state is the

only one that couples directly to the impurity.

For an arbitrary w(ε), one can generalize Wilson’s approach by defining

ψ+n0(ε) =

w(ε)/F+n for Λ−(n+1) ≤ ε ≤ Λ−n,

0 otherwise,(2–17)

ψ−n0(ε) =

w(ε)/F−

n for − Λ−n ≤ ε ≤ −Λ−(n+1),


29

with normalization factors

[F+n ]2 =

∫ Λ−nΛ−(n+1)

dε [w(ε)]2, [F−n ]2 =

∫ −Λ−(n+1)

−Λ−ndε [w(ε)]2. (2–19)

With this choice,

f0σ = F−1∑n

[F+n an0σ + F

−n bn0σ

]. (2–20)

In the new basis, the band term Eq. (2–7) is transformed into

Hband ≃ D∑n

[ε+n a

†n0σan0σ + ε

−n b

†n0σbn0σ

], (2–21)

where

ε+n =[F+n ]

−2∫ Λ−nΛ−(n+1)

dε ε[w(ε)]2, (2–22)

ε−n =[F−n ]

−2∫ −Λ−(n+1)

−Λ−ndε ε[w(ε)]2. (2–23)

Here, following the argument by Wilson, we ignore the p = 0 conduction band states

because these states interact only indirectly with the impurity, and fully decouple in the

continuum limit Λ→ 1.

2.2.2 Mapping onto a Semi-Infinite Chain

Using the Lanczos method [65], the band Hamiltonian Eq. (2–21) can be mapped to

the form

Hband = D

∞∑n=0

[εnf

†nσfnσ+τn

(f †nσfn−1,σ+f

†n−1,σfnσ

)], (2–24)

where τ0 ≡ 0. The operator fnσ exhibits only nearest-neighbor coupling to fn±1,σ,

representing the form of a semi-infinite chain. This new set of operators fnσ is constructed

from an0σ and bn0σ via an orthogonal transformation

fnσ =∑m

[µnmam0σ + νnmbm0σ

], (2–25)

where, µ0m = F+m /F and ν0m = F−m /F were already defined in Eq. (2–20). The remaining

coefficients µnm and νnm as well as the parameters εn and τn in the chain form Eq. (2–24)

30

are determined by recursive relations

εn =∑m

(µ2nmε+m + ν

2nmε

−m) , (2–26)

τn+1µn+1,m =(ε+m − εn)µnm − τnµn−1,m , (2–27)

τn+1νn+1,m =(ε−m − εn)νnm − τnνn−1,m , (2–28)

1 =∑m

(µ2n+1,m + ν2n+1,m), (2–29)

with τ0 ≡ 0. These recursive relations have the feature that εn = 0 for all n if the

weighting function (hybridization width) has the symmetry w(ε) = w(−ε) [Γ(ϵ) = Γ(−ϵ)].

In the special case of a constant density of states w(ε) = 1, Wilson derived an explicit

expression for the hopping coefficients

τn =(1 + Λ−1)(1− Λ−n−1)

2√1− Λ−2n−1

√1− Λ−2n−3

Λ−n/2 → 1

2(1 + Λ−1)Λ−n/2 for n ≫ 1. (2–30)

For more general weighting functions, the above recursive relations must be calculated

numerically. It is generally found that the hopping coefficients τn and the on-site energies

εn drop off as Λ−n/2 for large n.

2.2.3 Iterative Diagonalization

After the logarithmic discretization that divides the continuous band spectrum into

a sequence of intervals. and chain mapping that transforms the original band part

of the Anderson Hamiltonian into a semi-infinite chain, the single-impurity Anderson

Hamiltonian is transformed into

HA = D

∞∑n=0

[εnf

†nσfnσ+τn


†n−1,σfnσ

)]+ ϵd nd + Und↑nd↓ +

√Γ0D

πF (f †0σdσ + d

†σf0σ).

(2–31)

Because the chain parameters τn and εn decrease as Λ−n/2, it is convenient to introduce

scaled parameters

en = α−1Λn/2εn , tn = α−1Λn/2τn , (2–32)

31

where, α = 12(1 + Λ−1)Λ1/2 is a conventional factor. Then, one see that the Hamiltonian

Eq. (2–31) can be recovered as the limit of a series of finite Hamiltonians

HA = limN→∞

αΛ−N/2DHN , (2–33)

where HN satisfies the recursive relation

HN+1 = Λ1/2HN + eN+1f

†N+1,σfN+1,σ + tN+1(f

†N+1,σfN,σ + f

†N,σfN+1,σ). (2–34)

The initial Hamiltonian

H0 = e0f†0σf0σ + εdnd + Und↑nd↓ + Γ

1/2(f †0σdσ + d†σf0σ), (2–35)

which includes only the operators f0σ and dσ, with the scaled couplings

εd =εdαD, U =

U

αD, Γ =

Γ0F2

πα2D. (2–36)

Repeated use of the recursive relation Eq. (2–34) allows one to solve the series of

Hamiltonians HN . The procedure is basically as follows. Assume that we have already

solved the eigenvalue equation

HN |r ,N⟩ = E(r ,N)|r ,N⟩, (2–37)

and know all the eigenenergies E(r ,N) and matrix elements ⟨r ,N|f †nσ|r ′,N⟩. The basis

for HN+1 can be constructed as the direct product of the eigenstates |r ,N⟩ and the new

degrees of freedom of the added site. Explicitly, for each of the states |r ,N⟩, there are

four corresponding states

|1, r ,N⟩ = |0⟩N+1 ⊗ |r ,N⟩, (2–38)

|2, r ,N⟩ = |↑⟩N+1 ⊗ |r ,N⟩, (2–39)

|3, r ,N⟩ = |↓⟩N+1 ⊗ |r ,N⟩, (2–40)

|4, r ,N⟩ = |↑↓⟩N+1 ⊗ |r ,N⟩. (2–41)

32

Sandwiching Eq. (2–34) between these states gives the matrix elements of HN+1, viz.,

⟨i ′, r ′,N|HN+1|i , r ,N⟩ =Λ1/2E(r ,N)δi ,i ′δr ,r ′ + eN+1⟨i ′|f †N+1,σfN+1,σ|i⟩δr ,r ′

+ tN+1(⟨i ′|f †N+1,σ|i⟩⟨r′,N|fN,σ|r ,N⟩+ ⟨r ′,N|f †N,σ|r ,N⟩⟨i

′|fN+1,σ|i⟩).

(2–42)

Diagonalizing this matrix gives rise to E(r ,N + 1) and ⟨r ,N + 1|f †nσ|r ′,N + 1⟩, which will

be used to solve HN+2, and so on. Note that the ground-state energy is set to be zero

after each diagonalization.

The whole recursive procedure can be understood [8] in terms of a renormalization

group transformation R:

HN+1 = R[HN ], (2–43)

which transforms the Hamiltonian specified by a set of eigenenergies and matrix

elements into another Hamiltonian of the same form but with a new set of eigenenergies

and matrix elements. Actually, due to the odd-even alternation properties of fermionic

chains, R itself has no fixed point but R2 does have fixed points.

The numerical labor of diagonalizing the Hamiltonian can be significantly reduced

by taking advantage of symmetries. The Hamiltonian Eq. (2–31) commutes with the total

spin operator

S =1

2

∑n

f †n,σσσ,σ′fn,σ′ +1

2d †σσσ,σ′dσ′, (2–44)

where σσ,σ′ is the set of Pauli matrices, and with the total charge operator

Q = nd − 1 +∑n

(f †n↑fn↑ + f

†n↓fn↓ − 1

). (2–45)

At particle-hole symmetry (ϵd = −12U) only, the Hamiltonian also commutes with the

SU(2) isospin (axial charge) operators

Iz =1

2Q, I+ = −d †↑d

†↓ +

∑n

(−1)nf †n↑f†n↓ ≡

(I−)†. (2–46)

33

Consequently, the Hamiltonian can be diagonalized in subspaces labeled by conserved

quantum numbers S , Sz , Q, and (at particle-hole symmetry only) I .

The dimension of the Fock space grows by a factor of four at each successive

iteration. After a few iterations, this becomes too big to allow all states to be computed,

so a truncation must be introduced. The Ns eigenstates with the lowest energies are

kept after each diagonalization. Provided that Ns is chosen large enough, the NRG

solution at iteration N allows accurate evaluation of static thermodynamic properties

at temperatures T ≃ (D/kB)Λ−N/2 and of zero-temperature dynamical properties at

frequencies |ω| ≃ (D/~)Λ−N/2. For the Anderson model, Ns ≃ 500 is big enough to

accurately calculate the eigenenergies, while much larger Ns (≃ 2000) is needed to

calculate the impurity contribution to the entropy and the spin magnetic susceptibility.

Although this truncation may seem to be questionable, it is successful in practice

because the discarded high-energy eigenstates have very little influence on the

low-energy eigenstates of the next iteration.

2.3 Numerical Renormalization Group with a Bosonic Bath

We will describe the bosonic NRG [72] using the example of the spin-boson model,

which describes tunneling within a two-state system coupled to a bosonic bath [75].

The model has many proposed applications, including frictional effects on biological and

chemical reaction rates [76], cold atoms in a quasi-one-dimensional optical trap [77], a

quantum dot coupled to Luttinger-liquid leads [78], and study of entanglement between

a qubit and its environment [79, 80]. In many cases, the dissipative bosonic bath can be

described by a spectral density [formally defined in Eq. (2–48) below] that is proportional

to ωs at low frequencies ω. The spin-boson model with an Ohmic (s = 1) bath has long

been known [75] to exhibit a Kosterlitz-Thouless QPT between delocalized and localized

phases. The existence of a QPT for sub-Ohmic (0 < s < 1) baths was the subject of

debate for many years [81]. However, clear evidence for a continuous QPT has been

34

provided by the NRG [72], by perturbative expansion in ϵ = s about the delocalized fixed

point [82], and through exact-diagonalization calculations [83].

The Hamiltonian of the spin-boson model reads

HSB =Hspin + Hbath + Hspin−bath

=∆Sx +∑

q

ω†qϕ

†qϕq +

1√NqSz∑

q

g†q(ϕq + ϕ†q), (2–47)

where ∆ is the tunneling between spin-up and spin-down states, ϕ†q and ϕq are

the creation and annihilation operators for a harmonic oscillator of energy ωq, Nq

is the number of oscillators in the bath (the number of distinct values of q), and

gq parameterizes the coupling between the impurity spin z component and the

displacement of the oscillator with wavevector q. The interaction between the spin

and the bath is entirely determined by the spectral function, taken to have the power-law

form

B(ω) ≡ π

Nq

∑q

g2q δ(ω − ωq) =

B0Ω

1−sωs for 0 ≤ ω ≤ Ω,


which is characterized by an upper cutoff Ω, an exponent s that must satisfy s > −1 to

ensure normalizability, and a dimensionless prefactor B0.

The bosonic NRG is very similar to the fermionic NRG described in the last section,

except that the spectral function Eq. (2–48) is nonzero only for positive frequencies

whereas the hybridization-width function Eq. (2–5) is defined for both positive and

negative frequencies.

Introducing a dimensionless scale y = ω/Ω, the bath and spin-bath terms of the

Hamiltonian can be written as

Hbath = Ω

∫ 10

dy yϕ†yϕy , (2–49)

and

Hspin−bath = Ω

√B0B2πSz(b0 + b

†0). (2–50)

35

Here, the new operator b0 is defined as

b0 = B−1∫ 10

dy W (y)ϕy , (2–51)

with a weighting function

W (y) =

√B(Ωy)

B0Ω, (2–52)


B2 =∫ 10

dy [W (y)]2. (2–53)

The continuous bath spectrum 0 ≤ y ≤ 1 (0 ≤ ω ≤ Ω) is divided into a sequence

of intervals, the nth of which extends over the energy range from Λ−(n+1) to Λ−n (n =

0, 1, 2, ...). Within each interval, a complete set of orthonormal functions similar to that of

the fermionic NRG can be introduced. The bath term in the Hamiltonian is approximated

Hbath ≃ Ω∑m

ωmϕ†mϕm , (2–54)

with

ωm = B−2m

∫ Λ−mΛ−(m−1)

dy y [W (y)]2, (2–55)

ϕm = B−1m

∫ Λ−mΛ−(m−1)

dy W (y)ϕy , (2–56)


B2m =∫ Λ−mΛ−(m−1)

dy [W (y)]2. (2–57)

Defining a new basis

bn =∑m

unmϕm , (2–58)

where u0m = Bm/B, Eq. (2–54) can be mapped via the Lanczos procedure [65] onto a

tight-binding Hamiltonian

Hbath = Ω

∞∑n=0

[εBn b

†nbn+τ

Bn

(b†nbn−1+b

†n−1bn

)], (2–59)

36

where τB0 ≡ 0. The remaining coefficients unm, εBn , and τBn in the chain form Eq. (2–59)

are determined by the recursive relations

εBn =∑m

u2nmωm , (2–60)

τBn+1un+1,m =(ωm − εBn )unm − τBn un−1,m , (2–61)

1 =∑m

u2n+1,m , (2–62)

with ωm given by Eq. (2–55). As a result of the one-sided form of the bath spectral

function, εBn and τBn decrease as Λ−n for large n.

The chain Hamiltonian can be solved recursively as in the fermionic NRG. Another

approximation must be introduced because the presence of bosons adds the further

complication that the Fock space is unbounded even for a single-site chain, making it

necessary to restrict the maximum number of bosons per chain site to a finite number

Nb. We must ensure that Nb is sufficiently large to produce reliable physical properties.

Finally, in order to solve problems with both fermionic and bosonic chains, we must

keep in mind that the spirit of the NRG is to treat fermions and bosons of the same

energy scale at the same iteration. Since the bosonic coefficients εBn and τBn (∝ Λ−n) in

Eq. (2–59) decay with site index n twice as fast as the fermionic coefficients εn and τn

(∝ Λ−n/2) in Eq. (2–24), after a few iterations the iterative procedure requires extension

of the bosonic chain only for every second site added to the fermionic chain.

37

CHAPTER 3RESULTS FOR CHARGE-COUPLED BOSE-FERMI ANDERSON MODEL

This chapter is based on a published paper. All the published contents are reprinted

with permission granted under the copyright policy of the American Physical Society

from Mengxing Cheng, Matthew T. Glossop, and Kevin Ingersent, Phys. Rev. B 80,

165113 (2009). Copyright (2009) by the American Physical Society.

3.1 Introduction

This chapter presents our investigation of a charge-coupled Bose-Fermi An-

derson (BFA) model in which the impurity not only hybridizes with conduction-band

electrons but also is coupled, via its electron occupancy, to a bath representing acoustic

phonons or other bosonic degrees of freedom whose dispersion extends to zero energy.

The model was introduced more than 30 years ago [48, 49] in connection with the

mixed-valence problem. A spinless version of the model was also discussed in the same

context [84]. More recently, very similar models have been shown to arise as effective

impurity problems in the extended DMFT for one- and two-band extended Hubbard

models [46, 47].

Our NRG study of the charge-coupled BFA model with bosonic baths characterized

by exponents 0 < s ≤ 1 reveals a crossover with increasing electron-boson (e-b)

coupling from a spin Kondo effect to a charge Kondo effect, very similar to that noted

previously in the Anderson-Holstein model [62, 85]. However, under conditions of

strict particle-hole symmetry, further increase in the e-b coupling leads to complete

suppression of Kondo physics at a quantum critical point. Beyond the critical e-

b coupling lies a localized phase in which charge fluctuations on the impurity site

are frozen. For sub-Ohmic baths (0 < s < 1), the QPT is continuous and the

numerical values of the critical exponents describing the response of the impurity

charge to a locally applied electric potential demonstrate that the transition belongs

to the same universality class as that of the spin-boson and Ising BFK models. For

38

Ohmic baths (corresponding to s = 1), the QPT is found to be of Kosterlitz-Thouless

type. Particle-hole asymmetry acts in a manner analogous to a magnetic field at a

conventional ferromagnetic ordering transition, smearing the discontinuous change

in the ground-state as a function of e-b coupling into a smooth crossover. Signatures

of the symmetric quantum critical point remain in the physical properties at elevated

temperatures and/or frequencies.

The rest of this chapter is organized as follows. Section 3.2 introduces the

charge-coupled BFA Hamiltonian and summarizes the NRG method used to solve

the model. Section 3.3 contains a preliminary analysis of the model, focusing on the

bosonic renormalization of the effective electron-electron interaction within the impurity

level. Numerical results for the symmetric model with sub-Ohmic (0 < s < 1) dissipation

are presented and interpreted in Sec. 3.4. Section 3.5 treats the symmetric model with

Ohmic (s = 1) dissipation. Section 3.6 discusses the effects of particle-hole asymmetry.

The work’s conclusions are presented in Sec. 3.7.

3.2 Model and Solution Method

This section introduces the charge-coupled Bose-Fermi Anderson model and briefly

reviews the NRG method (see Chapter 2 for the details) we use to solve the model.

3.2.1 Charge-Coupled Bose-Fermi Anderson Hamiltonian and Related Models

In this work, we investigate the charge-coupled Bose-Fermi Anderson model

described by the Hamiltonian

HCCBFA = Himp + Hband + Hbath + Himp-band + Himp-bath, (3–1)

where

Himp = ϵd nd + Und↑nd↓ , (3–2)

Hband =∑k,σ

ϵk c†kσckσ , (3–3)

39

Hbath =∑

q

ωq a†qaq (3–4)

Himp-band =1√Nk

∑k,σ

(Vk c

†kσdσ + V

∗k d

†σckσ

), (3–5)

Himp-bath =1√Nq(nd − 1)

∑q

λq(aq + a

†−q). (3–6)




repulsion between two electrons in the impurity level. Vk is the hybridization between the

impurity and a conduction-band state of energy ϵk annihilated by fermionic operator ckσ,

and λq characterizes the coupling of the impurity occupancy to bosons in an oscillator

state of energy ωq annihilated by operator aq. Nk is the number of unit cells in the host

metal and, hence, the number of inequivalent k values. Correspondingly, Nq is the

number of oscillators in the bath, and the number of distinct values of q. Without loss

of generality, we take Vk and λq to be real and non-negative. Note that, throughout the

chapter in all mathematical expressions and labels, we drop all factors of the reduced

Planck constant ~, Boltzmann’s constant kB , the impurity magnetic moment gµB , and the

electronic charge e.

Again, the model describes a magnetic impurity that hybridizes with metallic host

and is coupled, via the impurity charge, to a bath of dispersive bosons representing

dissipative environments. To focus on the most interesting physics of the model, we

assume a constant hybridization Vk = V and a flat conduction-band density of states

(per unit cell, per spin-z orientation)

ρ(ϵ) ≡ 1

Nk

∑k

δ(ϵ− ϵk) =

ρ0 = (2D)

−1 for |ϵ| ≤ D,

0 otherwise,(3–7)

40

defining the hybridization width Γ = πρ0V2. The bosonic bath is completely specified by

its spectral density, which we take to have the pure power-law form

B(ω) ≡ π

Nq

∑q

λ2q δ(ω − ωq) =

(K0λ)

2Ω1−sωs for 0 ≤ ω ≤ Ω,

0 otherwise,(3–8)

characterized by an upper cutoff Ω, an exponent s that must satisfy s > −1 to ensure

normalizability, and a dimensionless prefactor K0λ. In this work, we present results

only for the case Ω = D in which the bath and band share a common cutoff. We also

adopt the convention that K0 is held constant while one varies λ, which we term the

electron-boson (e-b) coupling. It should be emphasized, though, that the key features of

the model are a nonvanishing Fermi-level density of states ρ(0) > 0 and the asymptotic

behavior B(ω) ∝ ωs for ω → 0. Relaxing any or all of the remaining assumptions laid out

in this paragraph will not alter the essential physics of the model, although it may affect

nonuniversal properties, such as the locations of phase boundaries.

For many purposes, it is convenient to rewrite [66, 67] the impurity part of the

Hamiltonian (dropping a constant term ϵd )

Himp = δd(nd − 1) +U

2(nd − 1)2, (3–9)

where δd = ϵd + U/2. Most of the results presented below were obtained for the

symmetric model characterized by ϵd = −U/2 or δd = 0, for which the impurity states

nd = 0 and nd = 2 are degenerate in energy. Section 3.6 addresses the behavior of the

asymmetric model.

In any realization of HCCBFA involving coupling of acoustic phonons to a magnetic

impurity or a quantum dot, the value of the bath exponent s will depend on the precise

interaction mechanism. However, phase space considerations suggest that any such

system will lie in the super-Ohmic regime s > 1. Models closely related to HCCBFA

have also been considered in the context of extended DMFT [47], a technique for

41

systematically incorporating some of the spatial correlations that are omitted from the

conventional DMFT of lattice fermions [86]. Extended DMFT maps the lattice problem

onto a quantum impurity problem in which a central site interacts with both a fermionic

band and one or more bosonic baths, the latter representing fluctuating effective fields

due to interactions between different lattice sites. The charge-coupled BFA model

serves as the mapped impurity problem for various extended Hubbard models with

nonlocal density-density interactions [46, 47]. In these settings, the effective bath

exponent s is not known a priori, but is determined through self-consistency conditions

that ensure that the central site is representative of the lattice as a whole. The extended

DMFT treatment of other lattice models [26, 44, 45] gives rise to exponents 0 < s < 1,

and we expect this also to be the case for the extended Hubbard models.

At the Hartree-Fock level [48], the impurity properties of Hamiltonian (3–1) are

identical to those of the Anderson-Holstein Hamiltonian,

HAH = HA + ω0a†a + λ0(nd − 1)(a + a†), (3–10)

which augments the well-studied Anderson impurity model [9],

HA = Himp + Hband + Himp-band, (3–11)

with a Holstein coupling [63] of the impurity charge to a single phonon mode of energy

ω0. At several points in the sections that follow, we compare and contrast our results for

HCCBFA with those obtained previously for HAH.

3.2.2 Numerical Renormalization-Group Treatment

Here, we briefly summarize the NRG treatment of the charge-coupled Bose-Fermi

Anderson model. Those who would like to know more about technical details may read

Chapter 2.

We solve the charge-coupled BFA model using the NRG method [8, 64, 66, 67], as

recently extended to treat models involving both dispersive bosons and dispersive

42

fermions [73, 74]. The full range of conduction-band energies −D ≤ ϵ ≤ D

(bosonic-bath energies 0 ≤ ω ≤ Ω) is divided into a set of logarithmic intervals

bounded by ϵ = ±DΛ−k (ω = ΩΛ−k ) for k = 0, 1, 2, ..., where Λ > 1 is the Wilson

discretization parameter. The continuum of states within each interval is replaced by a

single state, namely, the particular linear combination of band (bath) states within the

interval that enters Himp-band (Himp-bath). The discretized model is then transformed into a

tight-binding form involving two sets of orthonormalized operators: (i) fnσ (n = 0, 1, 2, ...)

constructed as linear combinations of all ckσ having |ϵk| < DΛ−n; and (ii) bm (m = 0, 1,

2, ...) mixing all aq such that 0 < ωq < ΩΛ−m. This procedure maps the last four parts of

Hamiltonian (3–1) to

HNRGband = D

∞∑n=0σ

[ϵnf

†nσfnσ+τn


†n−1,σfnσ

)], (3–12)

HNRGbath = Ω

∞∑m=0

[emb

†mbm + tm

(b†mbm−1 + b

†m−1bm

)], (3–13)

HNRGimp-band =

√2ΓD

π(f †0σdσ + d

†σf0σ) , (3–14)

HNRGimp-bath =

ΩK0λ√π(s + 1)

(nd − 1)(b0 + b

†0

). (3–15)

Here, τ0 = t0 = 0, while the remaining coefficients ϵn, τn, em, and tm, which include

all information about the conduction-band density of states ρ(ϵ) and the bosonic

spectral density B(ω), are calculated via Lanczos recursion relations [74]. For a

particle-hole-symmetric density of states such as that in Eq. (3–7), ϵn = 0 for all

n.

