Kondo Physics in a Dissipative Environment: SpinSpin ...kevin/research/dresden07_2.pdf · Matthew Matthew GlossopGlossop ,,,,MengxingCheng & Nasim Khoshkhou ... •BoseBoseBose- ---Fermi

Kondo Physics in a Dissipative Environment - Dresden 8/22/07 1

Kondo Physics in a Dissipative Environment:Kondo Physics in a Dissipative Environment:Kondo Physics in a Dissipative Environment:Kondo Physics in a Dissipative Environment:SpinSpinSpinSpin----Boson Physics and BeyondBoson Physics and BeyondBoson Physics and BeyondBoson Physics and Beyond

Kevin Ingersent (Univ. of Florida)

CollaboratorsMatthew Matthew Matthew Matthew GlossopGlossopGlossopGlossop,,,, Mengxing Cheng & Nasim Khoshkhou

Chung-Hou Chung, Lars Fritz, Marijana Kirćan & Matthias VojtaQimiao Si

Q. What happens to the Q. What happens to the Q. What happens to the Q. What happens to the Kondo effect in the Kondo effect in the Kondo effect in the Kondo effect in the presence of friction?presence of friction?presence of friction?presence of friction?


Outline Outline Outline Outline

• BoseBoseBoseBose----Fermi Kondo/Anderson models with a bath spectrumFermi Kondo/Anderson models with a bath spectrumFermi Kondo/Anderson models with a bath spectrumFermi Kondo/Anderson models with a bath spectrum‣ Motivation for study: quantum dots, lattice problems.‣ Exhibit destruction of the Kondo effect at a boundary QPT.‣ Bosons dominate the QPT; fermions merely renormalize

parameters of an effective spin-boson model.

• BoseBoseBoseBose----Fermi models with a Fermi models with a Fermi models with a Fermi models with a fermionic fermionic fermionic fermionic density of statesdensity of statesdensity of statesdensity of states‣ Fermionic pseudogap assists destruction of the Kondo effect.‣ Critical behavior can escape the spin-boson stranglehold.‣ Insight from a spinless resonant-level model.

• BoseBoseBoseBose----Fermi Kondo model applied to heavy fermionsFermi Kondo model applied to heavy fermionsFermi Kondo model applied to heavy fermionsFermi Kondo model applied to heavy fermions‣ Mean-field theory for the Kondo lattice.‣ Magnetic fluctuations destroy the heavy-fermion state.‣ Supports a picture of the local quantum criticality.

ω∝ s

ε∝ r


• Describes a local spin-half SSSS coupled both to a conduction band and one, two, or three dissipative baths.

• Spin-isotropic model has the Hamiltonian

where (for )

• Anisotropic versions distinguish betweenand and

• This work focuses on the Ising-symmetry caseand

I. BoseI. BoseI. BoseI. Bose----Fermi Kondo (BFK) ModelFermi Kondo (BFK) ModelFermi Kondo (BFK) ModelFermi Kondo (BFK) Model

'0', '†02

1σσσ

ασσσα σ ccs ∑= †

00 ααα aau +=

σσσ ε kkk k ccH †,band ∑= ααα ω qqq q aaH †

,bath ∑=

zg ⊥== ggg yx

bathband HgHJH +⋅++⋅= uSsS

KondoH )0(bosonspin =∆−H

zJ ⊥== JJJ yx

zyx ,,=α

0=⊥g⊥= JJz


BoseBoseBoseBose----Fermi Kondo model: Bath spectraFermi Kondo model: Bath spectraFermi Kondo model: Bath spectraFermi Kondo model: Bath spectra

• Take a flatflatflatflat conductionband density of states:

• Assume a powerpowerpowerpower----lawlawlawlawbosonic spectral density:

for

• Dimensionless couplings: and .

( ) 0ρερ = for D<ε

( )ccKB ωωωω /)( 20= s

J0ρρρρ gK0

cωω <<0


PerturbativePerturbativePerturbativePerturbative solutions of the BFK modelsolutions of the BFK modelsolutions of the BFK modelsolutions of the BFK model

• Model has been solved via expansion in[Si & Smith (1999), Sengupta (2000), Zhu & Si (2002), Zaránd & Demler (2002)].

• For > 0, a quantum critical pointquantum critical pointquantum critical pointquantum critical pointseparates KondoKondoKondoKondo & localizedlocalizedlocalizedlocalized regimes.

• Critical point couplings andare of order .

