Qimiao Si Rice University

Qimiao Si

Rice University

KITPC, July 6, 2007

Quantum Criticalityand

Fermi Surfaces

http://www.welch1.org/index.asp

S. Paschen P. Gegenwart R. KüchlerT. Lühmann S. Wirth Y. Tokiwa,N. Oeschler T. Cichorek K. NeumaierO. Tegus O. Trovarelli C. GeibelJ. A. Mydosh F. Steglich P. ColemanE. Abrahams

Stefan Kirchner, Seiji Yamamoto, (Rice University)Silvio Rabello, J. L. SmithLijun Zhu (UC Riverside)Eugene Pivovarov (UC San Diego) Kevin Ingersent (Univ. of Florida)Daniel Grempel (CEA-Saclay)Jian-Xin Zhu (Los Alamos)Jianhui Dai (Zhejiang U.)

• Insulating Ising magnet– LiHoF4: transverse field Ising model

• Heavy fermion metals• ‘‘Simple’’ magnetic metals

– Cr1-xVx, Sr3Ru2O7, MnSi (1st order, but …) …

• High Tc superconductors • Mott transition

– V2O3, …: QCP? (magnetic ordering intervenes at low T!) – ultracold atoms: 2nd order?

• Field-driven BEC of magnons

• MIT/SIT/QH-QH in disordered electron systems

Materials (possibly) showing Quantum Criticality

Heavy fermions near an antiferromagnetic QCP:

CeCu6-xAux

H. v. Löhneysen et al, PRL 1994

AF Metal

TN

TN

Linearresistivity

(modern era of the heavy fermion field)

Heavy fermions near an antiferromagnetic QCP:

CeCu6-xAux

YbRh2Si2

H. v. Löhneysen et al, PRL 1994

J. Custers et al, Nature 2003

CePd2Si2N. Mathur et al, Nature 1998

AF Metal

TN

TN

TN

AF MetalSupercond.

Linearresistivity

(modern era of the heavy fermion field)

) x,m(

T=0 spin-density-wave transition

order parameter fluctuationsin space and (imaginary) time

scalingT no

) x,m(

Gaussian

zddeff ,4

T=0 spin-density-wave transition

order parameter fluctuationsin space and (imaginary) time

exponent MF

Dynamical and Static Susceptibilities in CeCu5.9Au0.1

A. Schröder et al., Nature ’00; PRL ’98; O. Stockert et al., PRL ’98; M. Aronson et al., PRL ‘95

• /T scaling

•Fractional exponent =0.75

• =0.75 `everywhere’ in q.

INS @ q=Q

q=Q

q=0

INS and M/H

T0.75

1/(q)

...

ω/T

• Quantum critical heavy fermions

– prototype for quantum criticality

– local quantum criticality – destruction of Kondo entanglement

• Fermi surfaces of Kondo lattice– Global phase diagram

• Extended-DMFT of Kondo Lattice

• Some recent experiments

• Bose-Fermi Kondo Model

– Kondo-destroying QCP

Single-impurity Kondo model

• Kondo singlet

• Kondo temperature TK0

• Kondo resonance: spin ½ charge e

• Kondo lattices:

Historical development of heavy Fermi liquid: Single-impurity: Anderson, Wilson, Nozières, Andrei, Wiegmann,

Coleman, Read & Newns, …Lattice: Varma, Doniach, Auerbach & Levin, Millis & Lee,

Rice & Ueda, …

Local Quantum Critical Point

• Destruction of Kondo screening (entanglement):

energy scale Eloc* 0 marks an electronic slowing down at the magnetic QCP

QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)

QS, J. L. Smith, and K. Ingersent, IJMPB 13, 2331 (1999)

Local Quantum Critical Point

• Spin damping: anomalous exponent and /T scaling

• Fermi surface jumps from “large” to “small”

• Destruction of Kondo screening (entanglement):

energy scale Eloc* 0 marks an electronic slowing down at the magnetic QCP


QS, J. L. Smith, and K. Ingersent, IJMPB 13, 2331 (1999)









Fermi surface inside the antiferromagnetic part of the Kondo lattice phase diagram

Kondo lattice

JK<<Irkky<<W

JK=0 as the reference point of expansion

Local Moments: Quantum non-Linear Sigma Model

Heisenberg model + coherent spin path integral QNLMHaldane (1983)Affleck (1985) :

Conduction electrons

Not important insideordered phase

Chakravarty, Halperin, NelsonPRB 39, 2344 (1989)

Effective fermi-magnoncoupling

N.B.: coupling irrelevant (see next slide)S. Yamamoto & QS, PRL (July 5, 2007);

cond-mat/0610001

Kondo lattice

Local-moment antiferromagnetism

At JK=0:

Conduction electrons

q

ω

Q

k

0

Effective Kondo coupling w/ magnons

e),( xQ nmxS i

Qq Q

Large momentum transfer scattering NOT allowed

kinematically.with AF order

Forward scatteringALLOWED.