The coefficients τn in Eq. (3–12) vary for large n as DΛ−n/2, while em and tm entering

Eq. (3–13) vary for large m as ΩΛ−m. Therefore, the problem can be solved iteratively

by diagonalization of a sequence of Hamiltonians HN (N = 0, 1, 2, ...) describing

tight-binding chains of increasing length. At iteration N ≥ 0, Eq. (3–12) is restricted to

0 ≤ n ≤ N, while Eq. (3–13) is limited to 0 ≤ m ≤ M(N). The spirit of the NRG is to treat

43

fermions and bosons of the same energy scale at the same iteration. Since the bosonic

coefficients decay with site index twice as fast as the fermionic coefficients, after a few

iterations the iterative procedure requires extension of the bosonic chain only for every

second site added to the fermionic chain. In this work, we have chosen for simplicity to

work with a single high-energy cutoff scale D ≡ Ω. It is then convenient to add to the

bosonic chain at every even-numbered iteration, so that the highest-numbered bosonic

site is M(N) = ⌊N/2⌋, where ⌊x⌋ is the greatest integer less than or equal to x .

The NRG method relies on two additional approximations. Even for pure-fermionic

problems, it is not feasible to keep track of all the eigenstates because the dimensions

of the Fock space increase rapidly as we add sites to the chains. Therefore, only the

lowest lying Ns many-particle states can be retained after each iteration. The presence

of bosons adds the further complication that the Fock space is infinite-dimensional

even for a single-site chain, making it necessary to restrict the maximum number of

bosons per chain site to a finite number Nb. Provided that Ns and Nb are chosen to be

sufficiently large (as discussed in Sec. 3.4.1), the NRG solution at iteration N provides a

good account of the impurity contribution to physical properties at temperatures T and

frequencies ω of order DΛ−N/2.

Hamiltonian (3–1) commutes with the total spin-z operator

Sz =1

2(nd↑ − nd↓) +

1

2

∑n

(f †n↑fn↑ − f

†n↓fn↓

), (3–16)

the total spin-raising operator

S+ = d†↑d↓ +

∑n

f †n↑fn↓ ≡(S−)†, (3–17)

and the total “charge” operator

Q = nd − 1 +∑n

(f †n↑fn↑ + f

†n↓fn↓ − 1

), (3–18)

44

which measures the deviation from half-filling of the total electron number. One can

interpret

Iz =1

2Q, I+ = −d †↑d

†↓ +

∑n

(−1)nf †n↑f†n↓ ≡

(I−)† (3–19)

as the generators of an SU(2) isospin symmetry (originally dubbed “axial charge” in

Ref. [69]). Since [Himp-bath, I±] = 0, the charge-coupled BFA model does not exhibit full

isospin symmetry. However, this symmetry turns out to be recovered in the asymptotic

low-energy behavior at certain renormalization-group fixed points.

As described in Ref. [66], the computational effort required for the NRG solution

of a problem can be greatly reduced by taking advantage of these conserved quantum

numbers. In particular, it is possible to obtain all physical quantities of interest while

working with a reduced basis of simultaneous eigenstates of S2, Sz , and Q with

eigenvalues satisfying Sz = S . With one exception noted in Sec. 3.4.7, any Ns value

specified below represents the number of retained (S ,Q) multiplets, corresponding to a

considerably larger number of (S ,Sz ,Q) states.

Even when advantage is taken of all conserved quantum numbers, NRG treatment

of the charge-coupled BFA model remains much more demanding than that of the

Anderson model [Eq. (3–11)] or the Anderson-Holstein model [Eq. (3–10)]. Being

nondispersive, the bosons in the last model enter only the atomic-limit Hamiltonian H0,

allowing solution via the standard NRG iteration procedure. For Bose-Fermi models

such as HCCBFA, the need to extend a bosonic chain as well as a fermionic one at every

even-numbered iteration N > 0, expands the basis of HN from 4Ns states to 4(Nb + 1)Ns

states, and multiplies the CPU time by a factor ∼ (Nb + 1)3. Since we typically use

Nb = 8 or 12 in our calculations, the increase in computational effort is considerable.

The choice of value for the NRG discretization parameter Λ involves trade-offs

between discretization error (minimized by taking Λ to be not much greater than 1)

and truncation error (reduced by working with Λ ≫ 1). Experience from similar kind of

problems [58, 73, 74] indicates that critical exponents can be determined very accurately

45

using quite a large Λ. Most of the results presented in the remaining sections of the work

were obtained for Λ = 9, with Λ = 3 being employed in the calculation of the impurity

spectral function. For convenience in displaying these results, we set Ω = D = 1 and

omit all factors of ρ0 and K0.

3.3 Preliminary Analysis

We begin by examining the special cases in which the impurity level is decoupled

either from the conduction band or from the bosonic bath. Understanding these cases

allows us to establish some expectations for the behavior of the full model described by

Eq. (3–1).

3.3.1 Zero Hybridization

The key point of this section is to prove that the main effect of the bosons is to

reduce the Coulomb interaction U to an effective one Ueff, which might be negative for

sufficiently large e-b coupling. If one sets Γ = 0 in Eq. (3–1), then the conduction band

completely decouples from the remaining degrees of freedom and can be dropped from

the model, leaving the zero-hybridization model

HZH = δd(nd − 1) +U

2(nd − 1)2 +

∑q

ωqa†qaq +

1√Nq(nd − 1)

∑q

λq

(aq + a

†−q). (3–20)

The Fock space separates into sectors of fixed impurity occupancy (nd = 0, 1, or 2),

within each of which the Hamiltonian can be recast, using displaced-oscillator operators

and ,q = aq +λq√Nq ωq

(nd − 1), (3–21)

in the trivially solvable form

HZH(nd) = H′imp +

∑q

ωqa†nd ,qand ,q , (3–22)

where

H ′imp = δd(nd − 1) +

Ueff

2(nd − 1)2. (3–23)

46

The bosons act on the impurity to reduce the Coulomb interaction from its bare value U

to an effective value

Ueff = U − 2

Nq

∑q

λ2qωq= U − 2

π

∫ ∞

0

B(ω)

ωdω. (3–24)

For the bath spectral density in Eq. (3–8) with −1 < s ≤ 0, one finds that for any

nonzero e-b coupling λ, Ueff = −∞ and the singly occupied impurity states drop out of

the problem. For the remainder of this section, however, we will instead focus on bath

exponents s > 0, for which Eqs. (3–8) and (3–24) give

Ueff = U − 2(K0λ)2

πsΩ. (3–25)

For weak e-b couplings, Ueff is positive and the ground state of HZH lies in the sector

nd = 1 where the impurity has a spin z component ±12. However, Ueff is driven negative

for sufficiently large λ, placing the ground state in the sector nd = 0 or nd = 2 where the

impurity is spinless but has a charge (relative to half filling) of −1 or +1.

Figure 3-1 illustrates this renormalization of the Coulomb interaction for the

symmetric model (δd = 0), in which the nd = 0 and nd = 2 states always have the

same energy. In this case, all four impurity states become degenerate at a crossover e-b

coupling

K0λc0 =√πsU/2Ω . (3–26)

The impurity contributions to physical properties at this special point, which is characterized

by effective parameters Γ = U = ϵd = 0, are identical to those at the free-orbital fixed

point [66, 67] of the Anderson model.

For the general case of an asymmetric impurity, the sectors nd = 0 and 2 have a

ground-state energy difference E0(nd = 2) − E0(nd = 0) = 2δd for any value of λ. The

overall ground state of Eq. (3–20) is a doublet (nd = 1, S = ±12) for small e-b couplings,

crossing over to a singlet (nd = 0 for δd > 0, or nd = 2 for δd < 0) for large λ. At

K0λc0 =√πs(U/2− |δd |)/Ω, a point of three-fold ground-state degeneracy, the impurity

47

contributions to low-temperature (T ≪ |δd |) physical properties are identical to those at

the valence-fluctuation fixed point of the Anderson model [66, 67].

Using the NRG with only a bosonic chain [Eq. (3–13)] coupled to the impurity

site, we have confirmed the existence for δd = 0 of a simple level crossing from a

spin-doublet ground state for λ < λc0 to a charge-doublet ground state for λ > λc0. In

the former regime, the bosons couple only to the high-energy (nd = 0, 2) impurity states,

so the low-lying spectrum is that of free bosons obtained by diagonalizing HNRGbath given in

Eq. (3–13). Here, NRG truncation plays a negligible role provided that one works with

Nb ≥ 8 (say).

For λ > λc0, the low-lying bosonic excitations should, in principle, correspond

to noninteracting displaced oscillators having precisely the same spectrum as the

original bath. However, the occupation number a†qaq in the ground state of Eq. (3–22)

obeys a Poisson distribution with mean λ2q/(Nq ω2q). Thus, the total number of bosons

corresponding to operators aq satisfying ΩΛ−(k+1) < ωq < ΩΛ−k takes a mean value

⟨nk⟩0 =∫ ΩΛ−kΩΛ−(k+1)

dωB(ω)

πω2=

(K0λ)

2

πln Λ for s = 1,

(K0λ)2

π

(Λ1−s − 1

)(1− s)

Λ(1−s)k otherwise.

(3–27)

The bath states in the k th interval are represented by NRG chain states 0 ≤ m ≤ k ,

with the greatest weight being borne by state m = k . Thus, a faithful representation

of the displaced-oscillator spectrum requires inclusion of states having b†mbm up to

several times ⟨nm⟩0; based on experience with the Anderson-Holstein model [62], one

expects Nb ≥ 4⟨nm⟩0 to suffice. Given that ⟨nm⟩0 ∝ Λ(1−s)m, it is feasible to meet this

condition as m → ∞ so long as the bath exponent satisfies s ≥ 1. Indeed, for Ohmic

and super-Ohmic bath exponents, the NRG spectrum for λ not too much greater than

λc0 is found to be numerically indistinguishable from that for λ = 0. For s < 1, by

contrast, the restriction b†mbm ≤ Nb leads, for λ > λc0 and large iteration numbers, to

an artificially truncated spectrum that cannot reliably access the low-energy physical

48

properties. Nonetheless, observation of this “localized” bosonic spectrum serves as a

useful indicator, both in the zero-hybridization limit and in the full charge-coupled BFA

model, that the effective e-b coupling remains nonzero.

Another interpretation of Eq. (3–27) is that at the energy scale E = ΩΛ−k

characteristic of interval k , the e-b coupling takes an effective value λ(E) governed

by the renormalization-group equation

d λ

d ln(Ω/E)=1− s2

λ, (3–28)

which implies that the e-b coupling is irrelevant for s > 1, marginal for s = 1, and relevant

for s < 1. While the NRG method is capable of faithfully reproducing the physics of

HCCBFA for arbitrary renormalizations of ϵd , U, and Γ, its validity is restricted to the region

(K0λ

)2 . πNB4

1− sΛ1−s − 1

Λ→1−→ πNB4 ln Λ

. (3–29)

For Λ = 9 and NB = 8, as used in most of our calculations, the upper limit on the “safe”

range of K0λ varies from 1.7 for s = 1 to 0.9 for s = 0.

We now focus on the value of the crossover e-b coupling λc0 determined using the

NRG approach. Figure 3-2 shows for five different bosonic bath exponents s that K0λc0

has an almost linear dependence on the NRG discretization Λ in the range 1.6 ≤ Λ ≤ 4.

We believe that the rise in K0λc0 with Λ reflects a reduction in the effective value of K0

arising from the NRG discretization. It is known [66, 67] that in NRG calculations for

fermionic problems, the conduction-band density of states at the Fermi energy takes an

effective value

ρ(0) = ρ0 = ρ0/AΛ, (3–30)

where

AΛ =lnΛ

2

1 + Λ−1

1− Λ−1. (3–31)

49

Table 3-1. Crossover coupling λc0 for HZH [Eq. (3–20)] with U = 0.1, δd = 0, and fivedifferent values of the bath exponent s : Comparison between λc0(exact) givenby Eq. (3–26) and λc0(Λ→1), the extrapolation to the continuum limit ofnumerical values obtained for Ns = 200, Nb = 16, and 1.6 ≤ Λ ≤ 4.Parentheses surround the estimated nonsystematic error in the last digit.s 0.2 0.4 0.6 0.8 1.0λc0(exact) 0.177 0.251 0.307 0.355 0.396λc0(Λ→1) 0.188(4) 0.250(2) 0.307(2) 0.355(2) 0.397(3)

The general trend of the data in Fig. 3-2 is consistent with there being an analogous

reduction of the bosonic bath spectral density that requires the replacement of K0 by

K0 = K0/AΛ,s , (3–32)

when extrapolating NRG results to the continuum limit Λ = 1. However, we have not

obtained a closed-form expression for AΛ,s .

Table 3-1 lists values λc0(Λ → 1) extrapolated from the data plotted in Fig. 3-2. For

s ≥ 0.4, these values are in good agreement with Eq. (3–26). For s = 0.2, however, the

extrapolated value of λc0 lies significantly above the exact value, indicating that for given

λ the NRG underestimates the bosonic renormalization of U. This is most likely another

consequence of truncating the basis on each site of the bosonic tight-binding chain.

In analyzing our NRG results for the full charge-coupled BFA model, we attempt to

compensate for the effects of discretization and truncation by replacing Eq. (3–25) by

UNRGeff = U

[1− (λ/λc0)2

]. (3–33)

Here, λc0 is not the theoretical value predicted in Eq. (3–26), but rather is obtained from

runs carried out for Γ = 0 but otherwise using the same model and NRG parameters as

the data that are being interpreted.

3.3.2 Zero Electron-Boson Coupling

For λ = 0, the bosonic bath decouples from the electronic degrees of freedom,

which are then described by the pure Anderson model. In this section, we briefly review

50

aspects of the Anderson model that will prove important in interpreting results for the

charge-coupled BFA model. For further details concerning the Anderson model, see

Refs. [1] and [66, 67].

For any Γ > 0, and for any U and δd ≡ ϵd + U/2 (whether positive, negative, or

zero), the stable low-temperature regime of the Anderson model lies on a line of strong-

coupling fixed points corresponding to Γ = ∞. At any of these fixed points, the system

is locked into the ground state of the atomic Hamiltonian H0, and there are no residual

degrees of freedom on the impurity site or on site n = 0 of the fermionic chain; the NRG

excitation spectrum is that of the Hamiltonian [66, 67]

HNRGSC (V1) = D

∞∑n=1

∑σ

τn(f †nσfn−1,σ + f

†n−1,σfnσ

)+ V1

(∑σ

f †1σf1σ − 1

). (3–34)

The coefficients τn are identical to those entering HNRGband [Eq. (3–12)], except that here

τ1 = 0. Note that in Eq. (3–34), the sum over n begins at 1 rather than 0.

As shown in Ref. [66, 67], the strong-coupling fixed points of the Anderson model

are equivalent—apart from a shift of 1 in the ground-state charge Q defined in Eq.

(3–18)—to the line of frozen-impurity fixed points corresponding to ϵd = ∞, Γ = U = 0,

with NRG excitation spectra described by

HNRGFI (V0) = H

NRGband + V0

(∑σ

f †0σf0σ − 1

). (3–35)

The mapping between alternative specifications of the same fixed-point spectrum is

πρ0V0 = −(πρ0V1)−1, (3–36)

where ρ0 [see Eq. (3–30)] is the effective conduction-band density of states.

The fixed-point potential scattering is related to the ground-state impurity charge via

the Friedel sum rule,

⟨nd − 1⟩0 =2

πarccot

(πρ0V0

)=2

πarctan

(−πρ0V1

). (3–37)

51

For |δd |, Γ≪ U ≪ D, one finds that

⟨nd − 1⟩0 = − 8δdΓπAΛU2

, (3–38)

where AΛ is defined in Eq. (3–31).

Even though the stable fixed point of the Anderson model for any Γ > 0 is one of

the strong-coupling fixed points described above, the route by which such a fixed point

is reached can vary widely, depending on the relative values of U, δd , and Γ. For our

immediate purposes, it suffices to focus on the symmetric (δd = 0) model, for which

there is a single strong-coupling fixed point corresponding to V0 = ±∞ or V1 = 0. If the

on-site Coulomb repulsion is strong enough that the system enters the local-moment

regime (T , Γ ≪ U), then it is possible to perform a Schrieffer-Wolff transformation [10]

that restricts the system to the sector nd = 1 and reduces the Anderson model to the

Kondo model described by the Hamiltonian

HK =Hband +Jz4Nk(nd↑ − nd↓)

∑k,k′

(c†k↑ck′↑ − c

†k↓ck′↓

)+J⊥2Nk

∑k,k′

(d †↑d↓c

†k↓ck′↑ + H.c.

), (3–39)

where

ρ0Jz = ρ0J⊥ =8Γ

πU. (3–40)

The stable fixed point is approached below an exponentially small Kondo temperature

TK when the spin-flip processes associated with the J⊥ term in HK cause the effective

values of ρ0Jz and ρ0J⊥ to renormalize to strong coupling, resulting in many-body

screening of the impurity spin.

Motivated by the discussion in Sec. 3.3.1, we also consider the case of strong

on-site Coulomb attraction. In the local-charge regime (T , Γ ≪ −U), a canonical

transformation similar to the Schrieffer-Wolff transformation restricts the system to the

sectors nd = 0 and nd = 2, and maps the Anderson model onto a charge Kondo model

52

described by the Hamiltonian

HCK =Hband +WdNk(nd − 1)

∑k,k′

(c†k↑ck′↑ + c

†k↓ck′↓ − δk,k′

)+2WpNk

∑k,k′

(d †↑d

†↓ck↓ck′↑ + H.c.

), (3–41)

where

ρ0Wd = ρ0Wp =2Γ

π|U|. (3–42)

In this case, the stable fixed point is approached below an exponentially small (charge)

Kondo temperature TK when the charge-transfer processes associated with theWp term

in HCK cause the effective values of ρ0Wd and ρ0Wp to renormalize to strong coupling,

resulting in many-body screening of the impurity isospin degree of freedom [associated

with the d-operator terms in Eqs. (3–19)].

Between the opposite extremes of large positive U and large negative U is a

mixed-valence regime T , |U| ≪ Γ in which interactions play only a minor role. Here, the

stable fixed point is approached below a temperature of order Γ when the effective value

of√Γ/(2πD) scales to strong coupling, signaling strong mixing of the impurity levels

with the single-particle states of the conduction band.

3.3.3 Expectations for the Full Model

Insight into the behavior of the full charge-coupled BFA model described by Eqs.

(3–1)–(3–6) can be gained by performing a Lang-Firsov transformation HCCBFA →

H ′CCBFA = U

−1HCCBFAU with

U = exp

[(nd − 1)

∑q

λq√Nq ωq

(aq − a†q

)]. (3–43)

The transformation eliminates Himp-bath, leaving

H ′CCBFA = H

′imp + Hband + Hbath + H

′imp-band, (3–44)

53

where H ′imp is as defined in Eqs. (3–23) and (3–24), and

H ′imp-band =

1√Nk

∑k,σ

Vk exp

[∑q

λq(aq − a†q

)√Nq ωq

]c†kσdσ + V

∗k exp

[−∑

q

λq(aq − a†q

)√Nq ωq

]d †σckσ

.

(3–45)

In addition to renormalizing the impurity interaction from U to Ueff entering H ′imp, the

e-b coupling causes every hybridization event to be accompanied by the creation and

annihilation of arbitrarily large numbers of bosons.

In the case of the Anderson-Holstein model [Eq. (3–10)], various limiting behaviors

are understood [87]. In the instantaneous limit ω0 ≫ Γ, the bosons adjust rapidly to any

change in the impurity occupancy; for λ20/ω0 ≪ U ≪ ω0, the physics is essentially that of

the Anderson model with U → Ueff, while for λ20/ω0 ≫ D,U, Γ, there is also a reduction

from Γ to Γ exp[−(λ0/ω0)2] in the rate of scattering between the nd = 0 and nd = 2

sectors, reflecting the reduced overlap between the ground states in these two sectors.

In the adiabatic limit ω0 ≪ Γ, the phonons are unable to adjust on the typical time scale

of hybridization events, and neither U nor Γ undergoes significant renormalization.

Similar analysis for the charge-coupled BFA model is complicated by the presence

of a continuum of bosonic mode energies ω, only some of which fall in the instantaneous

or adiabatic limits. Nonetheless, we can use results for the cases Γ = 0 (Sec. 3.3.1)

and λ = 0 (Sec. 3.3.2), as well as those for the Anderson-Holstein model, to identify

likely behaviors of the full model. Specifically, we focus here on the evolution with

decreasing temperature of the effective Hamiltonian describing the essential physics of

the symmetric (ϵd = −U/2) model at the current temperature. This effective Hamiltonian

is obtained under the assumption that real excitations of energy above the ground state

E ≥ ηT—where η is a number around 5, say—make a negligible contribution to the

observable properties, and thus can be integrated from the problem.

Based on the preceding discussion, one expects that at high temperatures T ≫ Γ,

the physics of the charge-coupled BFA model will be very similar to that of the Anderson

54

model with U replaced by U(ηT ), where

U(E) = U − 2π

∫ ∞

E

B(ω)

ωdω. (3–46)

Note that U(0) is identical to Ueff defined in Eq. (3–24). For the bath spectral density in

Eq. (3–8) with s > 0,

U(E) = U − 2(K0λ)2

πs

[1− (E/Ω)s

]Ω. (3–47)

When analyzing NRG data, we instead use

UNRG(E) = U1− (λ/λc0)2

[1− (E/Ω)s

], (3–48)

where λc0 is the empirically determined value discussed in connection with Eq. (3–33).

If, upon decreasing the temperature to some value TLM, the system comes to

satisfy U(ηTLM) = ηmax(TLM, Γ), then it should enter a local-moment regime described

by the effective Hamiltonian HLM = HK + Hbath with the exchange couplings in HK

[Eq. (3–39)] determined by Eq. (3–40) with U → U(ηTLM), similar to what is found in

the Anderson-Holstein model [85]. Since they couple only to the high-energy sectors

nd = 0 and nd = 2 that are projected out during the Schrieffer-Wolff transformation, the

bosons should play little further role in determining the low-energy impurity physics. The

outcome should be a conventional Kondo effect where the e-b coupling contributes only

to a renormalization of the Kondo scale TK.

If, instead, at some T = TLC the system satisfies U(ηTLC) = −ηmax(TLC, Γ), then it

should enter a local-charge regime described by the effective Hamiltonian

HLC = HCK + Hbath + Himp-bath. (3–49)

Based on the behavior of the Anderson-Holstein model [85], one expectsWd in HCK [Eq.

(3–41)] to be determined by Eq. (3–42) with U → U(ηTLC), but withWp exponentially

depressed due to the aforementioned reduction in the overlap between the ground

states of the nd = 0 and nd = 2 sectors. The bosons couple to the low-energy

55

sector of the impurity Fock space, so they have the potential to significantly affect the

renormalization ofWd andWp upon further reduction in the temperature. In particular,

the λ term in HLC, which favors localization of the impurity in a state of well-defined nd =

0 or 2, directly competes with theWp double-charge transfer term that is responsible for

the charge Kondo effect of the negative-U Anderson model. This nontrivial competition

gives rise to the possibility of a QPT between qualitatively distinct ground states of the

charge-coupled BFA model.

Between these extremes, the system can enter a mixed-valence regime of small

effective on-site interaction. In this regime, one must retain all the impurity degrees of

freedom of the charged-coupled BFA model. The impurity-band hybridization competes

with the e-b coupling for control of the impurity, again suggesting the possibility of a

QPT.

Each of the regimes discussed above features competition between band-mediated

tunneling within the manifold of impurity states and the localizing effect of the bosonic

bath. Although the tunneling is dominated by a different process in the three regimes, it

always drives the system toward a nondegenerate impurity ground state, whereas the

e-b coupling favors a doubly-degenerate (nd = 0, 2) impurity ground state. In order to

provide a unified picture of the three regimes (and the regions of the parameter space

that lie in between them), we will find it useful to interpret our NRG result in terms of an

overall tunneling rate ∆, which has a bare value

∆ ≃√J2⊥ + 2ΓD/π + 16W

2p . (3–50)

Here,Wp is assumed to be negligibly small in the local-moment regime, and J⊥ to be

similarly negligible in the local-charge regime. If ∆ renormalizes to large values while the

e-b coupling λ scales to weak coupling, then one expects to recover the strong-coupling

physics of the Anderson model. If, on the other hand, λ becomes strong while ∆

becomes weak, the system should enter a low-energy regime in which the bath governs

56

the asymptotic low-energy, long-time impurity dynamics. Whether or not each of these

scenarios is realized in practice, and whether or not there are any other possible ground

states of the model, can be determined only by more detailed study. These questions

are answered by the NRG results reported in the sections that follow.