Exception: Ising symmetry ( ),for which

• At the QCP, shows power laws inω and T with -dependent exponents.

• Large-N multichannel version also studied [Zhu et al. (2004)].

*0Jρ

*0gK

0=⊥g

locχ

s−=1

( ).1~, *0

*0 OgKJρ

( )** J,g

localized

Kondo


• Coulomb blockade in a noisy quantum dotCoulomb blockade in a noisy quantum dotCoulomb blockade in a noisy quantum dotCoulomb blockade in a noisy quantum dot[K. Le Hur, PRL 92, 196804 (2004)]

‣ Consider a metallic box …… in a strong magnetic field,… grounded via a point contact,… subject to a noisynoisynoisynoisy gate voltage.

‣ Can map charge fluctuations onto a transverse BFK model with …… an Ohmic bath (s = 1);… , .

BFK model applied to quantum dotsBFK model applied to quantum dotsBFK model applied to quantum dotsBFK model applied to quantum dots

tJ ∝⊥ ( )0==∝ ωZRgz

‣ Predicts a Kosterlitz-Thouless transition at R = Rc.

• 1111----electron transistor with ferromagnetic leadselectron transistor with ferromagnetic leadselectron transistor with ferromagnetic leadselectron transistor with ferromagnetic leads [Kirchner et al. (2005)]

‣ Mapped onto an s = ½ Bose-Fermi Kondo model.‣ Predicts an interacting quantum critical point via gate voltage.


BFK model applied to the Kondo latticeBFK model applied to the Kondo latticeBFK model applied to the Kondo latticeBFK model applied to the Kondo lattice

• Extended dynamical mean-field theory includes some spatial fluctuations [Si and Smith (1996), Chitra & Kotliar (2000)].

• Maps the Kondo lattice modelto a BFK model:

‣ Fermionic band accounts forlocal dynamical correlations.

‣ Dissipative baths representa fluctuating magnetic fieldfluctuating magnetic fieldfluctuating magnetic fieldfluctuating magnetic fielddue to other local moments.

‣ Band and bath densitiesof states must be foundselfselfselfself----consistentlyconsistentlyconsistentlyconsistently.

• Previous work indicates we need nonperturbative solutions for baths with s = ½ and s = 0+ [Si et al. (2001)].


BoseBoseBoseBose----Fermi NRGFermi NRGFermi NRGFermi NRGGlossop & KI, PRL (2005); PRB (2007)

• Seek an iterative procedure that treats simultaneously fermionic andbosonic degrees of freedom of the same energy.

• As in the bosonic NRG, discretize bosonic bath, then map to a tight-binding chain [Bulla et al. (2003)].

• Slightly complication: different Λ dependences of fermionic and bosonic tight-binding coefficients.‣ Could use different discretizations, Λfermions = Λ2

bosons .‣ Instead, we add a bosonic site at every other iteration:


• Based on -expansion, expect an interacting QCP with a free energy

where or , and couples only to .

• Obtain the order-parameter exponent from for

νννν(s)(s)(s)(s) = spin= spin= spin= spin----boson valueboson valueboson valueboson valuefor all 0 < s< 1.

SubSubSubSub----OhmicOhmicOhmicOhmic BFK Model: NRG ResultsBFK Model: NRG ResultsBFK Model: NRG ResultsBFK Model: NRG ResultsGlossop & KI, PRL (2005); PRB (2007)

∆= 2/–1loc

/1imp T, ην

h

TfTF

cJJ −∆ ~ cgg − loch zSν∆∝*T :0loc=h


SubSubSubSub----ohmicohmicohmicohmic BFK model: Magnetic exponent BFK model: Magnetic exponent BFK model: Magnetic exponent BFK model: Magnetic exponent ηηηη

• Obtain η by calculating

• Find

with = spinspinspinspin----boson valueboson valueboson valueboson value

(agrees with exact -expansion result).

.lim)0(loc

0loc h

Sz

h−==

→ωχ

s1η −=

( ) η

cloc Tgg,ωTχ ∝== 0

0.5=s


SubSubSubSub----ohmic ohmic ohmic ohmic BFK model: Other critical exponentsBFK model: Other critical exponentsBFK model: Other critical exponentsBFK model: Other critical exponents

• Other exponents obey hyperscalinghyperscalinghyperscalinghyperscalinge.g., order parameter

where

• Local spin dynamics obey

consistent with ωωωω /T scalingscalingscalingscaling:

• Conclusion: the QCP is interacting.the QCP is interacting.the QCP is interacting.the QCP is interacting.

