Quantum Criticality and the Kondo Lattice

Qimiao Si

Rice University

International School on Heavy Fermions and Quantum PhaseTransitions, Center for Correlated Matter, Zhejiang University,

Hangzhou, April 14, 2015

Quantum Criticality and the Kondo Lattice

1.  Introduction to quantum critical point

2.  Introduction to heavy Fermi liquid

3.  Heavy fermion quantum criticality I

4.  Heavy fermion quantum criticality II

5.  Further topics and outlook

Q. Si, arXiv:1012.5440, a chapter in the book “Understanding Quantum Phase Transitions”, ed. L. D. Carr (2010).

Q. Si, J. H. Pixley, E. Nica, S. J. Yamamoto, P. Goswami, R. Yu & S. Kirchner,

arXiv:1312.0764 [J. Phys. Soc. Jpn. 83, 061005 (2014)].


a.  Symmetry breaking and order parameter

b.  From classical criticality to quantum criticality

c.  Materials for quantum criticality

d.  Spin density wave quantum critical point

e.  Quantum criticality & non-Fermi liquid physics

Phases and Phase Transitions

Disorder (T>Torder)

Order (T<Torder)

Continuous Phase Transitions: Criticality

Disorder (T>Torder)

Order (T<Torder)

Criticality -- fluctuations of order parameter in d dimensions

•  A: every spin (spontaneously) points up Order parameter:

•  B: every microstate equally probable: m=0

σσ I- H zj

ij

zi∑=

><

ordered state

tem

pera

ture

T

T=0 A

B

1N /M limlim m site0h site

==∞→→ + N









σσ I- H zj

ij

zi∑=

><

ordered state

control parameter δ

tem

pera

ture

T

T=0 A

B

C ∑i

xi) (I- σδ


==∞→→ + N

•  C: every spin points along the transverse field: m=0

Quantum Phase Transition



σσ I- H zj

ij

zi∑=

><

QCP

quantum critical

ordered state

control parameter δ

tem

pera

ture

T

T=0 A

B

C ∑i

xi) (I- σδ


==∞→→ + N

•  C: every spin points along the transverse field: m=0

TN

Linear resistivity

Quantum Criticality

•  Competing states due to competing interactions •  Quantum critical point •  Finite T: Quantum critical regime

Torder

TN

Linear resistivity Quantum scaling: ξτ ~ ξ

z

τ ~ / kTorder

Tr e-(/kT)H/

! ∞ @QCP

Torder

Quantum Scaling

TN

Linear resistivity

Quantum Criticality

Fluctuations of order parameter, , in d+z dimensions m(x, τ )

TN

Linear resistivity

Quantum scaling:

ξτ ~ ξz

Torder

Quantum Scaling of Free Energy

ξ ~ r−ν

L. Zhu, M. Garst, A. Rosch, & QS, PRL (2003)

TN

Linear resistivity

Thermodynamic Singularities near QCP =>Grüneisen ratio divergent at QCP:

L. Zhu, M. Garst, A. Rosch, & QS, PRL (2003)

=>Entropy accumulation near a QCP:

Quantum critical points

unusual excitations; emergent phases

enhanced entropy







TN

Linear resistivity

Heavy Fermion AF Metals -- prototypical Quantum Critical Points

SC

CeRhIn5

N. M

athu

r et a

l

T. P

ark

et a

l

H. v

. Löh

neys

en e

t al

J. C

uste

rs e

t al

R. Küchler et al, PRL 91, 066405 (2003)

TN

Linear resistivity

Divergence of Grüneisen Ratio

Heavy fermions

Kas

ahar

a et

al

TN

Superconductivity at the border of magnetism

Bro

un

Falte

rmei

er e

t al

Pnictides Organics

Cuprates

Mat

hur e

t al







TN

Linear resistivity Spin-density-wave

Fermi liquid Paramagnetic Fermi liquid

J. A. Hertz; A. J. Millis; T. Moriya

Consider Hubbard model at a generic filling

Spin density wave and Stoner ferromagnetism

Selection of Q for order [order from disorder]: Q≠0: SDW Q=0: Stoner ferromagnetism (stability problems beware)

Spin density wave and Stoner ferromagnetism

Spin-dependent potential

Change of Fermi surface by spin-dependent potential:

Spin density wave

T=0 spin-density-wave transition

§ Dynamic exponent z=2 §  d+z>4,

o Gaussian fixed point o No omega/T scaling

§  At QCP o hot k-spots develop non-Fermi liquid form o Cold k-regions retain Fermi liquid form







TN

Quantum Criticality vis-à-vis Non-Fermi liquid Electronic Excitations

•  Quantum criticality generates non-Fermi liquid

electronic excitations –  How can AF quantum criticality turn the entire Fermi

surface hot? •  Can the non-Fermi liquid excitations in turn change

the universality class of QCP?

Beyond-Landau Quantum Criticality

Inherent quantum modes may be important, beyond order-parameter fluctuations -- need to identify the additional critical modes

Critical Kondo Destruction --Local Quantum Critical Point

QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001) P. Coleman et al, JPCM 13, R723 (2001)

Critical Kondo Destruction (f-elec. Mott localization) at the T=0 onset of antiferromagnetism









2. Introduction to heavy Fermi liquid

a.  Single-impurity Kondo problem

b.  Kondo lattice

c.  Heavy Fermi liquid

Single-impurity Kondo Model:

fermion bath Local

moment

S: spin-1/2 moment at site 0

Single-impurity Kondo Model: –  resistivity minimum (scattering increases as T is

lowered!) – asymptotic freedom – Kondo screening (process of developing Kondo

singlet correlations as T is lowered)

Single impurity Kondo model

•  Kondo temperature:


•  Kondo entanglement: singlet ground state



•  Kondo effect (emergence of Kondo resonance): –  Kondo-singlet ground state yields an electronic resonance –  local moment acquires electron quantum number

due to Kondo entanglement

•  Kondo entanglement: singlet ground state




b.  Kondo lattice


Kondo lattices:

Kondo lattices:

M. Klein et al, PRL 101, 266404 (’08) S. Ernst et al, Nature 474, 362 (’11)

Kondo lattices:

heavy Fermi liquid:

• Kondo singlet • Kondo resonance



b.  Kondo lattice


•  The large Fermi surface applies to the paramagnetic phase, when the ground state is a Kondo singlet.

•  This can be seen through adiabatic

continuity of a Fermi liquid. •  It can also be seen, microscopically,

through eg slave-boson MFT (Auerbach & Levin, Millis & Lee, Coleman, Read & Newns)

Heavy Fermi Liquid (Kondo Lattice)

•  Kondo resonance …

•  … heavy electron bands


),(--1

ωεωω

kkG

kc Σ

=),(

pole in Σ

•  Kondo resonance …

•  … heavy electron bands


),(--1

ωεωω

kkG

kc Σ

=),(

pole in Σ

k-independent

Cond. electron band

Heavy Fermi Liquid Heavy electron bands

E1,2(k)

Kondo resonance

)(kε

Cond. electron band

Heavy Fermi Liquid Heavy electron bands

E1,2(k)

Kondo resonance

)(kε

Large Fermi surface

• Kondo lattices:

heavy Fermi liquid:

• Kondo singlet • Kondo resonance No symmetry breaking,

but macroscopic order

• Kondo lattices:

•  Competition between Kondo & RKKY (Doniach)

•  SDW of heavy Fermi liquid (Kondo

effect intact at the transition)

Critical Kondo Destruction -- Local Quantum Critical Point

Kondo Destruction (f-electron Mott localization) at the T=0 onset of antiferromagnetism


TN

Linear resistivity

JK >>W>>I •  xNsite tightly bound local singlets

(cf. If x were =1, Kondo insulator)

•  (1-x)Nsite lone moments:




–  projection: –  (1-x)Nsite holes with U=∞

(C. Lacroix)




–  projection: –  (1-x)Nsite holes with U=∞

•  Luttinger’s theorem:

(1-x) holes/site in the Fermi surface (1+x) electrons/site ---- Large Fermi surface!