3.4 Results: Symmetric Model with Sub-Ohmic Dissipation

This section presents results 1 for Hamiltonian (3–1) with U = −2ϵd > 0 and with

sub-Ohmic dissipation characterized by an exponent 0 < s < 1. Figure 3-3 shows a

schematic phase diagram on the λ–Γ plane at fixed U. There are two stable phases:

the localized phase, in which the impurity dynamics are controlled by the coupling to

the bosonic bath and the system has a pair of ground states related to one another by

a particle-hole transformation; and the Kondo phase, in which there is a nondegenerate

ground state. These phases are separated by a continuous QPT that takes place on the

phase boundary (solid line in Fig. 3-3), which we parametrize as λ = λc(Γ). Within the

Kondo phase, the nature of the correlations evolves continuously with increasing λ (at

fixed Γ) from a pure spin-Kondo effect for λ = 0 to a predominantly charge-Kondo effect

beyond a crossover (dashed line in Fig. 3-3) associated with the change in sign of Ueff

defined in Eq. (3–24).

As s decreases, and the e-b coupling becomes increasingly relevant—in a

renormalization-group sense [see Eq. (3–28)]—the phase boundary moves to the

left as the localized phase grows at the expense of the Kondo phase, which disappears

entirely for s ≤ 0. As will be seen in Sec. 3.5, the phase diagram of the Ohmic (s = 1)

problem has the same topology as Fig. 3-3, even though (as described in Sec. 3.5) the

1 Note that a number of the results presented in Sec. 3.4 were obtained usingunphysically large values of the hybridization Γ. These values were employed toaccelerate the convergence of the NRG levels to the critical spectrum, and thereby tominimize computational rounding errors.

57

nature of the QPT is qualitatively different than for 0 < s < 1. For s > 1, the e-b coupling

is irrelevant, and the system is in the Kondo phase for all Γ > 0.

The remainder of this section presents the evidence for the previous statements.

We first discuss the renormalization-group flows and fixed points. We then turn to

the behavior in the vicinity of the phase boundary, focusing in particular on the critical

response of the impurity charge to a local electric potential. Following that, we present

results for the impurity spectral function, and show that the low-energy scale extracted

from this spectral function supports the qualitative picture laid out in the paragraphs

above and summarized in Fig. 3-3.

3.4.1 NRG Flows and Fixed Points

Figure 3-4 plots the schematic renormalization-group flows of the couplings λ

entering Eq. (3–15) and ∆ defined in Eq. (3–50) for a symmetric impurity (U = −2ϵd )

coupled to bath described by an exponent 0 < s < 1. These flows are deduced

from the evolution of the many-body spectrum with increasing iteration number N, i.e.,

with reduction in the effective band and bath cutoffs D = Ω ≃ DΛ−N/2. A separatrix

(dashed line) forms the boundary between the basins of attraction of a pair of stable

fixed points, regions that correspond to the two phases shown in Fig. 3-3. Figure 3-4

also shows three unstable fixed points. In contrast to the situation at other points on

the flow diagram, each of the fixed points exhibits a many-body spectrum that can be

interpreted as the direct product of a set of bosonic excitations and a set of fermionic

excitations.

As an example, Fig. 3-5 demonstrates evolutions of the low-lying many-body

energies EN as a function of even iteration number N for s = 0.2, Γ = 0.5, and five

different couplings λ. See caption for discussion.

The Kondo fixed point corresponds in the renormalization-group language of Fig.

3-4 to effective couplings λ = 0 and ∆ =∞. The many-body spectrum decomposes into

the direct product of (i) the excitations of a free bosonic chain described by Eq. (3–13)

58

alone, and (ii) the strong-coupling excitations of the Kondo (or symmetric Anderson)

model, corresponding to free electrons with a Fermi-level phase shift of π/2. This

spectrum, which exhibits SU(2) symmetry both in the spin and charge (isospin) sectors,

is identical to that found throughout the Kondo phase of the particle-hole-symmetric

Ising BFK Hamiltonian [73, 74] (a model in which the bosons couple to the impurity’s

spin rather than its charge).

The schematic RG flow diagram in Fig. 3-4 shows a localized fixed point corresponding

to λ = ∞ and ∆ = 0. However, this is really a line of fixed points described by HLC [Eq.

(3–49)] with effective couplings λ = ∞,Wp = 0, and 0 ≤ Wd < ∞. SinceWp = 0,

the impurity occupancy takes a fixed value nd = 0 or 2. (It is important to distinguish nd ,

used to characterize the fixed-point excitations, from the physical expectation value of

nd . The latter quantity is discussed in Sec. 3.4.5.1.)

Each fixed point along the localized line has an excitation spectrum that decomposes

into the direct product of (i) bosonic excitations identical to those at the localized fixed

point of the spin-boson model [72] with the same bath exponent s , and (ii) fermionic

excitations described by a Hamiltonian

HNRGL,f = H

NRGband +Wd(nd − 1)

(∑σ

f †0σf0σ − 1

), (3–51)

which is just the discretized version of HCK [Eq. (3–41)] withWp = 0 and the operator

nd replaced by the parameter nd . The low-lying many-body eigenstates of HNRGL,f appear

in degenerate pairs, one member of each pair corresponding to nd = 0 and the other to

nd = 2. The fixed-point couplingWd increases monotonically as the bare e-b coupling λ

decreases from infinity, and diverges on approach to the phase boundary. As illustrated

in Fig. 3-6, this divergence can be fitted to the power-law form

Wd ∝ (λ− λc)−β for λ→ λ+c . (3–52)

59

For reasons that will be explained in Sec. 3.4.5.1, the numerical value of β coincides, to

within a small error, with that of the order-parameter exponent β defined in Eq. (3–71).

The free-orbital fixed point (λ = λc0, ∆ = 0) is unstable with respect to a bare Γ = 0

or any deviation of λ from λc0 ≡ limΓ→0 λc(Γ). The local-moment fixed point (λ = ∆ = 0),

at which the impurity has a spin-12

degree of freedom decoupled from the band and from

the bath, is reached only for bare couplings Γ = 0 (hence, ∆ = 0) and λ < λc0.

Of greatest interest is the unstable critical fixed point that is reached for any bare

couplings lying on the boundary λ = λc(Γ) between the Kondo and localized phases.

At this fixed point, the low-lying spectrum can be constructed as the direct product of (i)

the critical spectrum of the spin-boson model with the same bath exponent s, and (ii) the

strong-coupling spectrum of the Kondo (or symmetric Anderson) model. This spectrum

as shown in Fig. 3-5(c), which exhibits full SU(2) symmetry in both the spin and isospin

sectors, is identical to that at the critical point of the Ising-anisotropic Bose-Fermi Kondo

model,2 and is illustrated in Fig. 3(c) of Ref. [74].

The decomposition of the critical spectrum can be understood by considering

the flow of couplings entering the local-charge Hamiltonian HLC defined in Eq. (3–49).

The fixed-point value of the density-density coupling isWd = ∞ in the charge-Kondo

regime of the Kondo phase and diverges according to Eq. (3–52) in the localized phase.

It is therefore reasonable to assume that in the vicinity of the phase boundary,Wd

rapidly renormalizes to strong coupling, locking the impurity site and site n = 0 of the

fermionic chain into one of just two states, which we can write in a pseudospin notation

as |⇑⟩ = d †↑d†↓ |0⟩ and |⇓⟩ = f †0↑f

†0↓|0⟩, where |0⟩ is the no-particle vacuum. Hopping of

electrons on or off site n = 0 is forbidden, so the discretized form of HLC reduces to an

2 This decomposition of the BFK critical spectrum was not explicitly noted in Refs.[73] and [74]. However, it can be understood (following arguments analogous to thosepresented here for the charge-coupled BFA model) under the assumption that, near thephase boundary, the longitudinal exchange coupling renormalizes rapidly to Jz =∞.

60

effective Hamiltonian

HNRGLC (Wd =∞) = HNRG

SC (0) + HNRGSBM . (3–53)

Here, HNRGSC (0) [Eq. (3–34)] acts only on fermionic chain sites n ≥ 1, and yields the

Kondo/Anderson strong-coupling excitation spectrum, while

HNRGSBM = H

NRGbath + 2Wp

(|⇑⟩⟨⇓|+ |⇓⟩⟨⇑|

)+

ΩK0λ√π(s + 1)

(|⇑⟩⟨⇑| − |⇓⟩⟨⇓|

) (b0 + b

†0

)(3–54)

acts on the remaining degrees of freedom in the problem in a subspace of states all

carrying quantum numbers S = Sz = Q = 0. HNRGSBM is precisely the discretized form

of the spin-boson Hamiltonian with tunneling rate ∆ = 4Wp and dissipation strength

α = 2(K0λ)2/π. These two couplings compete with one another, with three possible

outcomes: (1) ∆ can scale to infinity and α to zero, resulting in flow to the delocalized

fixed point (the Kondo fixed point of the charge-coupled BFA model); (2) α can scale to

infinity and ∆ to zero, yielding flow to the localized fixed point; or (3) both couplings can

renormalize to finite values ∆ = ∆C , α = αC at the critical point. This picture implies

that the universal critical behavior of the charge-coupled BFA model should be identical

to that of the spin-boson model, the conduction-band electrons serving only to dress

the nd = 0, 2 impurity levels and to renormalize the impurity tunneling rate and the

dissipation strength.

Given that the NRG approach necessarily involves Fock-space truncation, it is

instructive to examine the dependence of the fixed-point spectra on the parameters Ns

and Nb denoting, respectively, the number of states retained from one NRG iteration

to the next and the maximum number of bosons allowed per site of the bosonic chain.

Figure 3-7 shows, for representative bath exponents s = 0.2 and s = 0.8, that the energy

of the lowest bosonic excitation at λ = λc converges rapidly with increasing Ns and Nb.

This behavior suggests that for Λ = 9, at least, Ns = 500 and Nb = 8 are sufficient for

studying the physics at the critical point.

61

By contrast, the lowest bosonic excitation energy for λ = 1.1λc , plotted in Fig.

3-8, converges only slowly with respect to Nb. This points to the failure of the truncated

bosonic basis deep inside the localized phase of the sub-Ohmic model, where the

mean boson number per site is expected to diverge according to Eq. (3–27). This

interpretation is confirmed by calculation of the expectation value of the total boson

number,

BN =

M(N)∑m

b†mbm, (3–55)

where M(N) denotes the highest labeled bosonic site present at iteration N. Our results

for ⟨B20⟩ vs Nb (not shown) are shown in Fig. 3-9, with convergence by Nb = 8 at the

critical point, but no evidence of such convergence for an e-b coupling 10% over the

critical value.

Recently, Bulla et al. applied a “star” reformulation of the NRG to the spin-boson

model [88]. While this approach provides a good description of the localized fixed point,

it does not correctly capture the physics of the delocalized phase (corresponding to the

Kondo phase of the present model) or of the critical point that separates the two stable

phases. For this reason, we prefer to work with the “chain” formulation summarized in

Sec. 3.2.

3.4.2 Critical Coupling

Figure 3-10 plots the critical e-b coupling λc(Γ) for fixed U = −2ϵd and four different

values of the bath exponent s. As expected, with increasing Γ, the critical coupling

increases smoothly from λc(Γ = 0) ≡ λc0, reflecting the fact that entry to the localized

phase requires an e-b coupling sufficiently large not only to drive Ueff negative, but also

to overcome the reduction in the electronic energy that derives from the hybridization.

We believe that the vertical slope of the s = 0.2 phase boundary as it approaches the

horizontal axis in Fig. 3-10 is an artifact stemming from the same source as the NRG

overestimate of λc0 for the same bath exponent. (See the discussion of Fig. 3-2 in Sec.

3.3.1.)

62

In the subsections that follow, we show that the critical properties of the charge-coupled

BFA model map, under interchange of spin and charge degrees of freedom, onto

those of the spin-coupled BFA model studied (along with the corresponding Ising BFK

model) in Ref. [74]. The spin-coupled model is described by Eqs. (3–1)–(3–5) and

(3–12)–(3–14), with Eqs. (3–6) and (3–15) replaced by

Himp-bath =1

2√Nq(nd↑ − nd↓)

∑q

gq(aq + a

†−q), (3–56)

and

HNRGimp-bath =

ΩK0g

2√π(s + 1)

(nd↑ − nd↓)(b0 + b

†0

). (3–57)

In light of the parallels between the universal critical behavior of the two models, it is

of interest to compare their critical couplings, making due allowance for the additional

prefactor of 12

that enters Eqs. (3–56) and (3–57).

Figure 3-11 plots the s dependence of λc and gc/2 for fixed values of U = −2ϵd

and Γ. For all 0 < s ≤ 1, λc is found to exceed gc/2. This fact can be understood by

noting the contrasting role of the e-b coupling in the two models. In the spin-coupled

BFA model, increasing g from zero immediately begins to localize the impurity in a state

of fixed Sz , and thereby to impede the spin-flip processes that are central to the Kondo

effect. In the charge-coupled model, by contrast, increasing λ from zero initially acts to

decrease the effective Coulomb repulsion and hence to enhance charge fluctuations

on the impurity site; only for λc & λc0 do further increases in the e-b coupling serve to

localize the impurity in a state of fixed charge, eventually leading to the suppression of

the charge Kondo effect at λ = λc .

3.4.3 Crossover Scale

Under the renormalization-group flows sketched in Fig. 3-4, the system passes, with

decreasing energy cutoff or decreasing temperature, between the regions of influence

of different renormalization-group fixed points. For bare parameters that place the

system near the boundary between the Kondo and localized phases, the free-orbital

63

fixed point typically governs the behavior at temperatures much greater than the Kondo

temperature TK of the Anderson model obtained by setting λ = 0 in Eq. (3–1). For

temperatures between of order TK and a crossover scale T∗, the system exhibits

quantum critical behavior controlled by thermal fluctuations about the unstable critical

point. Finally, the physics in the regime T . T∗ is governed by one or other of the two

stable fixed points: Kondo or localized.

For fixed values of all other parameters, one expects T∗ to vanish as the e-b

coupling approaches its critical value according to a power law:

T∗ ∝ |λ− λc |ν for λ→ λc , (3–58)

where ν is the correlation-length exponent [17]. The crossover scale can be determined

directly from the NRG solution via the condition T∗ ∝ Λ−N∗/2, where N∗ is the number

of the iteration at which the many-body energy levels cross over to those of a stable

fixed point. There is some arbitrariness as to what precisely constitutes crossover of

the levels. Different criteria will produce T∗(λ) values that differ from one another by

a λ-independent multiplicative factor. It is of little importance what definition of N∗ one

uses, provided that it is applied consistently.

Figure 3-12 shows typical dependences of T∗ on λc − λ in the Kondo phase.

Equation (3–58) holds very well over several decades, as demonstrated by the linear

behavior of the data on a log-log plot. We find that the numerical values of ν(s), some

of which are listed in Table 3-2, are identical (within small errors), to those of the

spin-boson and Ising BFK models for the same bath exponent s. This supports the

notion that the critical point of the charge-coupled BFA model belongs to the same

universality class as the critical points of the spin-boson and Ising BFK models.

However, to confirm this equivalence, we must compare other critical exponents, as

reported below.

64

Table 3-2. Correlation-length critical exponent ν vs bath exponent s for thecharge-coupled Bose-Fermi Anderson model (CC-BFA, this work) and for theIsing-anisotropic Bose-Fermi Kondo model (BFK, from Refs. [73] and [74]).Parentheses surround the estimated nonsystematic error in the last digit.

s 0.2 0.4 0.6 0.8ν(CC-BFA) 4.99(3) 2.52(2) 1.97(4) 2.12(6)ν(BFK) 4.99(5) 2.50(1) 1.98(3) 2.11(2)

3.4.4 Thermodynamic Susceptibilities

In this subsection, we consider the response of the charge-coupled BFA model to a

global magnetic field H and to a global electric potential Φ. These external probes enter

the Hamiltonian through an additional term

Hext = HSz +ΦQ, (3–59)

where Sz and Q are defined in Eqs. (3–16) and (3–18), respectively. In particular, we

focus on the static impurity spin susceptibility χs,imp = −∂2Fimp/∂H2 and the static

impurity charge susceptibility χc,imp = ∂2Fimp/∂Φ2. Here, Fimp = ∆(F ), where ∆(X ) is the

difference between (i) the value of the bulk property X when the impurity is present and

(ii) the value of X when the impurity is removed from the system. It is straightforward to

show that

Tχs,imp = ∆(⟨⟨S2z ⟩⟩ − ⟨⟨Sz⟩⟩2

), (3–60)

Tχc,imp = ∆(⟨⟨Q2⟩⟩ − ⟨⟨Q⟩⟩2

), (3–61)

where, for any operator A,

⟨⟨A⟩⟩ = Tr A exp(−H/T )Tr exp(−H/T )

. (3–62)

Note that with the above definitions, limT→∞Tχs,imp =18

but limT→∞Tχc,imp =12,

a factor of four difference that must be taken into account when comparing the two

susceptibilities. Since each Tχimp is calculated as the difference of bulk quantities,

its evaluation using the NRG method is complicated by significant discretization and

65

truncation errors. In order to obtain reasonably well-converged results for Tχimp, we

retain Ns = 2000 states after each NRG iteration. However, even this number is

insufficient to allow reliable extraction of χimp ≡ (Tχimp)/T as T → 0.

Figure 3-13 plots NRG results for Tχs,imp(T ) and 14Tχc,imp(T ), calculated for bath

exponent s = 0.8 and different values of the e-b coupling λ. For λ ≪ λc0 (see Sec.

3.3.1), both impurity susceptibilities behave very much as they do in the Anderson

model: with decreasing temperature, Tχc,imp quickly falls toward zero, signaling

quenching of charge fluctuations upon entry into the local-moment regime, whereas

Tχs,imp initially rises toward its local-moment value of 14, before dropping to zero for

T ≪ T∗ on approach to the Kondo fixed point. With increasing λ, the charge response

grows and the spin response is suppressed. The two susceptibilities are approximately

equivalent for λ = λc0, where the effective Coulomb interaction Ueff = 0. For still stronger

e-b couplings, Tχs,imp plunges rapidly as the temperature is decreased, whereas Tχc,imp

first rises on entry to the local-charge regime before dropping to satisfy

limT→0Tχc,imp(T ) = 0 for λ < λc . (3–63)

These trends are very similar to those exhibited [89] by the Anderson-Holstein model.

In that model, however, the drop in Tχc,imp(T ) takes place for strong e-b couplings

λ0 ≫√ω0U/2 around an effective Kondo temperature T eff

K ∼ D exp(−πλ40/Γω30) [85].

In the charge-coupled BFA model, by contrast, neither the spin susceptibility nor the

charge susceptibility exhibits any obvious feature that correlates with the vanishing of

T∗ as λ → λ−c . This can be understood by noting that the impurity susceptibilities are

determined purely by the fermionic part of the excitation spectrum, whose asymptotic

low-energy form is the same at the critical fixed point (which governs the behavior in the

quantum critical regime T∗ . T . TK ) as at the Kondo fixed point (which controls the

regime T . T∗).

66

The behavior of the static impurity spin susceptibility is qualitatively unchanged

upon crossing from the Kondo phase to the localized phase. However, for λ > λc ,

Tχc,imp approaches at low temperatures a nonzero value that can be inferred from the

effective Hamiltonian HNRGL,f [Eq. (3–51)]. Electrons near the Fermi level experience an

s-wave phase shift

δ(ω = 0) =

δ0 for nd = 0,

π − δ0 for nd = 2,(3–64)

where nd labels the two disconnected sectors of HNRGL,f , and

δ0 = arctan (πρ0Wd), 0 ≤ δ0 ≤ π/2, (3–65)

with ρ0 being the effective conduction-band density of states defined in Eq. (3–30). It is

then straightforward to show that

limT→0Tχc,imp(T ) = (1− 2δ0/π)2. (3–66)

Equations (3–52), (3–65), and (3–66) together imply that

limT→0Tχc,imp(T ) ∝ (λ− λc)

2β for λ→ λ+c . (3–67)

As this example illustrates, the thermodynamic susceptibilities contain signatures

of an evolution from a spin-Kondo effect to a charge-Kondo effect. Furthermore, Eqs.

(3–63) and (3–67) suggest that χc,imp may serve as the order-parameter susceptibility

for the QPT. However, neither susceptibility manifests the vanishing of the crossover

scale T∗ on approach to the transition from the Kondo side. Moreover, the conservation

of Q prevents χc,imp from acquiring an anomalous temperature dependence in the

quantum-critical regime [90]. Thus, one is led to conclude that the response to a global

electric potential Φ does not provide access to the critical fluctuations near the QPT.

67

3.4.5 Local Charge Response

Given the nature of the coupling in Hamiltonian (3–1) between the impurity and

the bosonic bath, we expect to be able to probe the quantum critical point through the

system’s response to a local electric potential ϕ that acts solely on the impurity charge,

entering the Hamiltonian via an additional term

Hc,loc = ϕ (nd − 1). (3–68)

A nonzero ϕ is equivalent to a shift in δd entering Eq. (3–9) away from its bare value

ϵd + U/2 = 0.

In this subsection we show that for sub-Ohmic bath exponents 0 < s < 1, (i)

the response to a static ϕ is described by critical exponents that satisfy hyperscaling

relations characteristic of an interacting quantum critical point, (ii) numerical values of

these critical exponents are identical to those of the spin-boson and Ising BFK models,

and (iii) the dynamical response is consistent with the presence of ω/T scaling in the

vicinity of the quantum critical point.

3.4.5.1 Static local charge response

The response to imposition of a static local potential ϕ is measured by the

thermodynamic average value of the impurity charge,

Qloc = ⟨⟨nd − 1⟩⟩, (3–69)

and through the static local charge susceptibility

χc,loc(T ;ω = 0) = − ∂Qloc

∂ϕ

∣∣∣∣ϕ=0

= − limϕ→0

Qloc

ϕ. (3–70)

In NRG calculations of limϕ→0Qloc(ϕ) and χc,loc, we use potentials in the range 10−13 ≤

|ϕ| ≤ 10−10.

As illustrated in Fig. 3-14, the “spontaneous impurity charge” limϕ→0Qloc(λ,ϕ;T =

0) indeed serves as an order parameter for the QPT between the Kondo and localized

68

phases. This quantity vanishes for all λ < λc and is nonzero for λ > λc , its onset being

described by the power law

limϕ→0Qloc(λ,ϕ;T = 0) ∝ (λ− λc)

β for λ→ λ+c . (3–71)

In the localized phase, the presence of an infinitesimal local potential restricts the

effective Hamiltonian (3–51) to just one nd sector: nd = 0 for ϕ > 0, or nd = 2 for ϕ < 0.

Then substituting Eq. (3–64) into the Friedel sum rule ⟨nd⟩0 = 2δ(0)/π yields

limϕ→0Qloc(ϕ;T = 0) = −2 sgnϕ

πacot

(πρ0Wd

). (3–72)

The latter relation explains the equality of the exponents β entering Eqs. (3–52) and

(3–71). It should also be noted that Eqs. (3–65), (3–66), and (3–72) together imply that

limϕ→0Q2loc(ϕ;T = 0) = lim

T→0Tχc,imp(T ). (3–73)

At the critical point, the response to a small-but-finite potential ϕ obeys another

power law,

Qloc(ϕ;λ = λc ,T = 0) ∝ |ϕ|1/δ. (3–74)

This behavior is exemplified in Fig. 3-15 for four different values of s.

Figure 3-16 shows a logarithmic plot of the static local charge susceptibility

χc,loc(T ;ω = 0) vs temperature T for bath exponent s = 0.4 and a number of e-

b couplings straddling λc . In the quantum-critical regime, the susceptibility has the

anomalous temperature dependence

χc,loc(T ;ω = 0) ∝ T−x for T∗ ≪ T ≪ TK, (3–75)

characterized by a critical exponent x . For T ≪ T∗(λ), the temperature variation

approaches that of one or other of the stable fixed points. In the Kondo phase, the

susceptibility is essentially temperature independent, signaling complete quenching

of the impurity, and the zero-temperature value diverges on approach to the critical

69

coupling as

χc,loc(λ;ω = T = 0) ∝ (λc − λ)−γ for λ→ λ−c . (3–76)

In the localized phase, by contrast,

χc,loc(T ,λ;ω = 0) = limϕ→0

Q2loc(λ,ϕ;T = 0)

Tfor λ > λc and T ≪ T∗ , (3–77)

indicative of a residual impurity degree of freedom. Precisely at the critical e-b coupling,

Eq. (3–75) is obeyed all the way down to T = 0.