β∆∝=≡ )0(TSM zloc

.2νηβ =

( ) ωωgg,Tω,χη

cloc sgn01−∝==′′

0.2=s

0.5=s

( ) .,,1

loc

Φ

==′′−

TT

TggTT

KcK

ωωχη


SubSubSubSub----ohmicohmicohmicohmic BFA model: SingleBFA model: SingleBFA model: SingleBFA model: Single----particle dynamicsparticle dynamicsparticle dynamicsparticle dynamics

• Throughout Kondo regime, resonance has shoulders at

and

• Throughout bosonic regime,

A(ω) has maxima atand

( ) .10 0Γ== πωA

,*T±=ω( ) .100 0Γ<=< πωA

,*T±=ω

• Solve the symmetric Bose-Fermi Anderson model

70.=s


Why do the bosons dominate the QCP?Why do the bosons dominate the QCP?Why do the bosons dominate the QCP?Why do the bosons dominate the QCP?

• Sub-ohmic Ising-symmetry BFK model has identical exponents to the spin-boson model ⇒ QCPsQCPsQCPsQCPs lie in same universality class.lie in same universality class.lie in same universality class.lie in same universality class.

• Explain via bosonization of fermions [Grempel & Si (2003), LeHur (2003)]:‣ Flat band maps onto an ohmic (sband= 1) bath.‣ Impurity spin sees a combined spectral density

‣ The ohmic component serves to renormalize the tunneling rate∆ → TK in a sub-ohmic spin-boson model

• The spinThe spinThe spinThe spin----boson attractor is very strong.boson attractor is very strong.boson attractor is very strong.boson attractor is very strong. Anderson impurity model coupled to bosons via exhibits spin-boson exponents in its response to a local electricelectricelectricelectric field [Cheng, Glossop, KI].

( ).† † ∑∑ ++++∆−=q

qqqzqqq

qzx aaSaaSSH λωε

( ) ( ) .)( bandbath cc JJJ s ωωωωω +=

( ) ( )∑ +−q qqqd aan †1 λ


II.II.II.II. PseudogapPseudogapPseudogapPseudogap BoseBoseBoseBose----Fermi Kondo ModelFermi Kondo ModelFermi Kondo ModelFermi Kondo Model

• Assume a pseudogappseudogappseudogappseudogap conductionband density of states:

( is a conventional metal)

• Also assume a powerpowerpowerpower----lawlawlawlawbosonic spectrum:

for

• Dimensionless couplings: and .

bathband HgHJH +⋅++⋅= uSsS

( ) ερερ 0= for D<ε

cωω <<0

Jρ0 gK0

r

0=r



J

Jc

0

pseudogappseudogappseudogappseudogap KondoKondoKondoKondo ( )0=g

rJc ≈ for 1<<r∞→ for −→ 2

1r

critical point is interacting[KI & Si (2002); Vojta & Fritz(2004-2006)]

: impurities in high1=rcT

at p-h symmetry

BoseBoseBoseBose----Fermi KondoFermi KondoFermi KondoFermi Kondo( )10,0 <<= sr

critical point lies in universalityclass of spin-boson model[Glossop & KI (2005)]

: quantum dots: heavy-fermion AFMs

1,,0 21=s

21,0=s

Pseudogap Pseudogap Pseudogap Pseudogap BoseBoseBoseBose----Fermi Kondo Model: AntecedentsFermi Kondo Model: AntecedentsFermi Kondo Model: AntecedentsFermi Kondo Model: Antecedents

pseudogappseudogappseudogappseudogap BoseBoseBoseBoseFermi KondoFermi KondoFermi KondoFermi Kondo

[isotropic case: Vojta &Kirćan (2003)]

J

Jc

0

pseudogappseudogappseudogappseudogap KondoKondoKondoKondo ( )0=g

rJc ≈ for 1<<r∞→ for −→ 2

1r

BoseBoseBoseBose----Fermi KondoFermi KondoFermi KondoFermi Kondo( )10,0 <<= sr


IsingIsingIsingIsing----Symmetry Symmetry Symmetry Symmetry PseudogapPseudogapPseudogapPseudogap BFK Model: NRG ResultsBFK Model: NRG ResultsBFK Model: NRG ResultsBFK Model: NRG ResultsGlossop, KI, and Khoshkhou (unpublished)

• We have solved the Ising-symmetry model at particle-hole symmetry.

• The plane divides into four regions:‣ K unstable for .‣ L unstable for .