(C. Lacroix)

Physics of the Kondo Effect

Consider a simplified problem – non-dispersive conduction electron

Projection to the lowest f-states (U is large)

Physics of the Kondo Effect (cont’d)

Projection à Kondo Hamiltonian

Ground state: Low-lying states:

Kondo resonance:

Composite fermion:

Description of the Kondo resonance

Kondo singlet ground state à

Kondo resonance: Landau quasiparticle with tiny but nonzero weight

TN

J. Custers et al, Nature 424, 52 (’03)

Non-Fermi liquid behavior at a Quantum Critical Point









3. Heavy fermion quantum criticality I

a.  Kondo effect vs antiferromagnetism b.  Collapse of Kondo scale:

Extended dynamical mean field theory c. Kondo destruction via Bose-Fermi Kondo

model d.  Kondo destruction and local quantum

criticality e.  Experimental evidence for local quantum

criticality

Kondo lattices:

Tuning parameter:

δ = TK0 / I

•  What happens to the Eloc* scale as the AF QCP is

approached from the PL side? –  Eloc* should decrease: the AF correlations among the local

moments reduces the strength of the Kondo singlet –  Does the Kondo effect become critical (beyond-Landau)?

•  Critical destruction of the Kondo effect, when Eloc*

continuously goes to zero •  Need methods that capture not only the AF order and

Kondo effect, but also the dynamical competition betwee the Kondo and RKKY interactions

Kondo Effect at the AF QCP






criticality

Extended-DMFT of Kondo Lattice

Mapping to a Bose-Fermi Kondo model:

(Smith & QS; Chitra & Kotliar)

+ self-consistency conditions

–  Electron self-energy Σ (ω) G(k,ω)=1/[ω – εk - Σ(ω)] –  “spin self-energy” M (ω) χ(q,ω)=1/[ Iq + M(ω)]

Extended-DMFT of Kondo Lattice

fermion bath

fluctuating magnetic field

Local moment

Kondo Lattice

Bose-Fermi Kondo Jk

g

+ self-consistency






criticality

ε-expansion of Bose-Fermi Kondo Model

JK

Kondo

Critical Kondo destruction

g

ε δ −1 −∑ ωwω pp

~)(

0<ε<1: sub-ohmic dissipation

Kondo destruction

QS, Rabello, Ingersent, Smith, Nature ’01; PRB ’03;

L. Zhu & QS, PRB ’02

ε-expansion of Bose-Fermi Kondo Model ε δ −1 −∑ ωwω p

p~)(




ε-expansion of Bose-Fermi Kondo Model

JK

Kondo

Critical Kondo destruction

g

ε δ −1 −∑ ωwω pp

~)(

Critical:

Crucial for LQCP solution


Kondo destruction








criticality

Dynamical Scaling of Local Quantum Critical Point

Continuous phase transition

δ ≡ IRKKY / TK0

J.-X. Zhu, D. Grempel, and QS, Phys. Rev. Lett. (2003)

J.-X. Zhu, S. Kirchner, R. Bulla & QS, PRL 99, 227204 (2007); M. Glossop & K. Ingersent, PRL 99, 227203 (2007)


α = 0.72

α = 0.83

α = 0.78

J-X Zhu, D. Grempel and QS, PRL (2003)

J-X Zhu, S. Kirchner, R. Bulla, and QS, PRL (2007)

M. Glossop & K. Ingersent, PRL (2007)

Physics of the Kondo destruction

Destruction of quasiparticles at the local Kondo-destruction QCP

Small Fermi Surface

Large

Fermi Surface






criticality

Local Quantum Critical Point (Kondo destruction)

•  ω/T scaling in χ(ω,T) and G(ω,T)

• Collapse of a large Fermi surface

• Multiple energy scales

Spin Dynamical Scaling in CeCu5.9Au0.1

A.  Schröder et al., Nature (’00); B.  O. Stockert et al; M. Aronson et al.

q=Q

q=0

INS and M/H

T0.75

1/χ(q)

. . .

INS @ AF Q

/Tω

Fractional exponent α=0.75

Fermi Surface Jump and Kondo-Destruction Energy Scale in YbRh2Si2

S. Friedemann, N. Oeschler, S. Wirth, C. Krellner, C. Geibel, F. Steglich, S. Paschen, S. Kirchner, and QS, PNAS 107, 14547 (2010) S. Paschen et al, Nature (2004); P. Gegenwart et al, Science (2007)

T*

2nd order transition across Bc

Crossover: isothermal Hall coeff.