Table 3-3 lists the numerical values of the critical exponents β, 1/δ, x , and γ, for four

different sub-Ohmic bath exponents s. For each s , these critical exponents are identical

within estimated error to those of the spin-boson and Ising BFK models. In all cases,

we find that x = s to within our estimated nonsystematic numerical error. We also note

that for s ≤ 12, the value of γ lies close to its mean-field value of 1. It is conceivable

that the deviations of γ from 1 are artifacts of the NRG discretization and truncation

approximations.

The exponents in Table 3-3 obey the hyperscaling relations

δ =1 + x

1− x, 2β = ν(1− x), γ = νx , (3–78)

which are consistent with the ansatz

F = Tf

(|λ− λc |T 1/ν

,|ϕ|

T (1+x)/2

)(3–79)

for the nonanalytic part of the free energy. Such hyperscaling suggests that the quantum

critical point is an interacting one [17].

3.4.5.2 Dynamical local charge susceptibility

The dynamical local charge susceptibility is

χc,loc(ω,T ) = i

∫ ∞

0

dt e−iωt⟨⟨[nd(t)− 1, nd(0)− 1]

⟩⟩. (3–80)

70

Table 3-3. Static critical exponents β, 1/δ, x , and γ defined in Eqs. (3–71) and(3–74)–(3–76), respectively, for four different values of the bosonic bathexponent s. Parentheses surround the estimated nonsystematic error in thelast digit.

s β 1/δ x γ

0.2 2.0005(3) 0.6673(1) 0.1997(2) 0.997(4)0.4 0.7568(2) 0.4283(2) 0.4002(4) 1.0117(6)0.6 0.3923(1) 0.2501(7) 0.600(2) 1.1805(5)0.8 0.2130(1) 0.1111(1) 0.800(2) 1.703(3)

Its imaginary part χ′′c,loc can be calculated within the NRG as

χ′′c,loc(ω,T ) =

π

Z(T )

∑m,m′

∣∣⟨m′|nd − 1|m⟩∣∣2(e−Em′/T − e−Em/T

)δ(ω − Em′ + Em). (3–81)

Here, |m⟩ is a many-body eigenstate with energy Em, and Z(T ) =∑m e

−Em/T is the

partition function. Equation (3–81) produces a discrete set of delta-function peaks

that must be broadened to recover a continuous spectrum. Following standard

procedure [91], we employ Gaussian broadening of delta functions on a logarithmic

scale:

δ(|ω|−|∆E |)→ e−b2/4

√π b |∆E |

exp

[−(ln |ω| − ln |∆E |)

2

b2

], (3–82)

with the choice of the broadening width b = 0.5 ln Λ.

(a) Zero temperature. Figure 3-17 plots χ′′c,loc(ω;T = 0) vs ω for bath exponent

s = 0.2 and a series of e-b couplings λ < λc . Whereas χ′′c,loc(ω;λ = 0,T = 0) ∝ ω for

|ω| ≪ TK (the usual Kondo result), we find that χ′′c,loc(ω; 0 < λ < λc ,T = 0) ∝ |ω|ssgn(ω)

as ω → 0, corresponding to a long-time relaxation behavior χc,loc(t) ∝ t−(1+s). Precisely

at the critical e-b coupling,

χ′′c,loc(ω;λ = λc ,T = 0) ∝ |ω|−ysgn(ω) for ω ≪ TK. (3–83)

Figure 3-18 shows χ′′c,loc(ω;λ = λc ,T = 0) vs ω and χc,loc(T ;λ = λc ,ω = 0) vs T for

representative bosonic bath exponents s = 0.2 and s = 0.8. These and all other data

71

that we have obtained are consistent with the relation

x = y = s for 0 < s < 1. (3–84)

For small deviations from the critical coupling, χ′′c,loc(ω;T = 0) exhibits the critical

behavior of Eq. (3–83) over the range T∗ ≪ |ω| ≪ TK, where T∗ is identical (up to a

constant multiplicative factor) to the crossover scale defined in Sec. 3.4.3 that vanishes

at the quantum critical point according to Eq. (3–58).

(b) Finite temperatures. Equation (3–84) is consistent with the presence of ω/T

scaling in the dynamical local charge susceptibility at the quantum critical point, viz

χ′′c,loc(ω,T ;λ = λc) = T

−sΨs(ω/T ). (3–85)

Figure 3-19 shows the collapse of data for χ′′c,loc(ω,T ;λ = λc) onto a single function

of ω/T within the critical regime. The Kondo temperature TK of the Anderson model

obtained by setting λ = 0 serves as a nonuniversal high-frequency cutoff on the critical

behavior; the curves have a common form for ω/T ≪ TK/T . It should be noted that the

NRG method is unreliable [58] for |ω| . T , preventing demonstration of complete ω/T

scaling.

Both the hyperscaling of the static critical exponents and what seems to be ω/T

scaling of the dynamical susceptibility are consistent with the QPT between the Kondo

and localized phases taking place at an interacting critical point below its upper critical

dimension.

3.4.6 Impurity Spectral Function

In this section, we present our results of the impurity spectral function, which is

closely related to probability of finding an electron in the impurity level. We find that the

resonances of the impurity spectral function exhibit signatures of the crossover from the

spin-Kondo effect to the charge-Kondo effect. The impurity spectral function is defined to

72

be

Aσ(ω,T ) = −π−1ImGdσ(ω,T ), (3–86)

where the retarded impurity Green’s function is

Gdσ(ω,T ) = −i∫ ∞

0

dt e iωt⟨⟨[dσ(t), d

†σ(0)

]+

⟩⟩. (3–87)

The spectral function can be calculated within the NRG using the formulation

Aσ(ω,T ) =1

Z(T )

∑m,m′

∣∣⟨m′|d †σ|m⟩∣∣2(e−Em′/T + e−Em/T

)δ(ω − Em′ + Em), (3–88)

where the notation is the same as in Eq. (3–81). To recover a continuous spectrum,

we have again applied Eq. (3–82) to the delta-function output of Eq. (3–88), choosing

the broadening factor b = 0.55 ln Λ that best satisfies the Fermi-liquid result Aσ(ω =

0,T = 0) = 1/πΓ for the Anderson model. In order to achieve satisfactory results, we

find it necessary to work with a smaller discretization parameter (Λ = 3 instead of the

value Λ = 9 employed for all the quantities reported above) and to retain more states

(Ns = 1200 rather than the 500 that typically suffices). Since the spectral functions

shown below are all spin-independent, we henceforth drop the index σ on Aσ. For

the particle-hole-symmetric model considered in this section, the spectral function is

symmetric about ω = 0.

Figure 3-20 plots Aσ(ω;T = 0) vs ω for s = 0.8 and a series of λ values. For

λ = 0, we recover the spectral function of the Anderson model, featuring a narrow Kondo

resonance centered at zero frequency and broad Hubbard satellite bands centered

around ω = ±12U. Increasing the e-b coupling from zero has two initial effects—a

displacement of the Hubbard bands to smaller frequencies, and a broadening of

the low-energy Kondo resonance—that can both be attributed to the boson-induced

renormalization of the Coulomb interaction described in Eq. (3–46).

We expect the Hubbard peak locations to obey ωH ≃ ±12Ueff for 0 ≤ λ ≪ λc0.

However, the peak locations plotted in Fig. 3-21(a) are better fitted by |ωH | = 0.4U −

73

λ2/(πs), which (given the discretization and truncation effects discussed in Sec. 3.3.1)

appears to represent a stronger bosonic renormalization than that predicted by |ωH | =12Ueff. We believe that this discrepancy arises primarily from the rapid broadening of the

Kondo resonance with increasing λ, which shifts the local maximum of the combined

spectral function (the sum of the Kondo resonance plus Hubbard satellite bands) to a

frequency smaller in magnitude than the central frequency of the Hubbard peak by itself.

The width 2ΓK of the Kondo resonance, plotted in Fig. 3-21(b), proves to be equal

(up to a multiplicative constant) to the crossover scale T∗ defined in Sec. 3.4.3. For

λ . λc0, the variation in both scales is well described by the replacement of U in the

expression [66, 67] for the Kondo temperature of the symmetric Anderson model by

UNRG(U/2) [given by Eq. (3–48)], the effective Coulomb interaction on entry to the

local-moment regime. The dashed line in Fig. 3-21(b) shows that the resulting formula,

ΓK = CK

√8UNRGΓ

πAΛexp

(−πAΛU

NRG

8Γ

), (3–89)

where AΛ is defined in Eq. (3–31), provides an excellent description of ΓK over almost

the entire range 0 ≤ λ < λc0 ≃ 0.369. This echoes the finding in the Anderson-Holstein

model that a weak e-b coupling serves primarily to reduce the impurity on-site repulsion,

leading to an increase in the Kondo scale [85].

Once the e-b coupling exceeds λc0, further increase in λ leads to suppression of the

Hubbard peaks (e.g., see the curves for λ = 0.4 and λ = 0.43 in Fig. 3-20) and to a rapid

narrowing of the Kondo resonance [see Fig. 3-21(b)]. In the Anderson-Holstein model,

the Kondo scale remains nonzero—although exponentially reduced—for arbitrarily large

e-b couplings [85]. In the charge-coupled BFA model, by contrast, the Kondo peak

collapses and ΓK extrapolates to zero as λ approaches its critical value λc . As shown in

Fig. 3-22, the central peak remains pinned to the Fermi-liquid result A(ω = 0,T = 0) =

1/πΓ even as the peak width vanishes for λ→ λ−c .

74

In the localized phase (λ > λc ), there is no vestige of the Kondo resonance, but

high-energy Hubbard-like peaks reappear; see the curves for λ = 0.5 and 0.6 in Fig.

3-20. In addition, there is a pair of low-energy peaks centered at ω ≃ ±T∗, as shown in

Fig. 3-22.

3.4.7 Spin-Kondo to Charge-Kondo Crossover

Based on the analysis of the zero-hybridization limit presented in Sec. 3.3.1, one

expects spin fluctuations to dominate the impurity behavior in the region λ ≪ λc0, but

charge fluctuations to be dominant for λc0 ≪ λ < λc . This picture is supported by the

behaviors of the thermodynamic susceptibilities discussed in Sec. 3.4.4. The evolution

from a spin-Kondo effect to a charge-Kondo effect can also be probed by comparing the

static local charge susceptibility [Eq. (3–70)] with its spin counterpart

χs,loc(T ;ω = 0) = − limh→0

⟨⟨nd↑ − nd↓⟩⟩2h

, (3–90)

where h is a local magnetic field that enters an additional Hamiltonian term

Hs,loc =h

2(nd↑ − nd↓). (3–91)

In particular, characteristic energy scales for the spin and charge Kondo effects are

expected to be 1/χs,loc(ω = 0,T = 0) and 4/χc,loc(ω = 0,T = 0), respectively [where the

factor of 4 accounts for the difference in conventions that ϕ couples to nd − 1, whereas

h couples to (nd↑ − nd↓)/2]. Figure 3-23 plots the λ dependence of these quantities for

the parameter set illustrated in Figs. 3-20 and 3-21. The Kondo resonance width 2ΓK

crosses over from paralleling 1/χs,loc(0, 0) for small λ to loosely tracking3 4/χc,loc(0, 0)

as λ approaches λc . In the intermediate region near λ = λc0, 2ΓK is much smaller than

3 From Eqs. (3–58) and (3–76), and Tables 3-2 and 3-3, one expects the vanishing ofΓK ∝ T∗ and 4/χc,loc(0, 0) on approach to the critical point to be governed by differentpowers of λc − λ.

75

either inverse static susceptibility, indicating that the Kondo effect has mixed spin and

charge character.

Figure 3-24 presents a λ-Γ phase diagram for s = 0.8 and fixed U = −2ϵd ,

showing data points along the phase boundary λ = λc(Γ) and along the crossover

boundary λ = λX(Γ), defined as the e-b coupling at which the Kondo resonance width

2ΓK is maximal for the given Γ. The fact that the latter line rises almost vertically from

λ = λc0 at Γ = 0 provides further confirmation of the picture of a crossover from a

spin-Kondo effect to a charge-Kondo effect resulting from the change in the sign of Ueff,

and establishes the validity of the schematic phase diagram (Fig. 3-3) presented in the

introduction to this section.

3.5 Results: Symmetric Model with Ohmic Dissipation

This section presents results for Hamiltonian (3–1) with U = −2ϵd > 0 and

an Ohmic bath (i.e., s = 1). We first discuss the behavior of the static local charge

susceptibility. We show that, in contrast with the sub-Ohmic case 0 < s < 1, the

crossover scale vanishes in exponential (rather than power-law) fashion as the e-b

coupling approaches its critical value from below, and there is no small energy scale

observed on the localized side of the transition. Therefore, the QPT for the Ohmic case

is of Kosterlitz-Thouless type. At the end of the section, we study the effects of the e-b

coupling on the impurity spectral function.

3.5.1 Fixed Points and Thermodynamic Susceptibilities

Figure 3-25 plots the schematic renormalization-group flows for a symmetric

impurity coupled to an Ohmic bath. The flows within the Kondo basin of attraction are

qualitatively very similar to those for the sub-Ohmic case depicted in Fig. 3-4. In the

localized regime, however, the e-b coupling flows not to λ = ∞, but rather to a finite

limiting value that varies continuously with the bare values of λ and Γ. What is shown as

a line of fixed points in Fig. 3-25 is really a plane of fixed points described by HLC [Eq.

(3–49)] with effective couplings λ > λc0,Wp = 0, and 0 ≤ Wd < ∞. Another important

76

departure from the sub-Ohmic case is that for s = 1 there is no longer a distinct critical

point reached by flow along the separatrix from the free-orbital fixed point; rather these

two fixed points merge as s → 1−, leaving a critical endpoint at λ = λc0, ∆ = 0. Strictly,

this is a line of critical endpoints described by HLC [Eq. (3–49)] with effective couplings

λ = λc0,Wp = 0, and 0 ≤ Wd < ∞. For a fixed bare value of Γ, the endpoint value

ofWd is just the limit of the localized fixed-point value ofWd as the bare coupling λ

approaches the phase boundary λc(Γ).

The behaviors of the static impurity spin and charge susceptibilities are qualitatively

very similar to those for a sub-Ohmic bath, as discussed in Sec. 3.4.4. The only

significant difference is that for s = 1, limT→0Tχc,imp(T ) undergoes a discontinuous

jump from its value of 0 for λ ≤ λc to a nonzero value for λ = λ+c . This jump can be

understood through Eqs. (3–65) and (3–66) as a consequence of the fact thatWd does

not diverge on approach to the critical coupling.

3.5.2 Static Local Charge Susceptibility and Crossover Scale

Figure 3-26 is a logarithmic plot of the static local charge susceptibility χc,loc(T ;ω =

0) vs temperature T for different e-b couplings λ. On the Kondo side of the phase

boundary, χc,loc(T ;ω = 0) is proportional to 1/T at high temperatures, but levels off for

T . T∗. We find it convenient to define

T∗ = 4/χc,loc(ω = T = 0) for λ→ λ−c , (3–92)

thereby removing the ambiguity in the definition of the crossover iteration N∗ (see Sec.

3.4.3) on the Kondo side of the s = 1 quantum phase transition.

For λ→ λ−c , the crossover scale vanishes according to (see Fig. 3-27)

T∗ ∝ exp

[− C∗√1− (λ/λc)2

]. (3–93)

In the localized phase, χc,loc(T ;ω = 0) satisfies Eq. (3–77) over the entire

temperature range T ≪ U. Since the critical and localized fixed points share the

77

same temperature variation, no crossover scale can be identified on the localized side

of the phase boundary. Moreover, the order parameter limϕ→0Qloc(ϕ;T = 0) does

not vanish continuously as λ → λ+c , but rather undergoes a discontinuous jump at the

transition, as shown in Fig. 3-27. The magnitude of this jump is nonuniversal, being

related via Eq. (3–72) to the value ofWd at the critical endpoint.

The properties described above are analogous to those of the Kondo model [Eq.

(3–39)] at the transition between the Kondo-screened phase (reached for J⊥ = 0 and

Jz > −|J⊥|) and the local-moment phase (reached for Jz ≤ −|J⊥|). Such behaviors are

characteristic of a Kosterlitz-Thouless type of QPT.

3.5.3 Impurity Spectral Function

Figure 3-28 shows the impurity spectral function A(ω;T = 0) for an Ohmic bath.

The behavior in the Kondo phase is similar to that in the sub-Ohmic case discussed in

Sec. 3.4.6: As the e-b coupling λ increases from zero, the Hubbard satellite bands are

initially displaced to smaller frequencies according to ωH ≃ ±12Ueff [Fig. 3-29(a)], while

the width 2ΓK of the Kondo resonance [Fig. 3-29(b)] first rises before falling sharply on

approach to λ = λc . Just as for 0 < s < 1, the variation in ΓK for λ . λc0 is well described

by Eq. (3–89) with UNRG [Eq. (3–48)] evaluated at E = U/2. Throughout the Kondo

phase, A(ω = T = 0) remains pinned at its Fermi-liquid value 1/πΓ.

For λ ≥ λc , however, the behavior of the spectral function is quite different for s = 1

than for 0 < s < 1. In the sub-Ohmic case, the Kondo-phase pinning extends to the

quantum critical point, i.e., πΓA(ω = T = 0,λ = λc) = 1, while in the localized phase

peaks appear at ω ≃ ±T∗. Figure 3-30 shows that the Ohmic spectral function instead

satisfies πΓA(ω = T = 0,λ = λc) < 1, and exhibits no feature in the localized phase at

energy scales much smaller than 12|Ueff|.

3.6 Results: Asymmetric Model

Sections 3.4 and 3.5 focused exclusively on results for a symmetric impurity

satisfying ϵd = −U/2 in Eq. (3–2) or, equivalently, δd = 0 in Eq. (3–9). We now turn

78

to the general situation of an asymmetric impurity, starting with the sub-Ohmic case

0 < s < 1.

For δd = 0 and small, nonzero values of λ, one expects the fermionic sector of

the charge-coupled BFA model to behave in essentially the same manner as in the

asymmetric Anderson model (reviewed in Sec. 3.3.2), with the exception that the

effective value of the Coulomb interaction U will be reduced by the coupling to the

bosonic bath. At temperatures well below TK, there will be no further renormalization

of the electronic degrees of freedom, the system will exhibit quasiparticle excitations

described by HNRGSC in Eq. (3–34), and the low-energy many-body states will share a

nonvanishing expectation value ⟨nd − 1⟩ [= Qloc(T = 0)]. The bosons will couple to

this impurity charge, yielding low-energy states described most naturally in terms of

displaced-oscillator states [cf. Eq. (3–21)] annihilated by operators

aq = aq +λq√Nq ωq

⟨nd − 1⟩. (3–94)

For s < 1, the e-b coupling is relevant so λ will scale to strong coupling below a

crossover temperature TL ≪ TK.

For δd = 0 and very large values of λ, one instead expects the bosons to localize

the impurity at a high temperature scale TL into a state with ⟨nd⟩ ≃ 0 (for δd > 0) or

⟨nd⟩ ≃ 2 (for δd < 0). For T . TL, the impurity degrees of freedom will be frozen, the

bosonic spectrum will rapidly approach strong coupling, and the conduction electrons

will have an excitation spectrum corresponding to HNRGFI in Eq. (3–35) with a small value

of |V0|.

Given the equivalence of HNRGSC and HNRG

FI , it seems likely that the low-energy

behavior of the asymmetric model will be the same in the small-λ and large-λ limits.

This suggests that the many-body eigenstates evolve adiabatically as the e-b coupling is

increased from λ = 0+ to λ→ ∞, without the occurrence of an intervening QPT.

79

For s = 1, the e-b coupling is marginal, rather than relevant. One again expects

a continuous evolution of the low-energy NRG spectrum with the bare value of λ.

However, in this Ohmic case, the bosonic excitations should correspond to noninteracting

displaced oscillators rather than the (truncated) strong-coupling spectrum found for

0 < s < 1.

The preceding arguments are supported by our NRG results. Here, we illustrate just

the case s = 0.4. Figure 3-31 shows the variation with λ of the ground-state expectation

value ⟨1 − nd⟩0 for several values of δd . In the symmetric case (the δd = 0 curve in

Fig. 3-31), the impurity charge vanishes throughout the Kondo phase, and grows in

power-law fashion on entry to the localized phase. Away from particle-hole symmetry,

by contrast, ⟨1 − nd⟩0 increases smoothly from its Anderson-model value at λ = 0 to

approach 1 as λ→ ∞.

For all nonzero values of δd , Γ and λ, the low-energy spectrum can be decomposed

into the direct product of the fermionic spectrum corresponding to HNRGSC (V1) [or

HNRGFI (V0)] and the same localized-phase bosonic spectrum as found for the symmetric

model. The potential scattering V1 (or V0) is tied to ⟨nd − 1⟩0 by Eq. (3–37), just as in the

Anderson model.

For small λ, the value of ⟨nd − 1⟩0 can be related to the corresponding quantity

in the Anderson model by making use of the effective Coulomb interaction introduced

in Sec. 3.3.1. In the asymmetric Anderson model, the ground-state charge becomes

frozen once the system passes out of its mixed-valence regime, i.e., somewhat below a

characteristic temperature Tf defined [67] for Γ≪ −ϵd ≪ U as the solution of

Tf = |ϵd | −Γ

πlnU

Tf. (3–95)

In the charge-coupled BFA model, U and ϵd in Eq. (3–95) should presumably be

replaced by U(Tf ) and δd − 12U(Tf ), respectively. However, it suffices for our purposes

to note that Tf can be expected to be of the same order as, but somewhat smaller than,

80

|ϵd |. It is then reasonable to hypothesize that ⟨nd − 1⟩0 in the asymmetric charge-coupled

BFA model should be close to the ground-state impurity charge of the Anderson model

with the same Γ and δd , but with U replaced by U(E) [Eq. (3–47)] evaluated at E ≃ Tf .

Our numerical results support this conjecture. For example, Fig. 3-31 shows that close

to particle-hole symmetry (ϵd = −U/2), the Anderson-model charge calculated for

UNRG(E) [Eq. (3–48)] with E = 0.3U (solid lines) reproduces quite well the value of

⟨nd − 1⟩0 (symbols) over quite a broad range of e-b couplings 0 ≤ λ . 23λc , where

λc ≃ 0.29835 is the critical coupling of the symmetric problem.

In the small-λ limit, one can also estimate the boson-localization temperature TL

by considering the evolution with decreasing T of the effective value of λ⟨nd − 1⟩0. The

impurity charge does not renormalize, while to lowest order the effective e-b coupling

obeys Eq. (3–28) [46]. Defining TL by the condition λ(TL)|⟨nd − 1⟩0| = CL, we find

TL ≃∣∣C−1L λ⟨nd − 1⟩0

∣∣2/(1−s). (3–96)

In Fig. 3-32, symbols represent TL values extracted from the crossover of bosonic

excitations in the NRG spectrum, while solid lines show the results of evaluating Eq.

(3–96) using CL = 3 and the ⟨nd − 1⟩0 values shown in Fig. 3-31. The algebraic relation

between the numerical values of TL and ⟨1 − nd⟩0 is well obeyed over a range of e-b

couplings that extends beyond λc of the symmetric problem.

Figure 3-33 plots the static local charge susceptibility calculated for s = 0.4 at the

critical e-b coupling of the symmetric model. For δd = 0, χc,loc follows the quantum

critical behavior χc,loc(T ;ω = 0) ∝ T−x from a high-temperature cutoff of order TK down

to a crossover temperature T∗, below which the susceptibility saturates. Based on Eq.

(3–79) with the identification ϕ ≡ δd , one expects T∗ ∝ |δd |2/(1+x) and, hence,

χc,loc(ϕ;λ = λc ,ω = T = 0) ∝ |δd |−2x/(1+x). (3–97)

81

The log-log plot in the inset of Fig. 3-33 has a slope 0.57 that is fully consistent with Eq.

(3–97).