RELEVENT ???

0q

Combined bosonic/fermionic* RGTree-level scaling (*ferminoc RG: Shankar, RMP ’94)

marginal!

note: remember for forward scattering

From NLM

From electrons near FS

RG at 1-loop & beyond:

Marginal even at 1-Loop and beyond.


~(dΛ)3/2



AF phase w/ “small” Fermi surface!

S. Yamamoto & QS,

PRL (July 5, 2007);

cond-mat/0610001


=

~


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


S. Yamamoto & QS,

PRL (July 5, 2007);

cond-mat/0610001

Large N:




Side: AFS: “hybridization” is finite at finite energies; it

only goes to zero upon reaching the fixed point.

Implications for the choice of variational wavefunctions [H. Watanabe and M. Ogata, arXiv:0704.1722]









Kondo lattice

JK

Kon

do

cou

pli

ng

J K

JK=0

Kondo lattice

JK

G

G ~ frustration, reduceddimensionality, etc.

Loc

al-m

omen

t m

agn

etis

m, I

rkk

y

Kon

do

cou

pli

ng

J K

JK=0

JK

G

AFS

JK<<Irkky<<W

Néel, without Kondo screening

JK

G

PML

paramagnet, w/ Kondo screening

JK >>W>>Irkky

Cf. A. C. Hewson, The Kondo Problem to Heavy Fermions

(Cambridge Univ. Press, 1993)

JK >>W>>Irkky

• xNsite tightly bound local singlets

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

• (1-x)Nsite lone moments:

JK >>W>>Irkky



• (1-x)Nsite lone moments: – projection:

– (1-x)Nsite holes with U=∞

JK >>W>>Irkky



• (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!

JK

G

AFS

PML

AFL

QS, Physica B 378, 23 (2006);

QS, J.-X. Zhu, D. Grempel, JPCM ‘05

I

II

Global phase diagram

SDW of PML

Type I transition

Type II transition

Hertz-Moriya-Millis fixed point for T=0 SDW transition

Second order if

Destruction of Kondo screening where magnetism sets in

Type I quantum transition (cont’d)

I

AFS AFL

PML

G









• Kondo lattices:

Extended-DMFT* of Kondo Lattice

• Mapping to a Bose-Fermi Kondo model:

(* Smith & QS; Chitra & Kotliar)

+ self-consistency conditions

– Electron self-energy Σ () G(k,ω)=1/[ω – εk - Σ(ω)]

– “spin self-energy” M () (q,ω)=1/[ Iq + M(ω)]

E-DMFT solution to the Kondo lattice

1 )/(1ln

1Im~~)(

iωwω p

p


• The self-consistent fluctuating field bath:

1~

1~loc

• Destruction of Kondo screening:

Divergent χloc(ω) locates the local problem on the critical manifold

JK

g

Critical

Kondo

Kondo lattice with Ising anisotropy: Extended-DMFT

Critical

KondoJK

g

Kondo lattice with Ising anisotropy: Quantum Monte Carlo of EDMFT

• At the QCP, I ≈ Ic

D. Grempel and QS, Phys. Rev. Lett. ’03

loc (n)

n

Critical

KondoJK

g

Kondo lattice with Ising anisotropy: Quantum Monte Carlo of EDMFT

• At the QCP, I ≈ Ic

= 0.72D. Grempel and QS, Phys. Rev. Lett. ’03

-1(Q,n)

n

loc (n)

n

Critical

KondoJK

g

Kondo lattice with Ising anisotropy

≡ IRKKY / TK0

Quantum Monte Carlo

J.-X. Zhu, D. Grempel, and QS,

PRL (2003)

≡ IRKKY / TK0

EDMFT of Anderson lattice with Ising anisotropy

EDMFT of

P. Sun and G. Kotliar, Phys.Rev.Lett. ’03

Jc1Jc2

Avoiding double-counting in EDMFT

QS, J.-X. Zhu, & D. Grempel, J. Phys.: Condens. Matter 17, R1025 (2005)

see also:

P. Sun & G. Kotliar, Phys. Rev. B71, 245104 (2005)

Quantum transition is second order only when ΔIq is kept the same on the paramagnetic and magnetic sides

Kondo lattice with Ising anisotropy

≡ IRKKY / TK0

Numerical RG (T=0)

Quantum Monte Carlo

J.-X. Zhu, S. Kirchner, QS, & R. Bulla,

cond-mat/0607567 (2006);

See also, M. Glossop & K. Ingersent, cond-mat/0607566 (2006)

J.-X. Zhu, D. Grempel, and QS,

Phys. Rev. Lett. (2003)

(T=0.01 TK0)









