Antoine GeorgesOlivier Parcollet

Nick ReadSubir Sachdev

Jinwu Ye

Mean field theories of quantum spin glasses

Talk online: Sachdev

Classical Sherrington-Kirkpatrick model

ij i ji j

H J S S

Jij : a Gaussian random variable with zero mean

: a unit length component vectoriS n

Two routes to quantization

21

Action = 2

i

i

dSd H

g d

A. Quantum rotor model

n=1: Ising model is a transverse field g

Spectrum at Jij=0 g

1

2j j j

1

2j j j

n=3: randomly coupled spin dimers

Spectrum at Jij=0 g

1

2 j j

1, ,

2j j j j

Two routes to quantization

Action = jj

j

dSd iSA S H

d

B. Heisenberg spins

Spectrum at Jij=0 ,j j

2

First term is kinematic Berry phase which ensures

, and ( 1)j k jk j jS S i S S S S

(2S+1)-fold degeneracy

Generalize model to SU(N) spins and explore phase diagram in N, S plane

Outline

A. Insulating quantum rotors.

B. Insulating Heisenberg spins

C. DMFT of a random t-J model

D. Metallic spin glasses: DMFT of a random Kondo lattice

A. Insulating quantum rotors

21

Action = 2

iij i j

i i j

dSd J S S

g d

A. Quantum rotor model

Jij : a Gaussian random variable with zero mean

g

Spin glass Paramagnet

T=0 phases 1/

00 n

T in j jd S S e

'' ~ EAq '' gapped

Local dynamic spin susceptibility

Specific heat C ~ T (?)

D.A. Huse and J. Miller, Phys. Rev. Lett. 70, 3147 (1993).J. Ye, S. Sachdev, and N. Read, Phys. Rev. Lett. 70, 4011 (1993).N. Read, S. Sachdev, and J. Ye, Phys. Rev. B 52, 384 (1995).A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. B 63, 134406 (2001).

'' ~

T > 0 phase diagram

g

1/ 2Quantum critical '' ~

/ ln 1/T T

J. Ye, S. Sachdev, and N. Read, Phys. Rev. Lett. 70, 4011 (1993).N. Read, S. Sachdev, and J. Ye, Phys. Rev. B 52, 384 (1995).

gc

B. Insulating Heisenberg spins

Action = jj ij i j

j i j

dSd iSA S J S S

d

B. Heisenberg spin glass

Jij : a Gaussian random variable with zero mean

2 : a SU( ) spin with 1 components and "length" jS N N S

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).

T=0 phase diagram

S

N

Spin glass order

'' ~ EAq

Specific heat C ~ T(C ~ T2 ?)

Quantum critical "spin slush" phase

with "marginal Fermi liquid" spectrum:

sgn '' ~

J

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. Lett. 85, 840 (2000). A. Camjayi and M. J. Rozenberg, Phys. Rev. Lett. 90, 217202 (2003).

Quantum critical phase is described by fractionalized S=1/2 neutral spinon excitations

†~S f f

Spinon spectral density

1

Bk T

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. Lett. 85, 840 (2000). A. Camjayi and M. J. Rozenberg, Phys. Rev. Lett. 90, 217202 (2003).

T > 0 phase diagram

'' ~ sgnBk T

C. Doping the quantum critical spin liquid

†Hamiltonian = ij i j ij i jij ij

t Pc c P J S S

C. DMFT of a random t-J model

Jij : a Gaussian random variable with zero mean

†1

2i i iS c c

O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).

O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).

2

* *

Quantum critical "incoherent" physics with universal / scaling above

a coherence scale ~ ~

B

F B

k T

tk T

J

= carrier density

O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).

Physical consequences of quantum criticality

1. Electron spectral function (photoemission)

*

1Momentum integrated electron spectral density at 0 :

1 1 as 0 and as

F

Tt

Momentum resolved spectral density

2

*Quasiparticle peak with residue ~Z

1

O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).

Physical consequences of quantum criticality

2. d.c Resistivity

dc 2 *

F

h TT

e

* !

F

T

2

*F

T

O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).

Physical consequences of quantum criticality

3. NMR 1/T1 relaxation rate

*1

1 1

F

T

T J

constant (MFL)

*Korringa

F

T

O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).

Physical consequences of quantum criticality4. Optical conductivity

**

*

*

In quantum critical regime, with , Re

for with Re

for

FF

B

F

F

T Jk T

T J

TT

O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).

Phenomenological phase diagram for cuprates

D. Metallic spin glasses

†

†

Hamiltonian =

2

ij i j ij i jij ij

Ki i i

i

t c c J S S

JS c c

C. DMFT of a random Kondo lattice model

Jij : a Gaussian random variable with zero mean

S. Sachdev, N. Read, and R. Oppermann, Phys. Rev. B 52, 10286 (1995).A. M. Sengupta and A. Georges, Phys. Rev. B 52, 10295 (1995).  

S. Sachdev, N. Read, and R. Oppermann, Phys. Rev. B 52, 10286 (1995).A. M. Sengupta and A. Georges, Phys. Rev. B 52, 10295 (1995).  

JK

1/ 2

3/ 2Quantum critical '' ~ sgn

T

1/ 2s~ g' n' EAq

OutlookOutlook

• Spin glass order is an attractive candidate for a quantum critical point in the cuprates, on both theoretical and experimental grounds. (Impurities break the translational symmetry associated with charge-ordered states, and the Imry-Ma argument then prohibits a quantum critical point associated with charge order in the presence of randomness in two dimensions)

• A simple mean-field theory of a doped Heisenberg spin glass naturally reproduces all the “marginal” phenomenology.

• Needed: better theory of fluctuations in low dimensions