The results of this work show that gaining direct access to the quantum critical point

of the charge-coupled BFA model requires simultaneous fine tuning of two parameters:

the e-b coupling λ as a function of the hybridization Γ and the on-site Coulomb repulsion

U; and the particle-hole asymmetry (determined in our calculations solely by δd =

ϵd + U/2, but in general also affected by the shape of the conduction-band density of

states). While it may prove very challenging, or even impossible, to achieve this feat in

any experimental realization of the model, it should be a more feasible task to carry out

a rough tuning of parameters that places the system in the quantum critical regime over

some window of elevated temperatures and/or frequencies.

3.7 Summary

In this chapter we have conducted a detailed study of the charge-coupled

Bose-Fermi Anderson model, in which a magnetic impurity both hybridizes with

a structureless conduction band and is coupled, via its charge, to a dissipative

environment represented by a bosonic bath having a spectral function that vanishes

as ωs for frequencies ω → 0. With increasing coupling between the impurity and the

bath, we find a crossover from a conventional Kondo effect—involving conduction-band

screening of the impurity spin degree of freedom—to a charge-Kondo regime in which

the delocalized electrons quench impurity charge fluctuations.

Under conditions of strict particle-hole symmetry, further increase in the impurity-bath

coupling gives rise for 0 < s ≤ 1 to a quantum phase transition between the Kondo

phase, in which the static charge and spin susceptibilities approach constant values

at low temperatures, and a localized phase in which the static charge susceptibility

exhibits a Curie-Weiss behavior indicative of an unquenched local charge degree of

freedom. For sub-Ohmic bosonic bath spectra (described by an exponent s satisfying

0 < s < 1), the continuous quantum phase transition is governed by an interacting

82

critical point characterized by hyperscaling relations of critical exponents and ω/T

scaling in the dynamical local charge susceptibility. Moreover, the continuous phase

transition of the present model belongs to the same universality class as the transitions

of the spin-boson and the Ising-anisotropic Bose-Fermi Kondo models. For an Ohmic

(s = 1) bosonic bath spectrum, the quantum phase transition is of Kosterlitz-Thouless

type.

In the presence of particle-hole asymmetry, the quantum phase transition described

in the previous paragraph is replaced by a smooth crossover, but for small-to-moderate

asymmetries, signatures of the symmetric quantum critical point remain in the physical

properties at elevated temperatures and/or frequencies.

83

λ0 2

2

E

U

n

c02

n d = 0, 2

d

λ

= 1

U12 eff

Figure 3-1. Symmetric, zero-hybridization model defined by HZH in Eq. (3–20) withδd = 0: evolution with e-b coupling λ2 of the lowest eigenenergy in the spinsector (nd = 1, solid line) and in the charge sector (nd = 0, 2, dashed line). Alevel crossing occurs at λ = λc0 specified in Eq. (3–26).

84

1.0 2.0 3.0 4.0Λ

0.2

0.3

0.4

λ c0

0 0.5 100.51Γ = 0

00.5100.51

00.5100.51

s = 0.2

00.5100.51

00.5100.510.4

00.5100.51

00.5100.510.6

00.5100.51

00.5100.510.8

00.5100.51

00.5100.511

Figure 3-2. Dependence of the level-crossing coupling λc0 on the discretization Λ for theNRG solution of HZH [Eq. (3–20)] with U = 0.1, δd = 0, Ns = 200, Nb = 16,and five different values of the bath exponent s. Dashed lines show linear fitsto the data.

85

λ

Kondo

spin charge

localized

Γ

U

λ c 0

U eff eff( > 0) ( < 0)

Figure 3-3. Schematic phase diagram of the symmetric charge-coupled BFA model forbath exponents 0 < s < 1. The solid curve marks the boundary between theKondo phase, in which the impurity degrees of freedom are screened byconduction electrons, and the localized phase, in which the impuritydynamics are controlled by the coupling to the dissipative bath. The dashedvertical line represents a crossover from a regime in which Kondo screeningtakes place primarily in the spin sector to a regime in which a charge-Kondoeffect is predominant.

86

0L

∆

λ

K

C

LMFO

λ c0

Figure 3-4. Schematic renormalization-group flows on the λ-∆ plane for the symmetriccharge-coupled BFA model with a bath exponent 0 < s < 1. Trajectories witharrows represent the flow of the couplings λ entering Eq. (3–15) and ∆defined in Eq. (3–50) under decrease of the high-energy cutoffs on theconduction band and the bosonic bath. Between the basins of attraction ofthe Kondo fixed point (K) and the localized fixed point (L) lies a separatrix,along which the flow is away from the free-orbital fixed point (FO) located atλ = λc0, ∆ = 0 and toward the critical fixed point (C). For ∆ = 0 only, there isflow from FO toward the local-moment fixed point (LM) at λ = 0.

87

0 20 40

N (even iterations)

0

0.5

1.0

1.5

2.0

EN

0 20 40 0 20 40 0 20 40 0 20 40

(a) λ = 0 (d) λ = 0.5301(b) λ = 0.53 (c) λ = λc (e) λ = 0.7

Figure 3-5. Low-lying many-body energies EN vs even iteration number N for s = 0.2,Γ = 0.5, and five different couplings λ, whose critical value λc ≃ 0.530086.(a) The spectrum of the Kondo fixed point is reached for λ = 0, where thebosonic bath is decoupled from the impurity. (b) As λ approaches λc fromthe Kondo side, the quantum critical regime is accessed (6 ≤ N ≤ 26) beforethe energy levels cross over to the Kondo fixed point. (c) The spectrum ofthe quantum critical point. (d) When λ is close to its critical value but slight inthe localized phase, the energy levels cross over from the quantum criticalpoint (6 ≤ N ≤ 30) to the localized phase. (e) Deep in the localized phase,the spectrum of the localized fixed point is reached at small N. Thecrossover in the energy flows defines a low-energy scale T∗ that vanishes asg → gc in the fashion of Eq. (3–58). Note that the localized phase is actuallynot captured by a unique fixed point but corresponds to a line of fixed pointswith differentWd . See text for discussion.

88

10−7

10−6

10−5

10−4

10−3

10−2

10−1

λ−λc

100

102

104

106

Wd

s = 0.2

s = 0.4

s = 0.6

s = 0.8

Figure 3-6. (Color) Fixed-point couplingWd entering Eq. (3–51) vs e-b coupling λ− λcin the localized phase near the phase boundary at λ = λc . Results areshown for U = −2ϵd = 0.1, Λ = 9, Ns = 500, Nb = 8, four different values ofthe bath exponent s, and Γ = 0.5, 1.0, 10, and 50 for s = 0.2, 0.4, 0.6, and0.8, respectively (see footnote 1 on page 57). The power-law divergence ofWd as λ→ λ+c [Eq. (3–52)] is reflected in the linear behaviors of data on alogarithmic scale. The numerical values of the exponent β obtained here areidentical (to within small errors) to those listed in Table 3-3.

89

4 6 8 10 12N

b

0.100

0.104

0.108

0.112

E

0.2454

0.2456

0.2458

E

200 300 400 500 600N

s

00.5100.51

00.5100.51

s = 0.800.51

00.5100.51

00.51s = 0.8

00.5100.51

00.5100.51

s = 0.200.51

00.5100.51

00.51s = 0.2

Figure 3-7. Dependence of the energy of the first bosonic excitation at the critical point(λ = λc ) on the NRG truncation parameters Nb and Ns . Results are shownfor U = −2ϵd = 0.1, Γ = 0.01, Λ = 9, and bath exponents s = 0.2 ands = 0.8. In the left panels, Ns = 500, while in the right panels Nb = 8.

90

4 6 8 10 12N

b

0.54

0.56

0.58

0.60

0.62

E

1.10

1.20

1.30

1.40

1.50

1.60

E

200 300 400 500 600N

s

00.5100.51

00.5100.51

s = 0.800.51

00.5100.51

00.51s = 0.8

51015201.11.21.31.41.51.6

51015201.11.21.31.41.51.6

s = 0.200.51

00.5100.51

00.51s = 0.2

Figure 3-8. Dependence of the energy of the first bosonic excitation in the localizedphase (λ = 1.1λc ) on the NRG truncation parameters Nb and Ns . All otherparameters are as in Fig. 3-7.

91

4 6 8 10 12

Nb

0

20

40

60

80

<B

20>

λ=λc

λ=1.1λc

4 6 8 10 12

Nb

λ=λc

λ=1.1λc

00.5100.51

00.5100.51

s = 0.800.51

00.5100.51

00.51s = 0.2

(a) (b)

Figure 3-9. Dependence of ⟨B20⟩, defined in Eq. (3–55) and evaluated at characteristictemperature scale of iteration N = 20, on the NRG truncation parameters Nbfor (a) s = 0.2 and (b) s = 0.8. See text for discussion.

92

0 0.2 0.4 0.6 0.8 1.0λ

0

0.02

0.04

0.06

0.08

0.10

Γ

s = 0.2

s = 0.4

s = 0.6

s = 0.8

Figure 3-10. (Color) Critical coupling λc vs hybridization width Γ for U = −2ϵd = 0.1,Λ = 9, Ns = 500, Nb = 8, and the bath exponents s listed in the legend.

93

0 0.2 0.4 0.6 0.8 1s

0

0.2

0.4

0.6

0.8

λc

λc0

gc /2

Figure 3-11. Variation with bath exponent s of the critical couplings λc and λc0 in thecharge-coupled BFA model (this work) and gc/2 in the spin-coupled BFAmodel (Ref. [74]). Results are shown for U = −2ϵd = 0.1, Γ = 0.01, Λ = 9,Ns = 500, and Nb = 8.

94

10−6

10−5

10−4

10−3

λc − λ

10−20

10−15

10−10

10−5

T *

s = 0.2

s = 0.4

s = 0.6

s = 0.8

Figure 3-12. (Color) Crossover scale T∗ vs λc − λ on the Kondo side of the critical pointfor four different values of the bath exponent s , with all other parameters asin Fig. 3-6. The slope of each line on this log-log plot gives thecorrelation-length exponent ν(s) defined in Eq. (3–58).

95

10−5

100

T

0

0.05

0.10

0.15

0.20

0.25

λ = 0λ = λc,0

10−10

10−5

100

T

λ = 1.02λc

λ = 4λc

λ = 0.98λc

λ = λc

Tχs,imp

1−4 Tχc,imp

s = 0.8

Figure 3-13. (Color) Temperature dependence of the impurity contribution to the staticspin (left) and charge (right) susceptibilities for s = 0.8, U = −2ϵd = 0.1,Γ = 0.01, Λ = 9, Ns = 2000, Nb = 8, and different values of the e-b couplingλ. Dotted curves correspond to e-b couplings lying between the λ valuesspecified in the legend for the adjacent nondotted curves. Forλ = λc0 ≃ 0.396, the spin and charge susceptibilities are equivalent:χs,imp(T ) ≃ 1

4χc,imp(T ). For λ < λc0, the spin response is stronger, while for

λ > λc0, the charge response dominates. For λ ≤ λc ≃ 0.5052181,limT→0Tχc,imp(T ) = 0, whereas for λ > λc , the limiting value is nonzeroand obeys Eqs. (3–65) and (3–66).

96

0 2 4 6 8 10

103(λ−λc)

0

0.05

0.10

0.15

0.20

0.25

Qlo

c(φ →

0; T

= 0

)

s = 0.2

10−5

10−4

10−3

10−2

λ−λc

10−6

10−4

10−2

1

s = 0.4

2.52.552.62.652.72.52.552.62.652.7

2.52.552.62.652.72.52.552.62.652.7

s = 0.6

4.464.484.54.524.544.464.484.54.524.54

4.464.484.54.524.544.464.484.54.524.54

s = 0.8

Figure 3-14. (Color) Impurity charge limϕ→0− Qloc(λ,ϕ;T = 0) vs e-b coupling λ− λc forfour different values of the bath exponent s. All other parameters are as inFig. 3-6. As λ approaches λc from above, limϕ→0− Qloc(λ,ϕ;T = 0)vanishes (left panel) in a power-law fashion (right panel) described by Eq.(3–71).

97

10−14

10−12

10−10

10−8

10−6

10−4

|φ|

10−8

10−6

10−4

10−2

100

Qlo

c(φ; λ

= λ

c , T

= 0

)

s = 0.2

s = 0.4

s = 0.6

s = 0.8

Figure 3-15. (Color) Impurity charge Qloc(ϕ;λ=λc ,T =0) vs local electric potential |ϕ|for four different values of the bath exponent s . All other parameters are asin Fig. 3-6. The dashed lines represent fits to the form of Eq. (3–74).

98

10−15

10−10

10−5

100

T

100

102

104

106

108

1010

χ c,lo

c(T; ω

= 0

)

λ = 0.5λ = 1.0λ = 1.028λ = 1.02905λ = 1.04λ = 5.0

0 0.5 100.51

s = 0.4

Figure 3-16. (Color) Static local charge susceptibility χc,loc(T ;ω = 0) vs temperature Tfor s = 0.4, U = −2ϵd = 0.1, Γ = 1.0 (see footnote 1 on page 57), Λ = 9,Ns = 500, Nb = 8, and for different values of the e-b coupling λ straddlingthe critical value λc ≃ 1.02905.

99

10−20

10−15

10−10

10−5

100

ω

10−8

10−4

100

104

χ″ c,lo

c(ω; T

= 0

)

λ = 0

λc−λ = 10−1

λc−λ = 10−2

λc−λ = 10−3

λc−λ = 10−4

λ = λc

s = 0.2

Figure 3-17. (Color) Imaginary part of the dynamical local charge susceptibilityχ′′c,loc(ω;T = 0) vs frequency ω for s = 0.2, U = −2ϵd = 0.1, Γ = 0.5 (see

footnote 1 on page 57), Λ = 9, Ns = 500, Nb = 8, and different e-bcouplings λ < λc on the Kondo side of the critical point, which is located atλc ≃ 0.53008. As λ→ λ−c , χ′′

c,loc(ω;T = 0) follows the quantum critical form[Eq. (3–83)] for T∗ ≪ ω ≪ TK, where TK is the Kondo scale of thepure-fermionic (λ = 0) problem.

100

10−12

10−8

10−4

100

T or ω

100

104

108

1012

χc,loc

(T; λ = λc, ω = 0)

χ″c,loc

(ω; λ = λc, T = 0)

00.5100.51

00.5100.51

s = 0.2

00.5100.51

00.5100.51

s = 0.8

Figure 3-18. (Color) Critical static and dynamical response: χc,loc(T ;λ = λc ,ω = 0) vsT (circles) and χ′′

c,loc(ω;λ = λc ,T = 0) vs ω (squares) for tworepresentative bath exponents s = 0.2 and s = 0.8. All other parametersare as in Fig. 3-6. The equality of the slopes of the static and dynamicalcharge susceptibilities for a given bath exponent s indicates that thecorresponding critical exponents satisfy x = y .

101

100

105

1010

1015

1020

ω / T

10−8

10−6

10−4

10−2

Tsχ″ c,

loc(ω

, T )

T/TK = 10−20

T/TK = 10−15

T/TK = 10−10

T/TK = 10−6

T/TK = 10−2

00.5100.51

s = 0.2

Figure 3-19. (Color) Scaling with ω/T of the imaginary part of the dynamical localcharge susceptibility χ′′

c,loc(ω,T ) at the critical e-b coupling λc ≃ 0.53008 fors = 0.2, U = −2ϵd = 0.1, Γ = 0.5 (see footnote 1 on page 57), Λ = 9,Ns = 500, Nb = 8, and different temperatures T ≪ TK = 0.425.

102

−0.04 0 0.04 0.08 0.12 0.16ω

0

10

20

30

A(ω

; T =

0)

λ = 0λ = 0.16

λ = 0.4λ = 0.43λ = 0.5λ = 0.6

00.5100.51

s = 0.8

Figure 3-20. (Color) Impurity spectral function A(ω;T = 0) vs frequency ω for s = 0.8,U = −2ϵd = 0.1, Γ = 0.01, Λ = 3, Ns = 1200, Nb = 8, and different values ofthe e-b coupling λ. For these parameters, Ueff defined in Eq. (3–33)changes sign at λc0 ≃ 0.369 and the critical coupling is λc ≃ 0.474.

103

0 0.1 0.2

λ

0.02

0.03

0.04ω

H

00.5100.51

s = 0.8

0 0.1 0.2 0.3 0.4 0.5

λ

0

0.004

0.008

0.012

0.016

2ΓK

00.5100.51

(a)

00.5100.51

(b)

Figure 3-21. (Color) Variation with e-b coupling λ of two characteristic energy scalesextracted from the zero-temperature impurity spectral function. Allparameters except λ are the same as in Fig. 3-20. (a) Location ωH of theupper Hubbard peak. The dashed line shows ωH = 0.4U − λ2/(πs). (b)Kondo resonance width (full width at half height) 2ΓK. The dashed line,representing the prediction of Eq. (3–89) with CK = 0.82 and with UNRG inEq. (3–48) evaluated at E = U/2 = |ϵd |, fits the data over almost the entirerange 0 ≤ λ < λc0 ≃ 0.369.

104

−5×10−7 0 5×10

−7

ω

15

20

25

30

A(ω

; T =

0)

λ = 0.473λ = 0.474λ = 0.47500.51

00.51s = 0.8

Figure 3-22. (Color) Detail of the impurity spectral function A(ω;T = 0) aroundfrequency ω = 0 for s = 0.8, U = −2ϵd = 0.1, Γ = 0.01, Λ = 3, Ns = 1600,Nb = 8, and different e-b couplings λ straddling the critical valueλc ≃ 0.47458. For λ ≤ λc , A(ω;T = 0) is pinned to the value predicted byFermi-liquid theory. For λ > λc , the Kondo resonance disappears, leaving apair of low-energy peaks centered at |ω| of order the crossover temperatureT∗ (≃1.4× 10−8 for λ=0.475).

105

0 0.1 0.2 0.3 0.4 0.5

λ

0

0.02

0.04

0.06

4/χc,loc

(0,0)

1/χs,loc

(0,0)

2ΓK

00.5100.51

00.5100.51

s = 0.8

Figure 3-23. (Color) Variation with e-b coupling λ < λc of the Kondo resonance width2ΓK, the inverse static local spin susceptibility 1/χs,loc(ω = 0,T = 0), andthe inverse static local charge susceptibility 4/χc,loc(ω = 0,T = 0). Theresults shown are for s = 0.8, U = −2ϵd = 0.1, Γ = 0.01, Λ = 3, Ns = 1200,and Nb = 8. For the calculation of the static local spin susceptibility via Eq.(3–90), the total spin S is not a good quantum number, so Ns specifies thenumber of (Sz ,Q) states retained after each iteration.

106

0.2 0.3 0.4 0.5λ

0

0.005

0.010

Γ

λc

λX

00.5100.51

00.5100.51

s = 0.8

Figure 3-24. Phase boundary λc(Γ) and crossover boundary λX(Γ) (defined in the text)for s = 0.8, U = −2ϵd = 0.1, Λ = 3, Ns = 1200, and Nb = 8. The data areconsistent with the schematic phase diagram shown in Fig. 3-3.

107

λ0 λ

LM

c 0

L

K

∆

FO

Figure 3-25. Schematic renormalization-group flows on the λ-∆ plane for the symmetricmodel with bath exponent s = 1. Trajectories represent the flow of thecouplings λ entering Eq. (3–15) and ∆ defined in Eq. (3–50) underdecrease in the high-energy cutoffs on the conduction band and thebosonic bath. A separatrix (dashed line) forms the boundary between thebasins of attraction of the Kondo fixed point (K) and a line of localized fixedpoints (L). Flow along the separatrix is toward the free-orbital fixed point(FO) located at λ = λc0. For ∆ = 0 only, there is flow away from FO towardthe local-moment fixed point (LM) at λ = 0.

108

10−10

10−8

10−6

10−4

10−2

100

T

100

102

104

106

108

χ c,lo

c(T; ω

= 0

)

λ = 0.1λ = 0.3λ = 0.5λ = 0.62λ = 0.64λ = 0.76λ = 0.84

00.5100.51

s = 1

Figure 3-26. (Color) Static local charge susceptibility χc,loc(T ;ω = 0) vs temperature Tfor s = 1, U = −2ϵd = 0.1, Γ = 0.01, Λ = 9, Ns = 800, Nb = 12, and differente-b couplings λ. On the Kondo side of the QPT (λ < λc ≃ 0.726), there is aclear crossover from quantum-critical to screened behavior around therenormalized Kondo temperature T∗ = 4/χc,loc(ω = T = 0). No suchcrossover is evident on the localized side (λ > λc ).

109

0 0.2 0.4 0.6 0.8

λ2

0

0.4

0.8

1.2

1/lnχc,loc

(0,0)

Qloc

(0)

00.5100.51

s = 1

Figure 3-27. Variation with e-b coupling λ of the local charge susceptibilityχc,loc(ω = T = 0) in the Kondo phase λ < λc ≃ 0.726 and of the orderparameter limϕ→0− Qloc(ϕ;T = 0) in the localized phase λ > λc , for s = 1,U = −2ϵd = 0.1, Γ = 0.01, Λ = 9, Ns = 800, and Nb = 12. The dotted lineshows a fit of the susceptibility data using Eqs. (3–92) and (3–93).

110

−0.04 0 0.04 0.08 0.12 0.16

ω

0

10

20

30

A(ω

; T =

0)

λ = 0λ = 0.2λ = 0.4λ = 0.5λ = 0.6λ = 0.68λ = 0.74

s = 1

Figure 3-28. (Color) Impurity spectral function A(ω;T = 0) vs ω for s = 1,U = −2ϵd = 0.1, Γ = 0.01, Λ = 3, Ns = 1200, Nb = 12, and different valuesof the e-b coupling λ. For these parameters, Ueff [Eq. (3–33)] changes signat λc0 ≃ 0.413 and the critical coupling is λc ≃ 0.669.

111

0 0.1 0.2

λ

0.030

0.035

0.040

ωH

00.5100.51

00.5100.51

(a)

0 0.2 0.4 0.6

λ

0

0.004

0.008

0.012

0.016

2ΓK

(b)

s = 1

Figure 3-29. (Color) Variation with e-b coupling λ of two characteristic energy scalesextracted from the zero-temperature impurity spectral function. Allparameters except λ are the same as in Fig. 3-28. (a) Location ωH of theupper Hubbard peak. The dashed line shows ωH(λ) = 0.4U − λ2/π. (b)Kondo resonance width (full width at half height) 2ΓK. The dashed line,representing the prediction of Eq. (3–89) with CK = 0.82 and with UNRG inEq. (3–48) evaluated at E = U/2 = |ϵd |, fits the data over almost the entirerange 0 ≤ λ < λc0 ≃ 0.413.

112

10−15

10−10

10−5

100

ω

0.1

1

10

100

A(ω

; T =

0)

λ = 0λ = 0.4λ = 0.6λ = 0.64λ = 0.66λ = 0.68λ = 0.7λ = 0.74

00.5100.51s = 1

Figure 3-30. (Color) Impurity spectral function A(ω;T = 0) vs frequency ω on alogarithmic scale for s = 1, U = −2ϵd = 0.1, Γ = 0.01, Λ = 3, Ns = 1200,Nb = 12, and different e-b couplings λ. For λ < λc ≃ 0.669, the behavior issimilar to that found for 0 < s < 1. However, for λ ≥ λc , the spectral functionis essentially featureless below the energy scale 1

2|Ueff| of the Hubbard

peaks.

113

0 0.1 0.2 0.3 0.4 0.5 0.6

λ

10−4

10−3

10−2

10−1

100

< 1

− n d

>0

δd = 1×10−2

δd = 5×10−3

δd = 5×10−4

δd = 5×10−5

δd = 0

s = 0.4

Figure 3-31. (Color) Variation in the magnitude ⟨1− nd⟩0 of the ground-state impuritycharge with e-b coupling λ for s = 0.4, U = 0.1, Γ = 0.01, Λ = 9, Ns = 500,and Nb = 8. Symbols represent results for five values of the impurityasymmetry δd = ϵd + U/2. The solid lines corresponding to each caseδd = 0 represent the impurity charge calculated by solving the Andersonmodel [Eq. (3–11)] for the same δd value but using an effective Coulombinteraction UNRG(0.3U) [Eq. (3–48)]. The δd = 0 symbols show values oflimϕ→0− Qloc(λ,ϕ;T = 0), connected by an interpolating line.