Crossover width vs. T


P. Gegenwart, T. Westerkamp, C. Krellner, Y. Tokiwa, S. Paschen, C. Geibel, F. Steglich, E. Abrahams, and QS, Science 315, 969 (2007)

S. Paschen et al, Nature (2004); S. Friedemann et al., PNAS (2010)

Crossover: Isothermal magnetostricton and magnetization


P. Gegenwart, T. Westerkamp, C. Krellner, Y. Tokiwa, S. Paschen, C. Geibel, F. Steglich, E. Abrahams, and QS, Science 315, 969 (2007)

S. Paschen et al, Nature (2004); S. Friedemann et al., PNAS (2010)

Crossover: Isothermal magnetostricton and magnetization

Jump of Fermi-surface – dHvA Measurements in CeRhIn5

H. Shishido, R. Settai, H. Harima, & Y. Onuki, JPSJ 74, 1103 (’05)

P1 Pc P1 Pc

Fermi surface jumps across Pc Mass tends to diverge at Pc

2nd order transition across Pc

Dynamical Kondo Effect

J-X Zhu, D. Grempel, QS, PRL (2003) QS & S. Paschen, Phys. Status Solidi (2013)

Quasiparticle weight à 0 as the QCP is approached from both sides

Dynamical Kondo Effect

P. Gegeneart et al., PRL (2002)





5.  Perspective and outlook




4. Heavy fermion quantum criticality II

a.  Antiferromagnetic order and Kondo destruction

b.  Global phase diagram c.  Experimental evidence for the global phase

diagram

Collapse of Kondo scale from the paramagnetic side

QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001); Phys. Rev. B68, 115103 (2003)

Opposite limit – when RKKY dominates over Kondo coupling what JK << I

δ = JK / I

Opposite limit – when RKKY dominates over Kondo coupling

• JK=0 as the reference point of expansion: • f- local moments: AF, QNLσM • conduction electrons: Fermi volume “x”

JK << I

Heisenberg model + coherent spin path integral QNLσM

Quantum non-linear Sigma Model Representation

SBerry not important deep inside ordered phase

RG for mixed Bosons and Fermions with a Fermi Surface

S. Yamamoto & QS, PRB 81, 205106 (2010)

All directions scales Only 1 direction scales (Shankar)

When RKKY dominates: inside AF order

• JK=0 as the reference point of expansion: • f- local moments: AF, QNLσM • conduction electrons: Fermi volume “x”

• JK Exactly Marginal • Kondo destruction -- AFS phase

S. Yamamoto & QS, PRL 99, 016401 (2007)

JK << I




diagram

JK

G

AFS

JK<<Irkky<<W

Néel, Kondo destruction

Distinct from

PL

Global Phase Diagram G: frustration, reduced dimensionaltiy, …

In contrast to: single boundary a la Landau

Q. Si, Physica B 378, 23 (2006); Phys. Status Solidi B247, 476 (2010) also, P. Coleman & A. Nevidomskyy, JLTP 161, 182 (2010)




diagram

Global Phase Diagram

E. D. Mun et al., PRB 87, 075120 (2013)

Co & Ir-doped YbRh2Si2 (S. Friedemann et al, Nat Phys 2009) Shastry-Sutherland Lattice Yb2Pt2Pb (Kim & Aronson, PRL 2013) Kagome lattice CePdAl (V. Fritsch et al, PRB 2014)

J. Custers, R. Yu et al., Nat. Mater. (2012)

Mini-review: QS & S. Paschen, Phys. Status Solidi B250, 425 (2013)









5. Further topics and outlook

a.  Diverse settings for global phase diagram

b.  Quantum criticality vis-à-vis superconductivity

c.  Beyond antiferromagnetic settings

d.  Broader contexts


J1

J2

filling x=0.5

G=J2/J1 Shastry-Sutherland lattice

J. Pixley, R. Yu & QS, PRL 113, 176402 (2014)

Role of Berry Phase term in QNLσM Approach

P. Goswami + QS, PRL 107, 126404 (2011) – One dimension;

PRB 89, 045124 (2014) – 2D honeycomb lattice


P. Goswami + QS, PRL 107, 126404 (2011) – One dimension;

PRB 89, 045124 (2014) – 2D honeycomb lattice

Berry phase of local moments: è Singlet phases (Spin Peierls, …)

Cond. electron spins locked to local moments: è Cancellation of Berry phase è Kondo singlet è Competition w/ spin peierls

Global Phase Diagram Tuning by S.O.C.