T. Park et al., Nature 440, 65 (‘06);

G. Knebel et al., PRB74, 020501 (‘06)

_

dHvA across Magnetic Transition in CeRhIn5

H. Shishido, R. Settai, H. Harima,

& Y. Onuki, JPSJ 74, 1103 (2005)

• crossover width smaller as T decreases

• T=0 (extrapolation): sharp jump @ QCP

S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004)

Hall Effect in YbRh2Si2

Multiple Energy Scales of QCP

P. Gegenwart, T. Westerkamp, C. Krellner, Y. Tokiwa, S. Paschen, C. Geibel,

F. Steglich, E. Abrahams, & QS, Science 315, 969 (2007)









Bose-Fermi Kondo model

2

1

0

1~)(

ωwω pp

~)(

ε-expansion of Bose-Fermi Kondo model:

Order ε: J. L. Smith & QS ’97; A. M. Sengupta ’97; Higher orders in ε: L. Zhu & QS ’02; G. Zarand & E. Demler ’02

J K = 0: S. Sachdev & J. Ye ’93 (large N); M. Vojta, C. Buragohain & S. Sachdev ‘00

JK

Kondo

Critical

g

ωwω pp

~)(

Critical:

ε-expansion of Bose-Fermi Kondo model:

Order ε: J. L. Smith & QS ’97; A. M. Sengupta ’97; Higher orders in ε and spin anisotropies: L. Zhu & QS ’02; G. Zarand & E. Demler ’02

J K = 0: S. Sachdev & J. Ye ’93 (large N); M. Vojta, C. Buragohain & S. Sachdev ‘00

JK

Kondo

Critical

g

ωwω pp

~)(

JKKondo

Critical

g

IsingSU(2) & XY

Critical:

Crucial for LQCP solution

Bose-Fermi Kondo model in a large-N limit

S. Kirchner and QS,

arXiv:0706.1783

At the Kondo-destroying QCP (g=gc)


T=0:

S. Kirchner and QS,

arXiv:0706.1783


S. Kirchner and QS,

arXiv:0706.1783

g=gc:Finite T:

Of the form of a boundary CFT


T=0:

Finite T:

S. Kirchner and QS,

arXiv:0706.1783

Spin susceptibilityat g=gc:

All these suggest: an underlying boundary CFT; enhanced symmetry at the QCP

Bose-Fermi Kondo modelwith Ising anisotropy

Bose-Fermi Kondo model at ε=2: exact solution

• General arguments and indirect calculations imply that, at ε=1-, the QCP of BFKM has zero residual entropy.

• However, large-N limit yields a finite residual entropy for the QCP of any 0<ε≤2, including at ε=1- (M. Kircan and M. Vojta, PRB69, 174421 ’04).

• Shedding light from an exact solution at ε=2

.1

~)( 2

1

0const

J-H Dai, C. Bolech, and QS, to be published


Bose-only:

SU(2): Ising:



Bose-only:

SU(2): Ising:

Contrast to large N:



Bose-Fermi Kondo(Ising anisotropic, Kondo Toulouse point):



• General arguments and indirect calculations imply that, at ε=1-, the QCP of BFKM has zero residual entropy.

• However, large-N limit yields a finite residual entropy for any 0≤ε≤2, including at ε=1- (M. Kircan and M. Vojta, PRB69, 174421 ’04).

• Shedding light from an exact solution at ε=2:

The large N limit misses the change of the fixed-point structure across ε=1: for singular bosonic bath, large N over-estimates transverse fluctuations & under-estimates longitudinal fluctuation.

SUMMARY

• Quantum critical heavy fermions metals– Non-Fermi liquid behavior

– Beyond the order parameter fluctuation picture

• Local quantum criticality– Critical Kondo screening at the magnetic QCP

• Global phase diagram – An antiferromagnetic metal with “small” Fermi surface

• Experiments: dynamical scaling, Fermi surface reconstruction, multiple energy scales.

• Relevance to other strongly correlated systems?

Beyond microscopcs • What is the field theory?

• For α<1, Smag is Gaussian; the q-dependence of M(q,ω) would be smooth.

• The coupling to Scritical-kondo makes a contribution to M(q,ω) which is expected to be smoothly q-dependent.

• The spatial anomalous dimension ηspatial=0.

QS, S. Rabello, K. Ingersent and J.L.Smith, Nature 413, 804 (2001); Phys. Rev. B 68, 115103 (2003)

Small Fermi surface in the AF metal phase

TN

TN

S. Araki, R. Settai, T. C. Kobayashi, H. Harima, & Y. Onuki,

Phys Rev. B 64, 224417 (2001)

e.g. CeRh2Si2

Qimiao Si Rice University

Documents

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spin charge e kondo

magnetic qcp qs

antiferromagnetic qcp

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nature 2003cepd2si2n

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