114

0 0.1 0.2 0.3 0.4 0.5 0.6

λ

10−30

10−20

10−10

100

TL

δd = 1×10−2

δd = 5×10−3

δd = 5×10−4

δd = 5×10−5

s = 0.4

Figure 3-32. (Color) Variation in the bosonic localization temperature TL with coupling λfor s = 0.4, U = 0.1, Γ = 0.01, Λ = 9, Ns = 500, Nb = 8, and variousimpurity asymmetries δd = ϵd + U/2. The solid lines were obtained byevaluating Eq. (3–96) with the ⟨1− nd⟩0 values shown in Fig. 3-31 and withCL = 3.

115

10−15

10−10

10−5

100

T

102

104

106

χ c,lo

c(T; λ

= λ

c , ω

= 0

)

s = 0.4

δd = 1×10−2

δd = 5×10−4

δd = 5×10−6

δd = 5×10−8

δd = 0

10−12

10−8

10−4

δd

104

106

108

χ c,lo

c(λ =

λc

, T =

ω =

0)

Figure 3-33. (Color) Static local charge susceptibility χc,loc(T ;ω = 0) vs temperature Tfor s = 0.4, U = 0.1, Γ = 0.01, λ ≃ 0.29835, Λ = 9, Ns = 500, Nb = 8, andvarious impurity asymmetries δd = ϵd + U/2. The e-b coupling equals thecritical coupling λc of the symmetric case δd = 0. Inset: zero-temperaturestatic local charge susceptibility χc,loc(ω = T = 0) vs δd .

116

CHAPTER 4RESULTS FOR PSEUDOGAP ANDERSON-HOLSTEIN MODEL

This chapter is based on a manuscript by Mengxing Cheng and Kevin Ingersent

currently in final preparation for submission to Phys. Rev. B.

4.1 Introduction

In this chapter, we theoretically investigate a pseudogap Anderson-Holstein (PAH)

model of a magnetic impurity level that hybridizes with a fermionic host whose density of

states vanishes as |ϵ|r at the Fermi energy (ϵ = 0) and is also coupled, via the impurity

charge, to a local-boson mode. The model is relevant to electron transport in nanoscale

devices, as we discussed in Sec. 1.5.

Our numerical renormalization-group (NRG) study of the pseudogap Anderson-Holstein

model reveals that: (i) For 0 < r < 1, the pseudogap Anderson-Holstein model exhibits

continuous quantum phase transitions with anomalous critical exponents. At fixed weak

bosonic couplings, as the hybridization increases from zero, continuous quantum phase

transitions occur between the local-moment and two strong-coupling phases. However,

at fixed strong bosonic couplings, increase in the hybridization leads to continuous

quantum phase transitions from the local-charge phase to another two strong-coupling

phases. In the vicinity of the quantum critical points (QCPs), the critical exponents

are found to be numerically identical to those of the pseudogap Anderson model.

(ii) For r = 2, the pseudogap Anderson-Holstein model can effectively describe a

device consisting of two quantum dots connected in parallel between source and drain

electrodes, where one dot has an unpaired spin in a Coulomb blockade valley and is

coupled to local-boson mode, while the other dot is tuned to be noninteracting and in

resonance with the leads. In this case, quantum phase transitions are manifested by

peak-and-valley features in the linear-response conductance through the device.

This chapter is organized as follows. Section 4.2 introduces the pseudogap

Anderson-Holstein model and reviews related models studied previously. NRG results

117

for the PAH model with 0 < r < 1 are presented and discussed in Sec. 4.3 and

Sec. 4.4. Section 4.5 presents results for the r = 2 case, which is associated with the

double-quantum-dot device mentioned above. Finally, Sec. 4.6 summarizes the main

findings.

4.2 Model Hamiltonian and Preliminary Analysis

In this section, we first introduce the pseudogap Anderson-Holstein model and then

review two closely related quantum impurity models, establishing expectations for the

behaviors of the full model described by Eq. (4–1).

4.2.1 Pseudogap Anderson-Holstein Model

The PAH model is described by the Hamiltonian

HPAH = Himp + Hband + Hboson + Himp-band + Himp-boson , (4–1)

where

Himp = δd(nd − 1) + 12U(nd − 1)2 , (4–2a)

Hband =∑k,σ

ϵkc†kσckσ , (4–2b)

Hboson = ω0 a†a , (4–2c)

Himp-band =1√Nk

∑k,σ

(Vk c†kσdσ + V

∗k d

†σckσ) , (4–2d)

Himp-boson = λ(nd − 1)(a + a†) . (4–2e)



ϵd = δd − 12U < 0 in the impurity level, nd =

∑σ d

†σdσ is the total impurity occupancy, and

U > 0 is the Coulomb repulsion between two electrons in the impurity level. Note that

the impurity term (4–2a) is equivalent, apart from an additive constant −ϵd , to the more

conventional form Himp = ϵd nd + Und↑nd↓ from the Anderson impurity model. Vk is the

hybridization between the impurity and a conduction-band state of energy ϵk annihilated

by fermionic operator ckσ, and λ characterizes the coupling of the impurity occupancy to

118

a boson mode of energy ω0 annihilated by operator a. Without loss of generality, we take

Vk and λ to be real and non-negative. For compactness of notation, we drop all factors

of the reduced Planck constant ~, Boltzmann’s constant kB , the impurity magnetic

moment gµB , and the electronic charge e.

The conduction-band dispersion ϵk and the hybridization Vk affect the impurity

degrees of freedom only through the scattering rate

Γ(ϵ) ≡ π

Nk

∑k

|Vk|2δ(ϵ− ϵk). (4–3)

To focus on the most interesting physics of the model, we assume a simplified form

Γ(ϵ) = Γ |ϵ/D|r Θ(D − |ϵ|), (4–4)

where Θ(x) is the Heaviside (step) function. In this notation, the case r = 0 represents

a conventional metallic scattering rate. This work focuses on cases r > 0 in which the

scattering exhibits a power-law pseudogap around the Fermi energy. We will assume

henceforth that such a scattering rate arises from a local hybridization (Vk = V ) and

a density of states varying as ρ(ϵ) ≡ (Nk)−1∑

k δ(ϵ − ϵk) = ρ0|ϵ/D|rΘ(D − |ϵ|), in

which case Γ = πρ0V2. However, the results below apply also to situations in which the

hybridization contributes to the energy dependence of Γ(ϵ).

The assumption that Γ(ϵ) exhibits a pure power-law dependence over the entire

width of the conduction band is a convenient idealization. More realistic scattering rates

in which the power-law variation is restricted to a region around the Fermi energy exhibit

the same qualitative physics, with modification only of nonuniversal properties such as

critical couplings and Kondo temperatures.

The properties of the Hamiltonian specified by Eqs. (4–1)–(4–4) turn out to depend

on whether or not the system is invariant under the particle-hole transformation ckσ →

c†−k,σ, dσ → −d †σ, a → −a, which maps δd → −δd , ϵk → −ϵ−k, Vk → V−k. For the

119

symmetric scattering rate given in Eq. (4–4), the condition for particle-hole symmetry is

δd = 0 corresponding to ϵd = −12U.

4.2.2 Review of Related Models

Before addressing the full PAH model described by Eq. (4–1), it is useful to review

two limiting cases that have been studied previously. One is the pseudogap Anderson

model in absence of the impurity-boson coupling and the other is the Anderson-Holstein

model with a metallic host corresponding to the case r = 0.

4.2.2.1 Pseudogap Anderson model.

For zero bosonic coupling λ = 0, the PAH model reduces to the pseudogap

Anderson model [71, 92–95] plus free local bosons. In the conventional (r = 0)

Anderson impurity model, the generic low-temperature limit is a strong-coupling (Kondo

or mixed valence) regime in which the impurity level is effectively absorbed into the

conduction band [66, 67]. In the pseudogap variant of the model, the depression of the

scattering rate around the Fermi energy gives rise to a competing local-moment phase

in which the impurity retains an unscreened spin degree of freedom all the way down to

the absolute zero of temperature. The T = 0 phase diagram of this model depends on

the presence or absence of particle-hole symmetry [71].

For the symmetric case δd = 0 (ϵd = −12U) and any band exponent 0 < r < 1

2,

there is a continuous quantum phase transition at a critical coupling Γ = Γc(r ,U)

between the local-moment (LM) phase and a symmetric strong-coupling (SSC) phase.

In the LM phase (Γ < Γc ), the impurity contributions to entropy and to the static

spin susceptibility approach the low-temperature limit Simp = ln 2 and Tχs,imp =14,

respectively, while conduction electrons at the Fermi energy experience an s-wave

phase shift δ0(ϵ = 0) = 0. In the SSC phase (Γ > Γc ), the corresponding properties are

Simp = 2r ln 2, Tχs,imp = r/8, and δ0(0) = −(1− r)(π/2) sgn ϵ (all indicative of incomplete

quenching of the impurity degrees of freedom). The quantum phase transition takes

place at an interacting quantum critical point [58, 96, 97]. For the symmetric case and

120

r ≥ 12, by contrast, the SSC fixed point is unstable[71, 93] and the system lies in the LM

phase for all values of Γ.

Away from particle-hole symmetry, the model remains in the LM phase described

above for |δd | < 12U (i.e., −U < ϵd < 0) and Γ < Γc(r ,U, δd) ≡ Γc(r ,U,−δd). As shown

schematically in Fig. 4-1, the critical hybridization width Γc increases monotonically from

zero as |δd | drops below 12U. For 0 < r < 1

2(left panel of Fig. 4-1), Γc(r ,U, δd) smoothly

approaches the symmetric critical value Γc(r ,U) as δd → 0. For r ≥ 12

(right panel of

Fig. 4-1), Γc(r ,U, δd) instead diverges as δd → 0, consistent with the symmetric case

discussed in the preceding paragraph.

For δd = 0 and Γ > Γc , the model lies in one of two asymmetric strong-coupling

(ASC) phases having low-temperature properties Simp = 0 and Tχs,imp = 0. For

δd > 0, the Fermi-energy phase shift is δ0(0) = −π sgn ϵ, while the ground-state charge

(total fermion number measured from half-filling) is Q = −1. For δd < 0, by contrast,

δ0(0) = +π sgn ϵ and Q = +1. We label these two phases ASC− and ASC+ according to

the sign of Q.

For r < r ∗ ≃ 3/8, the low-temperature physics on the phase boundary Γ =

Γc(r ,U, δd) is identical to that at particle-hole symmetry, whereas for r > r ∗, there are

two asymmetric quantum phase transitions (one for δd > 0, the other for δd < 0) [71].

The asymmetric transitions take place at interacting quantum critical points for r ∗ < r <

1, but at noninteracting critical points for 1 < r < 2; for r ≥ 2, the transitions are first

order [58, 97].

4.2.2.2 Anderson-Holstein model

For r = 0, the PAH model reduces to the Anderson-Holstein model [62, 85].

Insight into the physics of both models can be gained by considering the limit Γ = 0

of zero hybridization. Here, the Fock space of each model can be partitioned into

subspaces of fixed impurity occupancy nd = 0, 1, and 2, and the Hamiltonian can be

diagonalized by introducing a displaced-oscillator operator a = a + λ(nd − 1)/ω0, yielding

121

H = Himp + Hband + Hboson with U entering Eq. (4–2a) replaced by

Ueff = U − 2λ2/ω0. (4–5)

It can be seen from Eq. (4–5) that this effective on-site Coulomb interaction changes

sign at λ = λ0, where

λ0 =√ω0U/2 . (4–6)

For weak bosonic couplings λ < λ0, the interaction is repulsive, and for |δd | < 12Ueff

the impurity ground state is a spin doublet with nd = 1 and σ = ±12. For λ > λ0, by

contrast, the strong coupling to the bosonic bath yields an attractive effective on-site

interaction and for |δd | < −12Ueff the two lowest-energy impurity states are spinless but

have a charge (relative to half filling) Q = nd − 1 = ±1; note that these two states are

degenerate only under conditions of strict particle-hole symmetry (δd = 0).

The full Anderson-Holstein model with Γ = 0 exhibits a continuous evolution [62, 85]

of its physical properties with increasing bosonic coupling. The properties for a given λ

are essentially those of the conventional Anderson impurity model with U replaced by

Ueff(λ). In particular, for Γ ≪ |δd | ≪ U, there is a smooth crossover from a conventional

spin-sector Kondo effect for λ ≪ λ0 (and thus Ueff ≫ Γ) to a charge-sector analog of

the Kondo effect for λ ≫ λ0 (and −Ueff ≫ Γ). A primary goal of the present work was to

understand how this physics is modified by the presence of a pseudogap in the impurity

scattering rate.

4.3 Results: Symmetric PAH Model with Band Exponents 0 < r < 12

As mentioned in Sec. 4.2.2, the particle-hole-symmetric pseudogap Anderson

model with a band exponent 0 < r < 12

has a quantum phase transition at Γ = Γc(r ,U)

between local-moment and symmetric strong-coupling phases. In this section we

investigate the changes that arise from the Holstein coupling of the impurity charge

to a local boson mode. For bosonic couplings λ < λ0 [see Eq. (4–6)], we find that

the low-energy physics of the PAH model is the same as for the pseudogap Anderson

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model with U replaced by Ueff defined in Eq. (4–5). A quantum critical point located at

Γ = Γc1(r ,U,λ < λ0) ≃ Γc(r ,Ueff) has universal properties indistinguishable from those

at the critical point of the pseudogap Anderson model. For stronger bosonic couplings

λ > λ0, there is instead a quantum phase transition at Γ = Γc2(r ,U,λ > λ0) between the

symmetric strong-coupling phase and a local-charge phase in which the impurity has a

residual two-fold charge degree of freedom. The critical exponents describing the local

charge response at the the Γc2 critical point are identical to those characterizing the local

spin response at the Γc1 critical point.

All numerical results presented in this section were obtained for a symmetric PAH

model with U = −2ϵd = 0.5, for a bosonic energy ω0 = 0.1, for NRG discretization

parameter Λ = 3, and restricting the number of bosons to no more than Nb = 40. Except

where noted otherwise, Ns = 500 states were retained after each NRG iteration.

4.3.1 NRG Spectrum and Fixed Points

The first evidence for the existence of multiple phases of the symmetric PAH

model comes from the eigenspectrum of HN . This spectrum can be used to identify

stable and unstable renormalization-group fixed points, as well as temperature scales

characterizing crossovers between those fixed points.

(1) Weak bosonic couplings. Figure 4-2(a) shows—for r = 0.4, λ = 0.05 < λ0 ≃ 0.158,

and seven different values of Γ—the variation with even iteration number N of the energy

of the first excited multiplet having quantum numbers S = 1, Q = 0. For small values

of Γ, this energy EN at first rises with increasing N, but eventually falls toward the value

ELM = 0 expected at the local-moment (LM) fixed point corresponding to effective model

couplings Γ = λ = 0 and U = ∞. At this fixed point, the impurity nd = 1 doublet

asymptotically decouples from the tight-binding chain of length N + 1, leaving a localized

spin-12

degree of freedom. The impurity thermodynamic contributions Simp = ln 2,

Tχs,imp ≡ ⟨S2z ⟩ = 14, and Tχc,imp ≡ ⟨(nd − 1)2⟩ = 0 as well as the Fermi-energy phase

123

shift δ0(0) = 0, coincide with the corresponding properties at the LM fixed point of the

pseudogap Anderson model (see Sec. 4.2.2.1).

For large Γ, EN instead rises monotonically to reach a limiting value ESSC ≃ 1.119

characteristic of the symmetric strong-coupling (SSC) fixed point, corresponding to

effective couplings Γ = ∞, U = λ = 0. Here, the impurity level forms a spin singlet

with an electron on the end (n = 0) site of the tight-binding chain. The singlet formation

freezes out the end site, leaving free-fermionic excitations on a chain of reduced length

N, leading to a Fermi-energy phase shift δ0(0) = −(1 − r)(π/2) sgn ϵ. This phase shift,

along with the impurity contributions Simp = 2r ln 2, Tχs,imp = r/8, and Tχc,imp = r/2 are

identical to the SSC properties in the pseudogap Anderson model.

The LM and SSC fixed points describe the large-N (low-energy DΛ−N/2) physics for

all initial choices of the hybridization except Γ = Γc1 ≃ 0.3166805, in which special case

EN rapidly approaches Ec ≃ 0.6258 and remains at that energy up to arbitrarily large N.

This behavior can be associated with an unstable pseudogap critical point separating

the LM and SSC phases. The critical point corresponds to the PAH model with λ = 0

and Γ/U equaling an r -dependent critical value.

The low-energy NRG spectrum at each of the renormalization-group fixed points

identified in the preceding paragraphs—LM, SSC, and critical—varies with the band

exponent r and the NRG discretization parameter Λ. However, for given r and Λ, the

fixed-point spectra are found to be identical to those of the particle-hole-symmetric

pseudogap Anderson model (as described in Sec. 4.2.2). Each of these fixed points can

be interpreted as corresponding to an effective boson coupling λ = 0 and exhibits the

SU(2) isospin symmetry that is broken in the full PAH model.

(2) Strong bosonic couplings. Figure 4-3(a) plots the energy at even iterations of

the first NRG excited state having quantum numbers S = Q = 0, for r = 0.4,

λ = 0.2 > λ0 ≃ 0.158, and seven different Γ values. For Γ > Γc2 ≃ 0.6878956, EN

eventually flows to the value ESSC ≃ 1.119 identified in the weak-bosonic-coupling

124

regime, and examination of the full NRG spectrum confirms that the low-temperature

behavior is governed by the same SSC fixed point.

For Γ < Γc2, EN flows to zero, the value found at the LM fixed point. In fact, all the

fixed-point many-body states obtained for λ > λ0 turn out to have the same energies

as states at the LM fixed point. However, the quantum numbers of states in the λ > λ0

spectrum and the LM spectra are not identical, but rather are related by the interchanges

S ↔ I and Sz ↔ Iz [See Eq. (2–46) for the definition of the isospin (axial charge)].

We therefore associate the λ > λ0 spectrum with a local-charge (LC) fixed point,

corresponding to Γ = λ = 0 and U = −∞, at which the impurity has a residual isospin-12

degree of freedom. This fixed point shares with its LM counterpart the properties

Simp = ln 2 and δ0(0) = 0, but is distinguished by having Tχs,imp = 0 and a Curie-like

charge susceptibility Tχc,imp ≡ ⟨(nd − 1)2⟩ = 1. (Due to the different definitions of the

spin and charge susceptibilities, it is most appropriate to compare 14χc,imp with χs,imp.)

For Γ = Γc2, EN rapidly approaches and remains at the same critical value Ec

as found for λ < λ0 and Γ = Γc1. Once again, however, the many-body spectrum is

related to that at the corresponding weak-bosonic-coupling fixed point by interchange of

spin and isospin quantum numbers, leading to the interpretation of this fixed point as a

charge analog of the critical point of the conventional pseudogap Anderson model.

The properties described above are summarized in the schematic renormalization-group

flow diagram shown in Fig. 4-4. Arrows indicate the direction in which the effective

values of the coupling Ueff and Γ evolve with increasing NRG iteration number N, i.e.,

under progressive reduction of the effective band cutoff DΛ−N/2. The high-temperature

limit of the model is governed by the free-orbital (FO) fixed point, corresponding to bare

model parameters U = Γ = λ = 0, at which Simp = ln 4, Tχs,imp =18, Tχc,imp =

12,

and δ0(0) = 0. Dashed lines mark the separatrices between the basins of attraction

of the LM, LC, and SSC fixed points described above. Flow along each separatrix is

from FO toward one or other of two quantum critical points—either the conventional spin

125

pseudogap critical point Cs reached for Ueff > 0, or its charge analog Cc reached for

Ueff < 0.

4.3.2 Phase Boundaries and Crossover Scales

Figure 4-5 shows the phase boundaries Γc1 and Γc2 vs λ for U = −2εd = 0.5,

ω0 = 0.1, and four different values of the band exponent r . For each r , Γc1 and Γc2

are both found to vanish at λ0 = 0.15812(1), close to the value√ω0U/2 ≃ 0.158114

predicted by Eq. (4–6).

With decreasing |Γ − Γc1|, EN in Fig. 4-2(a) remains close to Ec over an increasing

number of iterations before heading either to ELM or to ESSC. To quantify this effect, it

is useful to define threshold energy values E± where ELM < E− < Ec < E+ < ESSC.

The passage of EN below E− [above E+] at some N∗1 (determined by interpolation of

the NRG data at even integer values of N) can be taken to mark the crossover around

temperature T ∗1 ≃ DΛ−N∗

1 /2 from a high-temperature quantum-critical regime dominated

by the unstable critical point to a low-temperature regime controlled by the stable LM

[SSC] fixed point. This crossover scale is expected to vanish for Γ → Γc1, as shown

schematically in Fig. 4-2(b). Figure 4-2(c) plots values of T ∗1 determined by the criterion

E− = 0.3, E+ = 0.8. These data are consistent with the relation

T ∗1 ∝ |Γ− Γc1|ν1 as Γ→ Γc1, (4–7)

where ν1 is the correlation-length exponent at the quantum critical point. The precise

values of E± are arbitrary. Other choices of these threshold energies yield different

numerical values of T ∗1 but the same exponent ν1. These exponents are shown for three

representative band exponents r in Table 4-1

The passage of EN outside a range E− < EN < E+ can also be used to define a

crossover scale near the Γc2 critical point. This scale T ∗2 is expected to vanish at the

critical point according to

T ∗2 ∝ |Γ− Γc2|ν2 as Γ→ Γc2, (4–8)

126

Table 4-1. Correlation-length critical exponents ν1 and ν2 vs band exponent r for theparticle-hole-symmetric PAH model. Parentheses surround the estimatednonsystematic error in the last digit.

r ν1 ν2

0.2 6.22(1) 6.22(1)0.3 5.14(1) 5.14(1)0.4 5.84(1) 5.84(1)

a behavior that is sketched qualitatively in Fig. 4-3(b) and is confirmed quantitatively

in Fig. 4-3(c). As shown in Table 4-1, for all the values of r that we have studied, the

numerical values of ν1 and ν2 coincide to within our estimated errors.

4.3.3 Thermodynamic Properties

In the weak bosonic coupling regime, Figure 4-6 plots the dependence of Tχs,imp,

14Tχc,imp, and Simp on temperature for r = 0.4, λ = 0.05, and various Γ straddling the

critical value Γc1. The low-temperature behavior of Tχs,imp clearly shows that a QCP

separates the SSC and LM phases. Exactly at the critical point (blue line), Tχs,imp

exhibits renormalization-free behavior at low temperatures and maintains a critical value

≃ 0.1348. When Γ deviates slightly from the critical value, Tχs,imp traces the critical

behavior until it crosses below a certain temperature to either 1/4 for the LM phase

(solid symbols) or r/8 for the SSC phase (empty symbols). There is a weaker reflection

of the QCP in 14Tχc,imp, which has a critical value ≃ 0.0158 and, as the temperature

decreases, falls toward zero in the LM phase but rises toward r/8 in the SSC phase.

Simp has a critical value ≃ 0.6945 and reaches either ln 2 for the LM phase or 2r ln 2 for

the SSC phase at low temperatures. (The critical value of Simp is close to but apparently

distinct from ln 2. This will be investigated further in future work.)

In the strong bosonic coupling regime, Figure 4-7 plots the dependence of Tχs,imp,

14Tχc,imp, and Simp on temperature for r = 0.4, λ = 0.2, and various Γ straddling the

critical value Γc2. Here, it is the low-temperature behavior of Tχc,imp that most clearly

shows that a QCP separates the SSC and LC phases. Exactly at the critical point

Γ = Γc2 (blue line), Tχc,imp exhibits renormalization-free behavior at low temperatures.

127

When Γ deviates slightly from the critical value, 14Tχc,imp traces the critical behavior

≃ 0.1348 until it crosses over below a certain temperature to either 1/4 for the LC phase

(solid symbols) or r/8 for the SSC phase (empty symbols). In this case, Tχs,imp traces

its critical value ≃ 0.0158 until falls to zero for the LC phase and rises toward r/8 for the

SSC phase.The behavior of Simp upon decreasing temperatures is very similar to that in

Fig. 4-6

4.3.4 Local Response and Universality Class

In order to investigate in greater detail the properties of the spin and charge variants

of the pseudogap critical point (Cs and Cc in Fig. 4-4), it is necessary to identify an

appropriate order parameter for each quantum phase transition.