• Focus on the pseudogap BFK critical point ( ).

),( sr

21>r

1>s


• Local-magnetic exponent η everywhere satisfies ,the sub-ohmic spin-boson model result found for[c.f. Kirćan & Vojta (2004)].

• Rather than determining ν via

it is numerically preferableto compute β via

PseudogapPseudogapPseudogapPseudogap BFK QCP: Static critical exponentsBFK QCP: Static critical exponentsBFK QCP: Static critical exponentsBFK QCP: Static critical exponents

0=rs1η −=

ν∆∝*T

β∆∝=≡ )0(TSM zloc


PseudogapPseudogapPseudogapPseudogap BFK QCP: Finally escape the SB black holeBFK QCP: Finally escape the SB black holeBFK QCP: Finally escape the SB black holeBFK QCP: Finally escape the SB black hole

For s > 1–2r, …

… β depins from its SB value… hyperscaling still obeyed… have a new universalitynew universalitynew universalitynew universality………… class of interacting class of interacting class of interacting class of interacting QCPsQCPsQCPsQCPs

For s < 1–2r, …

… β takes its SB value… bosons dominate the QCPbosons dominate the QCPbosons dominate the QCPbosons dominate the QCP


Insight from Insight from Insight from Insight from perturbativeperturbativeperturbativeperturbative RG AnalysisRG AnalysisRG AnalysisRG Analysis

• Combining perturbative RG equations for the Bose-Fermi Kondo model and the pseudogap Kondo model, get

• The pseudogap Bose-Fermi QCP is at

• (i.e., pert. RG is OK) for s > 1 – 2r, but for s < 1 –2r.

• NRG spectrum shows that for s< 1 – 2r.

⊥⊥⊥ +−= JJrJdldJ z/

zzz JJJrJdldJ 2212/ ⊥⊥ −+−=

( )gJsdldg 221 1/ ⊥−−=

,1* sJ −=⊥ ,rs

sJ *

z 21)2(1

+−−= ).,(** srgg =

1* <zJ 1* >zJ

∞=*zJ

( ) ⊥⊥⊥ −+− JgJJJ z2

2122

41


Summary: The Summary: The Summary: The Summary: The PseudogapPseudogapPseudogapPseudogap BoseBoseBoseBose----Fermi Kondo ZooFermi Kondo ZooFermi Kondo ZooFermi Kondo Zoo

spin-bosonQCP

novel QCP

pseudogap Kondo QCP

localized

free spin

s

1

0 21

Kondo phase

r

no Kondo

localizedphase

no localized


Simpler Model: A Resonant Level with DissipationSimpler Model: A Resonant Level with DissipationSimpler Model: A Resonant Level with DissipationSimpler Model: A Resonant Level with DissipationChung et al. (2007)

• Describes a resonant level hybridizing with a pseudogapped band and a bosonic bath:

• Assume a pseudogappseudogappseudogappseudogap conductionband density of states:

• Also assume a powerpowerpowerpower----lawlawlawlawbosonic spectrum:

• Dimensionless couplings: ,

( ) ( ) ( ) bathband†

21† H.c. HHaangcdVnH

k qqqdkdd +++−+++= ∑ ∑ε

for

D<ε

cωω <<0

20Vρπ=ΓΓΓΓ gK0

r


( ) ερερ 0= for


Parameter space of modelParameter space of modelParameter space of modelParameter space of model

• Can perform expansions about r = 1, s= 1 and about s = 2r –1.

????QPT

• NRG shows that ν depins from the spin-boson value at s = 1–2r .


Preliminary explanation for Preliminary explanation for Preliminary explanation for Preliminary explanation for depinning depinning depinning depinning (M. (M. (M. (M. VojtaVojtaVojtaVojta))))

• Bosonic fluctuation spectrum goes like

• Fermionic two-particle spectrum goes like

• When the bosonic fluctuations are slower, get spin-boson exponents.

• When fermionic fluctuations are slower, exponents depend on both rand s.

• Q. Why does same pinning criterion extend to the BFK model?Q. Why does same pinning criterion extend to the BFK model?Q. Why does same pinning criterion extend to the BFK model?Q. Why does same pinning criterion extend to the BFK model?

.sω

.21 r−ω


III. BFK Model and HeavyIII. BFK Model and HeavyIII. BFK Model and HeavyIII. BFK Model and Heavy----Fermion Fermion Fermion Fermion Quantum CriticalityQuantum CriticalityQuantum CriticalityQuantum Criticality

• Extended dynamical mean-field theory includes some spatial fluctuations [Si and Smith (1996), Chitra & Kotliar (2000)].

