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)].
1. Introduction to quantum critical point
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
1. Introduction to quantum critical point
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
• A: every spin (spontaneously) points up Order parameter:
• B: every microstate equally probable: m=0
σσ I- H zj
ij
zi∑=
><
ordered state
control parameter δ
tem
pera
ture
T
T=0 A
B
C ∑i
xi) (I- σδ
1N /M limlim m site0h site
==∞→→ + N
• C: every spin points along the transverse field: m=0
Quantum Phase Transition
• A: every spin (spontaneously) points up Order parameter:
• B: every microstate equally probable: m=0
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
1N /M limlim m site0h site
==∞→→ + 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
1. Introduction to quantum critical point
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
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
1. Introduction to quantum critical point
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
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
1. Introduction to quantum critical point
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
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
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)].
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:
Single impurity Kondo model
• Kondo entanglement: singlet ground state
• Kondo temperature:
Single impurity Kondo model
• 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
• Kondo temperature:
2. Introduction to heavy Fermi liquid
a. Single-impurity Kondo problem
b. Kondo lattice
c. Heavy Fermi liquid
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
2. Introduction to heavy Fermi liquid
a. Single-impurity Kondo problem
b. Kondo lattice
c. Heavy Fermi liquid
• 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
Heavy Fermi Liquid (Kondo Lattice)
),(--1
ωεωω
kkG
kc Σ
=),(
pole in Σ
• Kondo resonance …
• … heavy electron bands
Heavy Fermi Liquid (Kondo Lattice)
),(--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
Q. Si, arXiv:1012.5440, a chapter in the book “Understanding Quantum Phase Transitions”, ed. L. D. Carr (2010).
TN
Linear resistivity
JK >>W>>I • xNsite tightly bound local singlets
(cf. If x were =1, Kondo insulator)
• (1-x)Nsite lone moments:
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)
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=∞
• 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
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)].
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
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
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
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
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~)(
0<ε<1: sub-ohmic dissipation
QS, Rabello, Ingersent, Smith, Nature ’01; PRB ’03;
L. Zhu & QS, PRB ’02
ε-expansion of Bose-Fermi Kondo Model
JK
Kondo
Critical Kondo destruction
g
ε δ −1 −∑ ωwω pp
~)(
Critical:
Crucial for LQCP solution
0<ε<1: sub-ohmic dissipation
Kondo destruction
QS, Rabello, Ingersent, Smith, Nature ’01; PRB ’03;
L. Zhu & QS, PRB ’02
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
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)
Dynamical Scaling of Local Quantum Critical Point
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
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
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
Fermi Surface Jump and Kondo-Destruction Energy Scale in YbRh2Si2
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
Fermi Surface Jump and Kondo-Destruction Energy Scale in YbRh2Si2
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)
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. Perspective 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)].
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
4. Heavy fermion quantum criticality II
a. Antiferromagnetic order and Kondo destruction
b. Global phase diagram c. Experimental evidence for the global phase
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)
4. Heavy fermion quantum criticality II
a. Antiferromagnetic order and Kondo destruction
b. Global phase diagram c. Experimental evidence for the global phase
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)
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)].
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
Global Phase Diagram
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
Role of Berry Phase term in QNLσM Approach
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
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
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
T. Park et al., Nature 440, 65 (’06); G. Knebel et al., PRB74, 020501 (’06)
Superconductivity near Local Kondo-destruction QCP in CeRhIn5
Topic of
the research talk tomorrow
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
• 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
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
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
Superconductivity at the border of magnetism
• 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
Superconductivity at the border of magnetism
Bro
un
Falte
rmei
er e
t al
Pnictides Organics
Cuprates
Mat
hur e
t al
Dynamical Scaling of Local Quantum Critical Point
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
Global Phase Diagram
SDW of PL QS, Physica B 378, 23 (2006) S. Yamamoto & QS, PRL 2007
G: frustration, reduced dimensionaltiy, …
Global Phase Diagram
S. F
riede
man
n et
al,
N
at. P
hys.
5, 4
65 (’
09)
Pure and doped YbRh2Si2
Pure YbRh2Si2
Doped YbRh2Si2
Global Phase Diagram
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
Global Phase Diagram
J. Custers, R. Yu, et al., Nature Materials 11, 189 (2012)
Effect of dimensionality – the case of Ce3Pd20Si6
T *
4. Heavy fermion quantum criticality II
a. Antiferromagnetic order and Kondo destruction
b. Global phase diagram c. Experimental evidence for the global phase
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
Heisenberg model + coherent spin path integral QNLσM
Role of Berry Phase term in QNLσM Approach
Heisenberg model + coherent spin path integral QNLσM
Role of Berry Phase term in QNLσM Approach
Heisenberg model + coherent spin path integral QNLσM
Role of Berry Phase term in QNLσM Approach
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)
Kondo effect and Berry phase in 1D
JK<<Irkky, W
QNLσM -- chiral rotation:
Kondo effect and Berry phase in 1D
P. Goswami + QS, PRL 107, 126404 (’11)
JK<< Irkky,W
QNLσM -- chiral rotation:
Kondo effect and Berry phase in 1D
P. Goswami + QS, PRL 107, 126404 (’11)
Cancels the θ term of local moments, S[n] à spin gapped Kondo singlet
JK<< Irkky, W
QNLσM -- chiral rotation:
Kondo effect and Berry phase in 1D
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
T. Park et al., Nature 440, 65 (’06); G. Knebel et al., PRB74, 020501 (’06)
Superconductivity near Local Kondo-destruction QCP in CeRhIn5
H. Q. Yuan et al, to be published (’12)
Superconductivity near Local Kondo-destruction QCP in CeRhIn5
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)