(1) Weak bosonic coupling. In the pseudogap Kondo and Anderson models, the critical

properties manifest themselves [58] through the response to a local magnetic field h that

couples only to the impurity spin. Such a field enters the Anderson model through an

additional Hamiltonian term ∆H = 12h(nd↑ − nd↓). The order parameter for the pseudogap

quantum phase transition is the limiting value as h → 0 of the local moment

Mloc = ⟨12(nd↑ − nd↓)⟩, (4–9)

and the order-parameter susceptibility is the static local spin susceptibility

χs,loc = − limh→0

Mloc

h. (4–10)

Based on the similarities noted above between the pseudogap Anderson critical

point and the Cs critical point of the PAH model (i.e., the properties of the phases on

either side of each transition, the NRG spectrum at the transition, and the value of the

order-parameter exponent), we expect that the two quantum phase transitions also to

share the same order parameter. Accordingly, the behaviors of Mloc and χs,loc in the

vicinity of the critical hybridization width Γ = Γc1, should be described by exponents β1,

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γ1, δ1, and x1 defined in the following fashion:

Mloc(Γ < Γc1; h → 0,T = 0) ∝ (Γc1 − Γ)β1, (4–11a)

χs,loc(Γ > Γc1;T = 0) ∝ (Γ− Γc1)−γ1, (4–11b)

Mloc(h; Γ = Γc1,T = 0) ∝ |h|1/δ1, (4–11c)

χs,loc(T ; Γ = Γc1) ∝ T−x1. (4–11d)

The preceding expectations are proved correct by NRG calculations, as demonstrated

in Fig. 4-8 for r = 0.4 and λ = 0.05, the case treated in Fig. 4-2. The critical exponents

extracted as best-fit slopes of log-log plots are listed in Table 4-2 for three values of the

band exponent r . The values of individual exponents vary with r , but are independent

of other Hamiltonian parameters (U, ω0, and λ) and well converged with respect to the

NRG parameters (Λ, Ns , and Nb). To within their estimated accuracy, the exponents for a

given r obey the hyperscaling relations

δ1 =1 + x11− x1

, 2β1 = ν(1− x1), γ1 = ν1x1, (4–12)

which are consistent with the scaling ansatz

F = Tf

(|Γ− Γc1|T 1/ν1

,|h|

T (1+x1)/2

)(4–13)

for the nonanalytic part of the free energy at an interacting critical point [17].

(2) Strong bosonic coupling. We have seen above that the NRG spectrum and

low-temperature thermodynamics at the Cc fixed point are related to those at the Cs

Table 4-2. Exponents describing the local spin response at the critical point Cs of theparticle-hole-symmetric PAH model, evaluated for three values of the bandexponent r . A number in parentheses indicates the estimated random error inthe last digit of each exponent.

r β1 1/δ1 x1 γ1

0.2 0.15(1) 0.02630(2) 0.9488(2) 5.85(6)0.3 0.34(1) 0.07364(1) 0.8629(3) 4.41(3)0.4 0.90(1) 0.1845(1) 0.6885(2) 3.95(5)

129

fixed point by interchange of spin and charge degrees of freedom. One therefore

expects to be able to probe the critical properties via the response to a local electric

potential ϕ that enters the model through an additional Hamiltonian term ∆H = ϕ(nd −1).

The order parameter should be the ϕ→ 0 limiting value of the local charge

Qloc = ⟨nd − 1⟩, (4–14)

and the order-parameter susceptibility should be the static local charge susceptibility

χc,loc = − limϕ→0

Qloc

ϕ. (4–15)

In the vicinity of the critical point Γ = Γc2, one expects the following critical behaviors:

Qloc(Γ < Γc2;ϕ→ 0,T = 0) ∝ (Γc2 − Γ)β2, (4–16a)

χc,loc(Γ > Γc2;T = 0) ∝ (Γ− Γc2)−γ2, (4–16b)

Qloc(ϕ; Γ = Γc2,T = 0) ∝ |ϕ|1/δ2, (4–16c)

χc,loc(T ; Γ = Γc2) ∝ T−x2. (4–16d)

These expectations are borne out by the NRG results. Figure 4-9 demonstrates the

predicted critical behaviors for the case r = 0.4, λ = 0.2 treated in Fig. 4-3. The values

of the critical exponents β2, γ2, 1/δ2, and x2 are equal to those of β1, γ1, 1/δ1, and x1

listed in Table 4-2 with small errors, as demonstrated as follows.

(3) Comparison between weak and strong bosonic coupling. Figure 4-10(a) superimposes

the variation with Γ of the order parameter in the vicinity of the r = 0.2 and r =

0.4 critical points. The equality of the slopes of the log-log plots at the spin- and

charge-sector quantum phase transitions shows that β1 = β2. Similarly, Fig. 4-10(b)

shows that the temperature variation of the order-parameter susceptibilities is consistent

with x1 = x2. Indeed, for each value of r that we have examined, we find that all

exponents at the charge-sector critical point are indistinguishable (within our estimated

errors) from the corresponding exponents at the spin-sector critical point of the PAH

130

model and at the critical point of the pseudogap Kondo model (as given in Table I

of Ref. [58]). This leads us to conclude that all three critical points lie in the same

universality class.

4.4 Results: General PAH Model with Band Exponents 0 < r < 1

In the last section, we focus on the symmetric PAH model with 0 < r < 12. In this

section, we discuss the effects of particle-hole-asymmetry and finite local magnetic

fields on the PAH model with band exponents 0 < r < 1.

(1) Weak bosonic couplings. In this regime, Ueff > 0. Based on the discussion in

Sec. 4.2.2.2, one expects the PAH model to exhibit the properties as the pseudogap

Anderson model with Ueff replacing the bare Coulomb interaction U. Figure 4-11 plots

phase boundaries of the PAH model on the δd -Γ plane for r = 0.4 (left) and 0.6 (right),

and three weak bosonic couplings λ < λ0. The model remains in the LM phase for

|δd | < 12Ueff and Γ < Γc(r ,U, δd ,λ) ≡ Γc(r ,U,−δd ,λ); otherwise it lies in one or other

of the asymmetric strong-coupling phases described in Sec.4.2.2.1: ASC− for δd > 0

or ASC+ for δd < 0. The critical hybridization width Γc decreases monotonically as λ

increases, suggesting a contraction of the LM phase upon increasing bosonic coupling.

We can associate the transitions from the LM phase to the ASC± phases with

quantum critical points C±. In the vicinity of C±, the local spin responses exhibit

power-law behaviors as they do near the symmetric quantum critical point Cs . Table 4-3

lists critical exponents at C±. Comparison with Table 4-2 shows that Cs and C± have the

same low-temperature physics for r = 0.2 and 0.3 but not for r = 0.4, 0.6, and 0.8. This

is consistent with the pseudogap Anderson model, where the Cs and C± critical points

are identical for 0 < r < r ∗ ≃ 3/8 but not for r ∗ . r < 1.

(2) Strong bosonic couplings. In this regime, by contrast, Ueff < 0. Accordingly, as

discussed in Section 4.2.2.2, the lowest-energy impurity states are spinless but have

a charge. Here, we find that a finite local magnetic field h acts in the same manner as

particle-hole asymmetry δd does in the pseudogap Anderson model. Figure 4-12 plots

131

Table 4-3. Exponents describing the local spin response at the critical points C± of theparticle-hole-asymmetric PAH model, evaluated for five values of the bandexponent r . A number in parentheses indicates the estimated random error inthe last digit of each exponent.

r β1 1/δ1 x1 γ1

0.2 0.15(1) 0.02630(2) 0.9488(2) 5.85(6)0.3 0.34(1) 0.07364(1) 0.8629(3) 4.41(3)0.4 0.59(1) 0.1569(1) 0.7275(3) 3.12(2)0.6 0.188(1) 0.1173(2) 0.7896(4) 1.41(1)0.8 0.079(1) 0.0644(5) 0.879(1) 1.10(1)

phase boundaries of the symmetric PAH model on the h-Γ plane for r = 0.4 (left) and

0.6 (right), and three strong bosonic couplings λ > λ0. The model remains in the LC

phase for |h| < |Ueff| and Γ < Γc(r ,U, h,λ) ≡ Γc(r ,U,−h,λ), otherwise it lies in one or

other of two new asymmetric strong-coupling phases: ASC↓ for h > 0 or ASC↑ for h < 0.

Here, ↑ or ↓ indicates that the ground state has a spin z component Sz = ±12. The critical

hybridization width Γc increase monotonically as λ increases, indicating an expansion of

the LC phase upon increasing bosonic coupling.

In the vicinity of the critical points C↑,↓ marking the transitions from the LC phase

to the ASC↑,↓ phases, the local charge responses exhibit power-law behaviors as they

do in the vicinity of Cc . The values of the critical exponents are equal to those listed in

Table 4-3 within small errors.

4.5 Results: Double Quantum Dots with U2 = 0

This section addresses the PAH model with a band exponent r = 2, a case of

particular interest because it has a possible realization in double quantum dots. Below

we present results not only for the impurity contributions to thermodynamic properties

but also for the linear conductance of such a double-dot system in the vicinity of its spin-

and charge-sector quantum phase transitions.

4.5.1 Effective Pseudogap Model for Double Quantum Dots

The motivation for focusing on the case r = 2 comes from studies [56, 57] of

two quantum dots coupled in parallel to left (L) and right (R) leads, and gated in such

132

a manner that the low-energy physics is dominated by just one single-particle state

on each dot. It is assumed that one of the dots (dot 1) is small and hence strongly

interacting, while the other (dot 2) is larger, has a negligible charging energy, and can

be approximated as a noninteracting resonant level. This setup can be described by the

two-impurity Anderson Hamiltonian

HDD =∑i ,σ

ϵi niσ + U1n1↑n1↓ +∑ℓ,k,σ

ϵℓkc†ℓkσcℓkσ +

∑i ,ℓ,k,σ

Viℓ(d †iσcℓkσ + H.c.

). (4–17)

Here, diσ annihilates an electron of spin z component σ and energy ϵi in the dot i (i = 1,

2), niσ = d†iσdiσ is the number operator for such electrons, and cℓkσ annihilates an

electron of spin z component σ and energy ϵℓk in lead ℓ (ℓ = L, R). For simplicity, the

leads are assumed to have the same dispersion ϵLk = ϵRk, corresponding to a “top-hat”

density of states ρ(ϵ) = ρ0Θ(D − |ϵ|) with ρ0 = (2D)−1, and to hybridize symmetrically

with the dots so that ViL = ViR . Under these conditions, the dots couple only to the

symmetric combination of lead electrons annihilated by ckσ = (cLkσ + cRkσ)/√2 with

effective hybridizations Vi =√2ViL.

A key feature of Eq. (4–17) is the vanishing of the dot-2 Coulomb interaction U2

associated with a Hamiltonian term U2n2↑n2↓. This allows one to integrate out dot 2 to

yield an effective Anderson model for a single impurity characterized by a level energy

ϵ1, an on-site interaction U1, and a scattering rate [56]

Γ1(ϵ) =(ϵ− ϵ2)

2

(ϵ− ϵ2)2 + Γ22Γ1Θ(D − |ϵ|), (4–18)

where Γi = πρ0V2i for i = 1, 2. The presence of dot 2 in the original model manifests

itself here as a Lorentzian hole in Γ1(ϵ) of width Γ2 centered on ϵ = ϵ2. For ϵ2 = 0 (a

condition that might be achieved in practice by tuning a plunger gate voltage on dot

2), Γ1(ϵ) ∝ ϵ2 in the vicinity of the Fermi energy, providing a realization of the r = 2

pseudogap Anderson model [56].

133

In the remainder of this section, we consider the double-dot device introduced in

Ref. [56], augmented by a Holstein coupling between dot 1 and local bosons. Such a

system, modeled by a Hamiltonian HDD + ω0a†a + λ(n1 − 1)(a + a†), can be mapped

(following Ref. [56]) onto the effective single-impurity model

H =∑σ

ϵ1n1 + U1n↑n↓ +∑k,σ

ϵkc†kσckσ + ω0a

†a

+∑k,σ

V1(d †1σckσ + H.c.

)+ λ(n1 − 1)

(a + a†

)(4–19)

with the scattering rate Γ1(ϵ) = (Nk)−1π

∑k V

21 δ(ϵ− ϵk) as defined in Eq. (4–18).

All numerical results presented in this section were obtained for a strongly

interacting dot 1 having U1 = 0.5, for a bosonic energy ω0 = 0.1, for NRG discretization

parameter Λ = 2.5, and restricting the number of bosons to no more than Nb = 40.

Except where noted otherwise, Ns = 500 states were retained after each NRG iteration.

4.5.2 Impurity Thermodynamic Properties

In the weak bosonic coupling regime, we find a phase transition between the

ASC± and LM phases by tuning the dot-1 energy ϵ1. Figure 4-13 plots the temperature

dependence of the impurity contribution to the thermodynamic quantities Tχs,imp,

14Tχc,imp, and Simp for a weak bosonic coupling λ = 0.1 and various ϵ1 straddling the

critical value ϵ1c . The flows of Tχs,imp with decreasing temperatures clearly show a

quantum critical point C± separates the ASC± and LM phases. Exactly at ϵ1 = ϵ1c (blue

line), Tχs,imp exhibits renormalization-free behavior at low temperatures and maintains

a critical value 1/6 (the value of Tχs,imp for the valence-fluctuation fixed point). When ϵ1

deviates slightly from its critical value, Tχs,imp traces the critical behavior until it crosses

over below a certain temperature to either 1/4 for the LM phase (solid symbols) or zero

for the ASC± phase (empty symbols). By contrast, Tχc,imp has a critical value 1/18

and, as the temperature decreases, falls toward zero in both the LM and ASC± phases,

signaling suppression of charge fluctuations at the impurity site. The impurity entropy

134

Simp has a critical value ln 3 (the value of Simp for the valence-fluctuation fixed point) and

reaches either ln 2 for the LM phase or zero for the ASC± phase at low temperatures.

In the strong bosonic coupling regime, one can tune between the ASC↑,↓ and LC

phases using a local magnetic field h entering the Hamiltonian Eq. (4–19) through an

additional term ∆H = 12h(n1↑ − n1↓). Figure 4-14 plots the dependence of Tχs,imp,

14Tχc,imp, and Simp on temperature at particle-hole symmetry (ϵ1 = −1

2U1) for a

strong bosonic coupling λ = 0.2 and various h straddling the critical value hc . Here, in

contrast to Fig. 4-13, Tχs,imp falls to zero in both the LC and ASC↑,↓ phases, suggesting

suppression of spin fluctuations at the impurity site. However, the flows of Tχc,imp

with decreasing temperature clearly show that a quantum critical point Cσ (σ =↑, ↓)

separates the ASCσ and LC phases. Exactly at h = hc (blue line), Tχc,imp exhibits

renormalization-free behavior at low temperatures. When h deviates slightly from its

critical value, 14Tχc,imp traces the critical behavior until it crosses over below a certain

temperature to either 1/4 for the LC phase (solid symbols) or zero for the ASC↑,↓ phase

(empty symbols). The behavior of Simp with decreasing temperature is very similar to that

in Fig. 4-13

4.5.3 Phase Diagrams and Critical Couplings

Figure 4-15(a) shows the phase diagram on the ϵ1-λ2 plane for h = 0. For λ < λ0,

there is a LM phase bounded by the critical energies ϵ±1c , while, for λ > λ0, the LC phase

exists only on the line of particle-hole symmetry ϵ1 = −U1/2. The rest of the phase

space is filled by the ASC± phases. Figure 4-15(b) shows the phase diagram on the

h-λ2 plane at particle-hole symmetry ϵ1 = −U1/2. For λ < λ0, the LM phase exists

only on the line h = 0, while, for λ > λ0, the LC phase is bounded by the critical local

magnetic field ±hc . The ASC↑,↓ phases fill the rest of the phase space.

We find that both ϵ±1c and hc depend linearly on λ2, as shown in Fig. 4-15(a) and (b).

The linear relation between ϵ±1c and λ2 can be understood as follows. The pseudogap

Anderson model can be mapped via the Schrieffer-Wolff transformation [10] to the

135

pseudogap Kondo model [54] with a dimensionless exchange coupling

ρ0J = Γ1

(1

U1/2− δ1+

1

U1/2 + δ1

). (4–20)

The critical coupling Jc satisfies ρ0Jc = c [54], where the constant c depends on

the bath exponent r . Combining this condition with Eq. (4–20), and assuming that

ρ0J(δ1 = 0) = 4Γ1/U1 ≪ c , one concludes that the critical dot-1 energies of the

pure-fermionic pseudogap Anderson model satisfy

δ±1c ≃ ±(U12

− Γ1c

− Γ21c2U1

), (4–21)

or

ϵ+1c ≃ −Γ1c

− Γ21c2U1

and ϵ−1c ≃ −U1 +Γ1c+Γ21c2U1

. (4–22)

As we have discussed before in Sec. 4.2, the main effect of the boson mode in the PAH

model is to reduce the Coulomb repulsion. Therefore, U1 in Eq. (4–20) must be replaced

by its effective value U1,eff = U1 − 2λ2/ω0. This implies that as long as U1 ≫ 2λ2/ω0, the

dependences of the critical energies ϵ±1c on the bosonic coupling λ are

ϵ+1c(λ) ≃ −Γ1c

− Γ21c2U1

− λ2

ω0

(1− 2Γ21c2U21

), (4–23)

ϵ−1c(λ) ≃ −U1 +Γ1c+Γ21c2U1

+λ2

ω0

(1− 2Γ21c2U21

). (4–24)

Similar reasoning can be used to understand the linear relation between hc and

λ2. The locations of the critical local magnetic field hc are determined by considering

the mapping of the Anderson model with negative U1,eff to a charge-Kondo model with

dimensionless coupling

ρ0W = Γ1

(1

|U1,eff/2| − h/2+

1

|U1,eff/2|+ h/2

). (4–25)

By analogy with the spin case, the critical couplingWc is given by ρ0Wc = b, where the

constant b depends on the bath exponent r . Assuming that ρ0W (h = 0) = 4Γ1/|U1,eff| ≪

136

b, one can find the critical field

hc ≃2

ω0(λ2 − λ20)−

2Γ1b, (4–26)

where λ0 = (ω0U1/2)1/2 marks the level crossing between the weak and strong bosonic

coupling regimes.

4.5.4 Linear Conductance

The linear conductance at temperature T for the boson-coupled double-quantum-dot

system can be calculated from the Landauer formula [98]

g(T ) =2e2

h

∫ ∞

−∞dω(−∂f

∂ω)[−ImT (ω)], (4–27)

where f (ω) = [exp(ω/T ) + 1]−1 is the Fermi-Dirac distribution and the tunneling matrix is

defined to be T (ω) = πρ0∑i ,j ViGij(ω)Vj . The term −ImT (ω) entering Eq. (4–27) can be

expressed as [57]

−ImT (ω) =[1− 2πΓ2ρ2(ω)]πΓ1(ω)A11(ω) + πΓ2ρ2(ω)

+ 2π(ω − ϵ2)Γ1(ω)ρ2(ω)G′11(ω), (4–28)

where Γ1(ω) is as defined in Eq. (4–18), ρ2(ϵ) = (Γ2/π)[(ϵ − ϵ2)2 + Γ22]

−1 is a Lorentzian

of width Γ2 centered on energy ϵ2, A11(ω) = −π−1ImG11(ω) is the dot-1 spectral

function, and G ′11 = ReG11 (ω). Here, the dot-1 local Green’s function G11(ω) takes into

account both Coulomb electron-electron interaction and bosonic coupling. The NRG

allows us to calculate the impurity spectral function A11(ω) and we obtain G ′11(ω) via

the Kramers-Kronig transformation. Given A11(ω) and G ′11(ω), we then calculate the

conductance g using Eqs. (4–27) and (4–28). The technical details are as follows.

We calculate the impurity spectral function Aσ11(ω,T ) using the techniques

discussed in Section 3.4.6 with the choice of broadening width b = 0.655 ln Λ. From

Aσ11(ω,T ), we calculate the real part of the retarded impurity Green’s function via the

137

Kramers-Kronig transformation

ReGσ11(ω,T ) = −P

∫ ∞

−∞

Aσ11(ω

′,T )dω′

ω′ − ω. (4–29)

Thereafter, it is straightforward to use Eq. (4–28) to calculate the imaginary part of the

tunneling matrix −ImT (ω), which is substituted into the Landauer formula Eq. (4–27) to

obtain the linear conductance g.

Although the Green’s functions carry a spin index σ, we can use symmetry

properties to reduce the computational effort. (i) For the case of the weak bosonic

coupling λ = 0.1, the quantum phase transition is accessed at zero magnetic field,

so G ↑11(ω) = G

↓11(ω). (ii) For the case of the strong bosonic coupling λ = 0.2, a local

magnetic field is applied to tune the system to its critical point, but by time-reversal

A↑11(ω) = A

↓11(−ω) and hence [by Eq. (4–29)] ReG ↑

11(ω) = −ReG ↓11(−ω). As a result,

the tunneling matrix −ImT (ω) given by Eq. (4–28) is invariant under ω → −ω and ↑→↓.

This property, when combined with the invariance of the weight function −∂f /∂ω under

ω → −ω, leads via Eq. (4–27) to exact equality of the conductance g for the spin up and

down channels. (This is confirmed by our numerical calculations.)

Figure 4-16 shows the behaviors of the linear conductance g around the critical

point ϵ1 = ϵ+1c for a weak bosonic coupling λ = 0.1 and different temperatures T

expressed as multiples of TK0 = 7.0 × 10−4, which is the Kondo temperature for the

one-impurity Anderson model with U = −2ϵd = 0.5 and Γ = 0.05. In Fig. 4-16(a)

showing g vs ∆ϵ1 (∆ϵ1 = ϵ1 − ϵ+1c ), two features stand out: (i) At temperature T =

0.01TK0, the linear conductance g is structureless and reaches its unitary value 2e2/h,

signaling perfect electron transmission through the system; (ii) At higher temperatures,

however, the linear conductance g exhibits a maximum at ϵ1 = ϵ+1c and minima on

either side in the LM and ASC− phases. This peak-and-valley structure becomes more

prominent with increasing temperature. Furthermore, we observe a ∆ϵ1/T scaling of the

conductance g in the vicinity of the critical point, as shown in Fig. 4-16(b).

138

Figure 4-17 presents the linear conductance g around the critical point h = hc

for a strong bosonic coupling λ = 0.2. The linear conductance g exhibits very similar

behaviors as it does in Fig. 4-16. In particular, it exhibits ∆h/T scaling in the vicinity of

the critical field.

4.6 Summary

We have conducted a study of the pseudogap Anderson-Holstein model of a

magnetic impurity level that hybridizes with a fermionic host whose density of states

vanishes as |ϵ|r at the Fermi energy (ϵ = 0) and is also coupled, via the impurity charge,

to a local-boson mode. We found two regimes, depending on the strength of the bosonic

coupling, that exhibit distinctive quantum fluctuations. The weak bosonic coupling

regime is characterized by spin fluctuations while charge fluctuations are predominant in

the strong bosonic coupling regime.

For 0 < r < 1, we have found continuous quantum phase transitions with

anomalous critical exponents. At fixed weak bosonic couplings, as the impurity-band

hybridization increases from zero, continuous quantum phase transitions occur between

the local-moment and two strong-coupling phases. However, at fixed strong bosonic

couplings, increase in the impurity-band hybridization results in continuous quantum

phase transitions between the local-charge and another two strong-coupling phases.

Particle-hole asymmetry in the model with weak bosonic couplings acts in a manner

analogous to a local magnetic field applied to the model with strong bosonic couplings.

For r = 2, the pseudogap Anderson-Holstein model can effectively describe a particular

boson-coupled two-quantum-dot setup. In this case, quantum phase transitions are

manifested by peak-and-valley features in the linear-response conductance through the

device.

139

Γ

ASC ASC

LM LM

SSC

r 0 < <

δ δd-2U 0 U

2U2

- 0 U2

Γ

> r 112 2

LMLM

ASC ASC++ - -

d

Figure 4-1. Schematic Γ-δd phase diagrams of the pseudogap Anderson model [Eqs.(4–1)–(4–4) with λ = 0] for band exponents (a) 0 < r < 1

2, (b) r ≥ 1

2.

