Mean field theories of quantum spin glasses Antoine Georges Olivier Parcollet Nick Read Subir Sachdev Jinwu Ye Talk online: Sachdev
Classical Sherrington-Kirkpatrick model H = JS S i j ij i j J ij : a Gaussian random variable with zero mean S : a unit length n component vector i
A. Quantum rotor model Action = Two routes to quantization 1 dτ 2g n=1: Ising model is a transverse field g Spectrum at J ij =0 n=3: randomly coupled spin dimers Spectrum at J ij =0 g i ds i dτ = = 1 1 2 2 2 + H ( ) j j j ( + ) j j j 1, ( + ), 2 g 1 j j j j ( ) 2 j j
B. Heisenberg spins Action = Two routes to quantization τ ( ) j j d isa S H j ds i dτ First term is kinematic Berry phase which ensures 2 Sj, S α kβ = iδ jkεαβγsjγ and Sj = S( S+ 1) + Spectrum at J ij =0, (2S+1)-fold degeneracy j j 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
A. Quantum rotor model Action = 2 1 ds i dτ + J S S 2g i dτ i j ij i j J ij : a Gaussian random variable with zero mean
T=0 phases Local dynamic spin susceptibility χ ω = τ τ 1/ T ( ) ( ) i ( ) i n d S S 0 e ω τ n 0 j j ( + )( + )( + ) Spin glass χ ''( ω) ~ q ( ) EA ωδ ω + ω Specific heat C ~ T (?) Paramagnet χ'' ω gapped ( ) χ ''( ω) ~ ω g 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 ω Quantum critical χ'' ( ω) ~ ωφ 1/2 T /ln ( 1/ T) g g c 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).
B. Insulating Heisenberg spins
B. Heisenberg spin glass Action = ds dτ isa( S ) i + J S S j j ij i j j dτ i j J ij : a Gaussian random variable with zero mean 2 S j : a SU( N) spin with N 1 components and "length" S S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).
T=0 phase diagram S Spin glass order ( ) q ( ) χ '' ω ~ EA ωδ ω + ω Specific heat C ~ T (C ~ T 2?) Quantum critical "spin slush" phase with "marginal Fermi liquid" spectrum: ( ω) χ'' ~ sgn ( ω) J N 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 σ α αβ β Spinon spectral density 1 Ψ ω ω kt B ω S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).
T > 0 phase diagram ω χ'' ( ω) ~ sgn ( ω) Φ kt B 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).
C. Doping the quantum critical spin liquid
C. DMFT of a random t-j model Hamiltonian = tpc ij iαcjα P+ JS ij i Sj ij ij 1 S c c 2 σ i iα αβ iβ J ij : a Gaussian random variable with zero mean O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
= carrier density Quantum critical "incoherent" physics with universal ω/ ktscaling above * * a coherence scale ε F ~ kt B ~ ( δt) 2 O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999). J B
Physical consequences of quantum criticality 1. Electron spectral function (photoemission) 1 ω Momentum integrated electron spectral density at T = 0 : ρ( ω) = φ * t ε F 1 1 φϖ ( ) as ϖ 0 and φϖ ( ) as ϖ π ϖ Momentum resolved spectral density 2 δ Quasiparticle peak with residue Z ~ * δ 1 ω O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
Physical consequences of quantum criticality 2. d.c Resistivity h T ρdc ( T ) = ψ 2 * e ε F T ε! * F T ε * F 2 O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
Physical consequences of quantum criticality 3. NMR 1/T relaxation rate 1 1 1 T * T1 J ψ = ε F constant (MFL) Korringa T ε * F O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
Physical consequences of quantum criticality 4. Optical conductivity * * ε F ω In quantum critical regime, with εf < T < J, Re σ ( ω) = ϑ ω kt B ( ) with Re σ ω * ε F ω for T = * ε F for ω < T < ω < J T O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
Phenomenological phase diagram for cuprates O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
D. Metallic spin glasses
C. DMFT of a random Kondo lattice model Hamiltonian = ij tc c + JS S ij iα jα ij i j ij J K + S c c 2 i σ i iα αβ iβ J ij : 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).
( ) ~ sgn ( ) Quantum critical χ'' ω ω ω 1/2 ω Φ 3/2 T ( ) ~ q ( ) + sgn ( ) χ ' ω ωδ ω ω ω EA 1/2 J K 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).
Outlook 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 chargeordered 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