X.-Y. Feng, J. Dai, C-H Chung and QS, Phys. Rev. Lett. 111, 016402 (2013)

JK/λsoc

hybr

idiz

atio

n

Global Phase Diagram Tuning by S.O.C.

X.-Y. Feng, J. Dai, C-H Chung and QS, Phys. Rev. Lett. 111, 016402 (2013)

JK/λsoc

hybr

idiz

atio

n Eg: o  SmB6 under pressure o  Ce(Ni,Pd,Pt)Sn






T. Park et al., Nature 440, 65 (’06); G. Knebel et al., PRB74, 020501 (’06)

Superconductivity in CeRhIn5

CeRhIn5

High Tc : Tc / TF ≈ 0.1

SC


Superconductivity near Local Kondo-destruction QCP in CeRhIn5

Topic of

the research talk tomorrow






•  Kondo destruction (kicking one electron/site out of the Fermi surface)

•  What happens beyond the Kondo limit, w/ mixed-valency?

β-YbAlB4

-- H/T scaling ! interacting Y. Matsumoto et al, Science 331, 316 (2011)

-- Mixed valency M. Okawa et al, PRL 104, 247201 (2010)

Kondo destruction and valence fluctuations in pseudogapped asymmetric anderson model

J. Pixley, S. Kirchner, K. Ingersent and QS, PRL, to appear (2012)

Charge excitations part of the quantum-critical spectrum

spin susceptibility charge susceptibility

Kondo destruction and valence fluctuations in pseudogapped asymmetric anderson model

J. Pixley, S. Kirchner, K. Ingersent and QS, PRL, to appear (2012)

Charge excitations part of the quantum-critical spectrum

•  Field/temperature scaling






TN

Beyond-Landau Quantum Criticality

at the border of magnetism u Quantum criticality beyond order-

parameter fluctuations o  Local QCP via Kondo destruction

o  vs conventional: SDW QCP u Distinguishing phases beyond

spontaneous symmetry breaking o  Kondo entanglement and destruction:

characterize a global phase diagram

How can the order-parameter-flucutation

description break down?

•  Quantum temporal fluctuations may not simply be extra dimensions of classical order-parameter fluctuations:

The partition function for an individual configuration in space and time may not be positive semi-definite

(cf, the “minus sign problem” of quantum Monte Carlo).

)(~ configτ)(x, in configZZ ∑

TN

Kondo Lattice and Heavy Fermions

at the border of magnetism u Two complimentary perspectives o  A lattice of Kondo “impurities” (advantageous for

understanding Kondo entanglement) o  Quantum magnetism of local moments +

conduction electrons (two types of quantum fluctuations, G and JK)

u Prototype system for magnetism at the

boundary of localization and itinerancy

TN


•  Magnetic fluctuations a la Landau – glues for superconductivity

or •  Magnetism as proxy

– new excitations in normal state

Heavy fermions

Kas

ahar

a et

al

TN


Bro

un

Falte

rmei

er e

t al

Pnictides Organics

Cuprates

Mat

hur e

t al


AdS/CMT:

N. Iqbal, H. Liu, M. Mezei and QS, PRD 82, 045002 (’10) T. Faulkner, G. T. Horowitz and M. M. Roberts, arXiv:1008.1581 AdS and Kondo lattice: S. Yamamoto & QS, JLTP 161, 233 (’10) S. Sachdev, PRL 105, 151602 (’10)

Kondo lattice f- local moments: AF, QNLσM

RG for mixed Bosons and Fermions with a Fermi Surface

S. Yamamoto & QS, PRB 81, 205106 (2010)

All directions scales Only 1 direction scales (Shankar)

RG at 1-loop & beyond:

=

~

Marginal even at 1-Loop and beyond.

No pole in self energy. Fermi surface remains “small”.

AF phase w/ “small” Fermi surface!