Generically, the system falls into either a local-moment phase (LM) or one oftwo asymmetric strong-coupling phases (ASC±). However, there is also asymmetric strong-coupling phase (the line labeled SSC) that is reached onlyfor 0 < r < 1

2under conditions of strict particle-hole symmetry (δd = 0) and

for sufficiently large hybridization widths Γ.

140

0 20 40 60 80 100

N

0.0

0.5

1.0

1.5

EN

Γ−Γc1=10−2

0

10−3

10−4

S = 1, Q = 0

−10−2

−10−3

−10−4

Γ

T

LM

Quantumcritical

SSC

∧

>Γc1

r = 0.4, λ = 0.05

10−6

10−5

10−4

|Γ − Γc1|

10−35

10−30

10−25

10−20

T* 1

Γ > Γc1 Γ < Γc1

(a)

(c)

(b)

T *1

Figure 4-2. (Color) Symmetric PAH model at weak bosonic coupling: (a) NRG energyEN vs even iteration number N of the first excited multiplet having quantumnumbers S = 1, Q = 0, calculated for r = 0.4, U = −2ϵd = 0.5, ω0 = 0.1,λ = 0.05 < λ0 ≃ 0.158, and seven values of Γ− Γc1 labeled on the plot. (b)Schematic Γ–T phase diagram for λ < λ0, showing the T = 0 transitionbetween the LM (Γ < Γc1) and SSC (Γ > Γc1) phases. Dashed lines mark thescale T ∗

1 of the crossover from the high-temperature quantum-critical regimeto one or other of the stable phases. (c) Crossover scale T ∗

1 vs |Γ− Γc1| inboth the LM phase and the SSC phase, showing the power-law behaviordescribed in Eq. (4–7). Here, T ∗

1 = DΛ−N∗

1 /2, where N∗1 is the interpolated

value of N at which EN in (a) crosses one or other of the horizontal dashedlines.

141

0 20 40 60 80 100

N

0.0

0.5

1.0

1.5

EN

Γ−Γc2=10−2

0

10−3

10−4

S = 0, Q = 0

−10−2

−10−3

−10−4

Γ

T

LC

Quantumcritical

SSC

∧

>Γc2

r = 0.4, λ = 0.2

10−6

10−5

10−4

|Γ − Γc2|

10−35

10−30

10−25

10−20

T* 2

Γ > Γc2Γ < Γc2

(a)

(c)

(b)

T *2

Figure 4-3. (Color) Symmetric PAH model at strong bosonic coupling: (a) NRG energyEN vs even iteration number N of the first excited state having quantumnumbers S = Q = 0, calculated for r = 0.4, U = −2ϵd = 0.5, ω0 = 0.1,λ = 0.2 > λ0 ≃ 0.158, and Γ− Γc2 values labeled on the plot. (b) SchematicΓ–T phase diagram for λ > λ0, showing the T = 0 transition between the LCand SSC phases and the scale T ∗

2 of the crossover from the quantum-criticalregime to a stable phase. (c) Crossover scale T ∗

2 vs |Γ− Γc2| in the LC andSSC phases, showing the power-law behavior described in Eq. (4–8). Here,T ∗2 = DΛ

−N∗2 /2, where N∗

2 is the interpolated value of N at which EN in (a)crosses one or other of the horizontal dashed lines.

142

0

cC

FO

Γ

effU

o o

SSC

o o ooLC LM

Cs

Figure 4-4. Schematic renormalization-group flows on the Γ-Ueff plane for the symmetricPAH model with a band exponent 0 < r < 1

2. Trajectories with arrows

represent the flows of the couplings under decrease of the conductionbandwidth. Dashed lines connecting unstable fixed points (open circles)separate the basins of attraction of stable fixed points (filled circles). See textfor a discussion of each fixed point.

143

0 0.05 0.1 0.15 0.2 0.25 0.3

λ

0

0.2

0.4

0.6

0.8

1.0

Γ

0.1 0.1 0.1 0.1

0.1 0.2 0.3 0.4

Γc1

Γc2

r

0.10.20.30.4

Figure 4-5. Phase boundaries of the symmetric PAH model: Variation with bosoniccoupling λ of the critical hybridization widths Γc1 (empty symbols, separatingthe LM and SSC phases) and Γc2 (filled symbols, separating the LC andSSC phases). Data are shown for U = −2εd = 0.5, ω0 = 0.1, and four bandexponents r listed in the legend.

144

0.0

0.1

0.2

Tχ s,

imp

10−2

10−3

10−4

0.00

0.02

0.04

1 − 4 Tχ c,

imp

ε1 = εc

1

10−25

10−20

10−15

10−10

10−5

T

0.5

0.6

0.7

S imp

10−2

10−3

10−4

(a)

(b)

(c)

|Γ − Γc1|1/20

1/20

1/4

0.8ln2

ln2

ln2

0

10−2

10−3

10−4

10−28

10−14

T

0.692

0.696

0.700S im

p

(d)

r = 0.4, λ = 0.05

Figure 4-6. (Color) Symmetric PAH model at weak bosonic coupling: Temperaturedependence of the impurity contribution to (a) the static spin susceptibilityTχs,imp, (b) the static charge susceptibility Tχc,imp, (c) the entropy Simp, and(d) zoom in of (c) around critical value for r = 0.4, U = −2ϵd = 0.5, ω0 = 0.1,λ = 0.05 < λ0 ≃ 0.158, and seven values of Γ− Γc1 labeled in the legend.Ns = 3000 states were retained after each NRG iteration.

145

0.00

0.02

0.04

Tχ s,

imp

10−2

10−3

10−4

0.0

0.1

0.2

1 − 4 Tχ c,

imp

ε1 = εc

1

10−30

10−25

10−20

10−15

10−10

10−5

T

0.5

0.6

0.7

S imp

10−2

10−3

10−4

(a)

(b)

(c)

|Γ − Γc2|1/20

1/20

1/4

0.8ln2

ln2

ln2

0

10−2

10−3

10−4

10−30

10−15

T

0.692

0.696

0.700S im

p

(d)

r = 0.4, λ = 0.2

Figure 4-7. (Color) Symmetric PAH model at strong bosonic coupling: Temperaturedependence of the impurity contribution to (a) the static spin susceptibilityTχs,imp, (b) the static charge susceptibility Tχc,imp, (c) the entropy Simp, and(d) zoom in of (c) around critical value for r = 0.4, U = −2ϵd = 0.5, ω0 = 0.1,λ = 0.2 > λ0 ≃ 0.158, and seven values of Γ− Γc2 labeled in the legend.Ns = 3000 states were retained after each NRG iteration.

146

10−3

10−2

10−1

Γ − Γc1

103

106

109

χ s,lo

c(T =

0)

10−3

10−2

10−1

100

Γc1 − Γ

10−3

10−2

10−1

Mlo

c(h →

0, T

= 0

)

0 0.2 0.4Γ

00.10.20.30.40.5

Mlo

c

10−9

10−6

10−3

h

10−3

10−2

10−1

100

Mlo

c(Γ =

Γc1

, T =

0)

10−12

10−10

10−8

10−6

10−4

T

102

104

106

108

χ s,lo

c(Γ =

Γc1

)

(a) (b)

(c) (d)

r = 0.4, λ = 0.05

Figure 4-8. (Color) Symmetric PAH model near the spin-sector quantum phasetransition: Local spin response for r = 0.4, U = −2ϵd = 0.5, ω0 = 0.1, andλ = 0.05, at or near the critical hybridization width Γc1 ≃ 0.3166805. Circlesare NRG data and dashed lines represent power-law fitting. (a) Static localspin susceptibility χs,loc(h → 0,T = 0) vs Γ− Γc1 in the SSC phase. (b) Localmagnetization Mloc(h → 0,T = 0) vs Γc1 − Γ in the LM phase. Inset:continuous vanishing of Mloc(h → 0,T = 0) as Γ approaches Γc1 from below.(c) Local magnetization Mloc(Γ = Γc1,T = 0) vs local magnetic field h. (d)Static local spin susceptibility χs,loc(T ; h → 0, Γ = Γc1) vs temperature T .

147

10−3

10−2

10−1

Γ − Γc2

103

106

109

χ c,lo

c(T =

0)

10−3

10−2

10−1

100

Γc2 − Γ

10−3

10−2

10−1

100

Qlo

c(φ →

0, T

= 0

)

0 0.4 0.8Γ

00.20.40.60.81.0

Qlo

c

10−9

10−6

10−3

φ

10−3

10−2

10−1

100

Qlo

c(Γ =

Γc2

, T =

0)

10−12

10−10

10−8

10−6

10−4

T

102

104

106

108

χ c,lo

c(Γ =

Γc2

)

(a) (b)

(c) (d)

r = 0.4, λ = 0.2

Figure 4-9. (Color) Symmetric PAH model near the charge-sector quantum phasetransition: Local charge response for r = 0.4, U = −2ϵd = 0.5, ω0 = 0.1, andλ = 0.2, at or near the critical hybridization width Γc2 ≃ 0.6878956. Circlesare NRG data and dashed lines represent power-law fitting. (a) Static localcharge susceptibility χc,loc(ϕ→ 0,T = 0) vs Γ− Γc2 in the SSC phase. (b)Local charge Qloc(ϕ→ 0,T = 0) vs Γc2 − Γ in the LC phase. Inset:Continuous vanishing of Qloc(ϕ→ 0,T = 0) as Γ approaches Γc2 from below.(c) Local charge Qloc(Γ = Γc2,T = 0) vs local electric potential ϕ. (d) Staticlocal charge susceptibility χc,loc(ϕ→ 0, Γ = Γc2) vs temperature T .

148

10−12

10−9

10−6

10−3

T

102

104

106

108

χs,loc

(Γ = Γc1)χ

c,loc(Γ = Γc2)

r = 0.2

r = 0.4

10−4

10−3

10−2

10−1

Γc − Γ

10−3

10−2

10−1

100

Mloc

(h → 0, T = 0)Q

loc(φ → 0, T = 0)

r = 0.2

r = 0.4

(a) (b)

Figure 4-10. (Color) Symmetric PAH model: (a) Dependence of the local magnetizationMloc(h → 0,T = 0) on Γ− Γc1 and of the local charge Qloc(ϕ→ 0,T = 0)on Γc2 − Γ, for U = −2ϵd = 0.5, ω0 = 0.1, λ = 0.05 (magnetic response) or0.2 (charge response), and band exponents r = 0.2 and r = 0.4. (b) Staticlocal spin susceptibility χs,loc(T ; Γ = Γc1) and static local chargesusceptibility χc,loc(T ; Γ = Γc2) vs temperature T . All parameters other thanΓ take the same values as in (a).

149

0 0.2 0.4

Γ

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

δ d

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Γ

λ = 0λ = 0.1λ = 0.1414

r = 0.4 r = 0.6

Figure 4-11. (Color) Phase boundaries of the PAH model on the δd -Γ plane for weakbosonic couplings λ < λ0 = 0.15812(1). Data are shown for r = 0.4 (left),r = 0.6 (right), U = 0.5, h = 0, ω0 = 0.1, and three weak bosonic couplingslisted in the legend.

150

0 0.4 0.8

Γ

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

h

0 0.4 0.8 1.2 1.6 2

Γ

λ = 0.1732λ = 0.1871λ = 0.2

r = 0.4 r = 0.6

Figure 4-12. (Color) Phase boundaries of the symmetric PAH model on the h-Γ planefor strong bosonic couplings λ > λ0 = 0.15812(1). Data are shown forr = 0.4 (left), r = 0.6 (right), U = −2ϵd = 0.5, ω0 = 0.1, and three strongbosonic couplings listed in the legend.

151

0

0.1

0.2

Tχ s,

imp

10−2

10−3

10−4

10−5

00.020.040.06

1 − 4 Tχ c,

imp

ε1 = εc

1

10−6

10−4

10−2

T

0

0.4

0.8

S imp

10−2

10−3

10−4

10−5

(a)

(b)

(c)

|ε1 − ε1c |1/6

ln3

ln2

1/18

0

10−2

10−3

10−4

10−5

Figure 4-13. (Color) U2 = 0 double-quantum-dot device at weak bosonic coupling:Temperature dependence of the impurity contribution to (a) the static spinsusceptibility Tχs,imp, (b) the static charge susceptibility Tχc,imp, and (c) theentropy Simp, for U1 = 0.5, Γ1 = 0.05, ϵ2 = 0, Γ2 = 0.02, h = 0, ω0 = 0.1,λ = 0.1, and different values of ϵ1 straddling the location ϵ+1c ≃ −0.124985 orϵ−1c ≃ −0.375015 of the spin-sector quantum phase transition. Ns = 3000states were retained after each NRG iteration. Properties at the transition(lines without symbols) are those expected at a level crossing between theLM (filled symbols) and ASC± (empty symbols) phases.

152

00.020.040.06

Tχ s,

imp

10−2

10−3

10−4

10−5

0

0.1

0.2

1 − 4 Tχ c,

imp

himp

= hcimp

10−6

10−4

10−2

T

0

0.4

0.8

S imp

10−2

10−3

10−4

10−5

(a)

(b)

(c)

|h − hc|

1/6

ln3

ln2

1/18

0

10−2

10−3

10−4

10−5

Figure 4-14. (Color) U2 = 0 double-quantum-dot device at strong bosonic coupling:Temperature dependence of the impurity contribution to (a) the static spinsusceptibility Tχs,imp, (b) the static charge susceptibility Tχc,imp, and (c) theentropy Simp, for U1 = −2ϵ1 = 0.5, Γ1 = 0.05, ϵ2 = 0, Γ2 = 0.02, ω0 = 0.1,λ = 0.2, and different values of the local magnetic field h straddling thelocation hc ≃ ±0.284959 of the charge-sector quantum phase transition.Ns = 3000 states were retained after each NRG iteration. Properties at thetransition (lines without symbols) are those expected at a level crossingbetween the LC (filled symbols) and ASC↑,↓ (empty symbols) phases.Comparison with Fig. 4-13 shows close similarities between the behavior ofthe entropy near the spin- and charge-sector quantum phase transitions,but an interchange of the spin and charge susceptibilities.

153

−0.5

−0.4

−0.3

−0.2

−0.1

0.0ε 1

0 0.02 0.04 0.06

λ2

−1.0

−0.5

0.0

0.5

1.0

h

LM

ASC−LC

ASC+

LMLC

ASC↑

ASC↓

(a)

(b)

Figure 4-15. (Color) U2 = 0 double-quantum-dot device: (a) Phase diagram on the ϵ1-λ2

plane for U1 = 0.5, Γ1 = 0.05, ϵ2 = 0, Γ2 = 0.02, ω0 = 0.1, and h = 0. (b)Phase diagram on the h-λ2 plane for U1 = −2ϵ1 = 0.5 Γ1 = 0.05, ϵ2 = 0,Γ2 = 0.02, and ω0 = 0.1. Note the linearity of the phase boundaries whenplotted against the square of the bosonic coupling.

154

−0.02 0 0.02

∆ε1

0.94

0.96

0.98

1.00

g(2

e2 /h)

0.010.51.01.52.0

−40 −20 0 20 40

∆ε1/T

0

0.005

0.010

0.015

0.020

(g−g

min)T

K0

/TT/T

K0

LM phase ASC− phase

(a) (b)

Figure 4-16. (Color) U2 = 0 double-quantum-dot device near the spin-sector quantumphase transition: (a) Linear conductance g vs ∆ϵ1 = ϵ1 − ϵ+1c for U1 = 0.5,Γ1 = 0.05, ϵ2 = 0, Γ2 = 0.02, ω0 = 0.1, λ = 0.1, and different temperaturesT specified in the legend as multiples of TK0 = 7× 10−4. The retention ofNs = 1000 states after each NRG iteration accounts for the small shift inϵ+c1 ≃ −0.1249871 relative to the case Ns = 3000 shown in Fig. 4-13. (b)The same data scaled as (g − gmin)TK0/T vs ∆ϵ1/T , where gmin is theminimum on the ASC− side for each curve.

155

−0.02 0 0.02

∆h

0.96

0.98

1.00

g(2

e2 /h)

0.010.51.01.52.0

−40 −20 0 20 40

∆h/T

0

0.001

0.002

0.003

(g−g

min)T

K0

/TT/T

K0

LC phase ASC↓ phase

(a) (b)

Figure 4-17. (Color) U2 = 0 double-quantum-dot device near the charge-sectorquantum phase transition: (a) Linear conductance g vs ∆h = h − hc forU1 = −2ϵ1 = 0.5, Γ1 = 0.05, ϵ2 = 0, Γ2 = 0.02, ω0 = 0.1, λ = 0.2, anddifferent temperatures T specified in the legend as multiples ofTK0 = 7× 10−4. The retention of Ns = 1000 states after each NRG iterationaccounts for the small shift in hc ≃ 0.2849588 relative to the caseNs = 3000 shown in Fig. 4-14. (b) The same data scaled as(g − gmin)TK0/T vs ∆h/T , where gmin is the minimum on the ASC↓ side foreach curve.

156

CHAPTER 5CONCLUSIONS AND FUTURE DIRECTIONS

5.1 Conclusions

In this work, using the numerical renormalization-group method, we have theoretically

studied dissipative quantum impurity models in which the magnetic impurity not only

hybridizes with a fermionic host but also is coupled, via the impurity charge, to bosonic

degrees of freedom. These models are relevant to quantum criticality of heavy-fermion

materials and electron transport in nanoscale devices. We find that the behavior of a

magnetic impurity is strongly modified when a dissipative environment represented by

bosonic degree of freedom is incorporated into the conventional Anderson model.

In general, the impurity-boson coupling reduces the Coulomb interaction between

two electrons in the impurity level from its repulsive bare value U to an effective value

Ueff. In the atomic limit of zero hybridization, and for weak impurity-boson coupling, Ueff

is positive and the ground state of the impurity models lies in the sector nd = 1 where

the impurity has a spin z component ±1/2. However, for sufficiently large impurity-boson

coupling, Ueff is driven negative, placing the ground state in the sector nd = 0 or nd = 2

where the impurity is spinless but has a charge of −1 or +1. In the more interesting

case of nonzero hybridization, the boson-induced renormalization of U can serve to

transfer the complex spin correlations of the conventional Kondo effect into the charge

sector, leading to a many-body Kondo quenching of a localized charge degree of

freedom.

The charge-coupled Bose-Fermi Anderson model describes a magnetic impurity

that hybridizes with a metallic host and is coupled to a dispersive bosonic bath with

spectral function proportional to ωs . At particle-hole symmetry, further increase in

the impurity-boson coupling may lead to a quantum phase transition from the Kondo

phase to a localized phase where the ground-state impurity occupancy nd acquires

an expectation value ⟨nd⟩0 = 1. For a sub-Ohmic bath characterized by 0 < s < 1,

157

the response of the impurity occupancy to a locally applied electric potential features a

continuous phase transition. In addition, the hyperscaling of critical exponents and ω/T

scaling suggest an interacting critical point. For the Ohmic case s = 1, the transition

is instead of Kosterlitz-Thouless type. Away from particle-hole symmetry, the quantum

phase transition is replaced by a smooth crossover, but signatures of the symmetric

quantum critical point remain in the physical properties at elevated temperatures and/or

frequencies.

In the pseudogap Anderson-Holstein model of a magnetic impurity that hybridizes

with a pseudogapped host and is coupled to a local boson mode, the pseudogap inhibits

Kondo screening and gives rise to phases in which the impurity retains unquenched

degrees of freedom. For weak impurity-boson couplings, the response of the system

to a locally applied magnetic field reveals transitions between two strong-coupling

(Kondo) phases and a local-moment phase. For strong impurity-boson couplings, by

contrast, the response of the system to a locally applied electrical potential indicates

transitions between another two strong-coupling phases and a local-charge phase.

Particle-hole asymmetry in the model with weak impurity-boson couplings acts

in a manner analogous to a local magnetic field applied to the model with strong

impurity-boson couplings. Critical exponents characterizing the four critical points are

found to be numerically identical, suggesting that they belong to the same universality

class as that of the pseudogap Anderson model. One specific case of the pseudogap

Anderson-Holstein model effectively describes a charge-coupled double-quantum-dot

device, in which the phase transitions are manifested in the finite-temperature linear

electrical conductance.

5.2 Future Directions

Dissipative quantum impurity problems describe interesting and exciting physics.

They are of intrinsic theoretical interest as tractable toy models for bulk quantum phase

transitions and for the novel universality classes of quantum phase transitions that

158

they exemplify. These problems also describe nanoscale systems of relevance to

nanoelectronics and quantum computing, two areas of the forefront of the technological

revolution. Here, we summarize several issues that might be addressed in the future.

(1) As we discussed in Sec. 1.4, the impurity model we studied in Chapter 3

is associated with the lattice model Eq. (1–6) within the framework of the extended

dynamical mean-field theory. To fully understand this lattice model, self-consistency

ought to be imposed properly. This might be pursued in future work.

(2) One straightforward generalization of our study is to replace the metallic host

in the charge-coupled Bose-Fermi Anderson model Eq. (3–1) with a pseudogapped

host. Preliminary work on this model indicates that there is still an interacting quantum

critical point separating the Kondo and localized phases. For band exponents r and

bath exponents s satisfying s + 2r < 1, the quantum critical point remains in the

universality class of the spin-boson model. However for s + 2r > 1, the model shows

low-temperature behavior very different from that of the spin-boson model [99]. This

novel regime requires further investigation.

(3) In Chapter 3, we only considered a magnetic impurity coupled to one bosonic

bath. A possible extension is to investigate models in which the impurity is coupled to

multiple bosonic baths. The simplest such model is the XY-anisotropic spin-boson model

HXYSB = ∆Sz +∑

q

ωq(ϕx†q ϕxq + ϕ

y†q ϕ

yq) + g⊥

∑q

[Sx(ϕxq + ϕ

x†−q) + Sy(ϕ

yq + ϕ

y†−q)], (5–1)

where the impurity is coupled, via spin x and y components, to two independent bosonic

baths described by ϕxq and ϕyq. This model exhibits the novel phenomenon of quantum

frustration of decoherence [100, 101]. Another model of interest is the XY-anisotropic

Bose-Fermi Kondo model

HXYBFKM =∑k,σ

ϵkc†kσckσ +

∑q

ωq(ϕx†q ϕxq + ϕ

y†q ϕ

yq)

159

+ J S · sc + g⊥∑

q

[Sx(ϕxq + ϕ

x†−q) + Sy(ϕ

yq + ϕ

y†−q)], (5–2)

which may describe the quantum criticality of YbRh2Si2−xGex [102] and certain

quantum-dot system [53]. The NRG treatment of these models is very challenging

because of the large number of states that must be retained in the calculation.

(4) A limitation of our research in Chapter 4 is that we were only able to calculate

the linear-response conductance, describing the behavior in the limit of an infinitesimal

bias between the electrical leads. However, many experiments show finite-bias Kondo

features [60, 103]. Thus, it is of great interest to develop nonperturbative methods

to treat strong correlations in non-equilibrium situations. One such method is the

time-dependent NRG method [104, 105], which is restricted to calculating the transient

response of a system initially at equilibrium to a sudden change in its Hamiltonian from

Hi to Hf (both time-independent). An important goal for future work is the development

of a nonperturbative numerical approach for treating more general non-equilibrium

problems.

160

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165

BIOGRAPHICAL SKETCH

Mengxing Cheng was born in July 1981, in Kaiyuan, Yunnan Province, China. He

attended local public schools there through high school. In September 1999, he enrolled

at the Physics Department of Fudan University in Shanghai. In June 2003, he graduated

and obtained a Bachelor of Science. In September 2003, he joined Professor Sun Xin’s

group at the same department and in June 2005, he was awarded a Master of Science.

In August 2005, he flew over the Pacific Ocean and the North American continent to

Gainesville to start graduate school at the Department of Physics at the University of

Florida. In the summer of 2006, he joined Doctor Ingersent’s group to study magnetic

impurity models using the numerical renormalization group. He and his wife Huan

Huang first met in the fall of 1995 and married in July 2005. Their son Jason Cheng was

born in May 2009.

166

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QUANTUM PHASE TRANSITIONS OF MAGNETIC IMPURITIES IN ...ufdcimages.uflib.ufl.edu/UF/E0/04/23/86/00001/cheng_m.pdf · I dedicate this dissertation to my family, particularly, to myself