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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