Large N:

S. Yamamoto & QS, PRL 99, 016401 (2007)

JK

G

AFS

JK<<Irkky<<W

Néel, Kondo destruction

Distinct from

PL

Collapse of Kondo scale from the paramagnetic side

QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001); Phys. Rev. B68, 115103 (2003)

JK

G

AFS PL

AFL

I

II


SDW of PL QS, Physica B 378, 23 (2006) S. Yamamoto & QS, PRL 2007

G: frustration, reduced dimensionaltiy, …


S. F

riede

man

n et

al,

N

at. P

hys.

5, 4

65 (’

09)

Pure and doped YbRh2Si2

Pure YbRh2Si2

Doped YbRh2Si2


S. F

riede

man

n et

al,

N

at. P

hys.

5, 4

65 (’

09)

Pure and doped YbRh2Si2

Global Phase Diagram AFS: CeCu6-xAux (I) YbRh2Si2 (I) YbRh2-xCoxSi2 (II) AFL: CeRu2-xRhxSi2 (II)

PS: YbRh2-xIrxSi2 (III) YbAgGe


J. Custers, R. Yu, et al., Nature Materials 11, 189 (2012)

Effect of dimensionality – the case of Ce3Pd20Si6

T *




diagram

d.  Kondo effect from the ordered side: topological defects and Berry phase

Global Phase Diagram Motivates new theoretical questions and approaches:

Approach from the ordered side







Kondo lattice in one dimension

Tsvelik, PRL ’94 Sikkema, Affleck & White, PRL ’97 Zachar & Tsvelik, PRB ’01 Pivovarov & QS, PRB ’04 Berg, Fradkin & Kivelson, PRL ’10 Eidelstein, Moukouri, Schiller, ’11

JK<< Irkky, W

Kondo effect and Berry phase in 1D

QNLσM:

JK<< Irkky, W

QNLσM -- chiral rotation: (Tanaka & Machida)


JK<<Irkky, W

QNLσM -- chiral rotation:


P. Goswami + QS, PRL 107, 126404 (’11)

JK<< Irkky,W



P. Goswami + QS, PRL 107, 126404 (’11)

Cancels the θ term of local moments, S[n] à spin gapped Kondo singlet

JK<< Irkky, W



P. Goswami + QS, PRL 107, 126404 (’11)

Kondo vs spin-Peierls: large (PL) vs small (PS) Fermi-Surface phases in paramagnetic region

Global Phase Diagram Motivates new theoretical questions and approaches:

Approach from the ordered side

More generally, how to capture Kondo effect using bosonic representations of spin

5. Perspective and outlook

a.  Quantum criticality vis-à-vis superconductivity

b.  Beyond antiferromagnetic settings c.  Broader contexts



H. Q. Yuan et al, to be published (’12)


Superconductivity in CeCu2Si2

Exchange energy saving ≈ 20 times of SC condensation energy

" large kinetic energy loss

O.Stockert, S. Kirchner, et al., Nat. Phys. 7, 119 (2011)

O.Stockert, S. Kirchner, et al., Nat. Phys. 7, 119 (2011)

Superconductivity in CeCu2Si2

Exchange energy saving ≈ 20 times of SC condensation energy

" large kinetic energy loss due to transfer of spectral weight to higher energies

5. Perspective and outlook

a.  Quantum criticality vis-à-vis superconductivity

b.  Beyond antiferromagnetic settings c.  Broader contexts

Spin-glass QCP in heavy fermions?

S. Wilson, P. Dai et al, Phys. Rev. Lett. ’05

D. Gajewski, R. Chau, and M. B. Maple, Phys. Rev. B (’00)

Beyond AF systems

Ferromagnetic Kondo lattice systems: FS phase is also stable S. Yamamoto & QS, PNAS (’10)

Growing list of ferromagnetic heavy fermions:

--URu2-xRexSi2 N. P. Butch & M. B. Maple, PRL (’09) --YbNi4P2 A. Steppke et al (’12) --CeRu2Al2B E. Baumbach et al, PRB (’12)

Localization-delocalization in UGe2?

A.  Huxley et al, JPCM 15, B.  S1945 (2003)

R. Settai et al, JPCM 14, L29 (2002)

Documents

Quantum Criticality and the Kondo Lattice