• Maps the Kondo lattice modelto a BKF model:

‣ Fermionic band accounts forlocal dynamical correlations.

‣ Dissipative baths representa fluctuating magnetic fieldfluctuating magnetic fieldfluctuating magnetic fieldfluctuating magnetic fielddue to other local moments.

‣ Band and bath densitiesof states must be foundselfselfselfself----consistentlyconsistentlyconsistentlyconsistently.


• EDMFT equations have been solved using various impurity solvers.• -expansion finds two types of QCP [Si et al. (2001); (2003)]:

‣ conventional spinconventional spinconventional spinconventional spin----densitydensitydensitydensity----wavewavewavewave type for 3D spin fluctuations;‣ locally critical QCP locally critical QCP locally critical QCP locally critical QCP for 2D spin fluctuations—reproduces some

features of CeCu6-x Aux and YbRh2Si2, butbutbutbut corresponds to .• Quantum Monte Carlo yields conflicting results:

‣ Anderson lattice has no locally critical QPT; transition is 1st order[Sun & Kotliar (2003)].

‣ Kondo lattice has 1st order transition at T > 0, but a locally critical

QCP at T = 0 [Grempel & Si (2003), Zhu et al. (2004)].

• To resolve this discrepancy, need need need need nonperturbativenonperturbativenonperturbativenonperturbative T = 0 solutionssolutionssolutionssolutions....

What is the nature of the QPT in EDMFT?What is the nature of the QPT in EDMFT?What is the nature of the QPT in EDMFT?What is the nature of the QPT in EDMFT?

−=1


SelfSelfSelfSelf----consistent EDMFT at consistent EDMFT at consistent EDMFT at consistent EDMFT at T = 0: NRG solutions: NRG solutions: NRG solutions: NRG solutions

• At the gross level, 3D and 2D spin fluctuations yield similar results:

• To within numerical resolution, QPT is continuous in both cases.QPT is continuous in both cases.QPT is continuous in both cases.QPT is continuous in both cases.

3D 2D

2D

3D


EDMFT static selfEDMFT static selfEDMFT static selfEDMFT static self----consistencyconsistencyconsistencyconsistency

• Differences between 3D and 2D show up only very near the QPT:

• Bosonic NRG has been able to get closer to the QPT [Zhu et al. (2007)].


Anomalous exponent in the dynamicsAnomalous exponent in the dynamicsAnomalous exponent in the dynamicsAnomalous exponent in the dynamics

• Logarithmic divergence inimplies an anomalous exponentanomalous exponentanomalous exponentanomalous exponent

in the lattice susceptibility.

• Compares well with the experimentalvalue in CeCu6-x Aux:

• More about local quantum criticality in the next talk.

)0('loc →ωχ

)4(78.0=α

.75.0≈α


SummarySummarySummarySummary

• Bose-Fermi models represent a fundamental class of quantum impurity problems.

• Provide a tractable setting for exploring the interplay between strong electron correlations and dissipation.

• For sub-ohmic bosonic baths, the destruction of the Kondo effect takes place at an interacting quantum critical point.

• When bosonic fluctuations dominate, the criticality seems drawn to the spin-boson universality class.

• Slowing down the fermionic spin fluctuations may drive the quantum critical point into novel universality classes.

• Bose-Fermi models can also arise as mean-field theories for correlated lattice problems.

• One such theory provides support for the notion of local quantum criticality in heavy fermions.


Response Response Response Response χχχχloc to a local magnetic field to a local magnetic field to a local magnetic field to a local magnetic field hloc

• EDMFT gives the peak lattice susceptibility as

• Can get a divergent peak susceptibility if, and only if,

satisfies

• NRG typically violates this by 10–20%!• Can fix the problem in an unbiased manner by the correction

where c is chosen to satisfy (*).

[ ]∫∞ −=0loc ,)0(),()( zz

ti StSedti ωωχ

.lim)0(loc

0loc

loc h

Sz

h−=

→χ

( ) ( ) .00, loc1

loc

1

zS

h+= −− χχ Q

)()1()( NRGlocloc ωχωχ c+=

(*)

Documents

Kondo Physics in a Dissipative Environment: SpinSpin ...kevin/research/dresden07_2.pdf · Matthew Matthew GlossopGlossop ,,,,MengxingCheng & Nasim Khoshkhou ... •BoseBoseBose- ---Fermi