From localized to itinerant spin glasses: Grassmann field theory and mean-field solutions

Abstract

We introduce a new method to describe (quantum) spin glasses which is based on a Grassmann field representation of spins. Five spin glass models are considered in detail. We distinguish between Ising/Heisenberg spin glass models ( I s H s ) on spin space and Ising/ Heisenberg spin glasses ( I f H f ) on Fock space. To demonstrate the effect of the two different underlying spaces we calculate T c, the spin glass order parameter q, and the replica-diagonal average q = 〈σ 2〉 . The free energy is derived both for the replica-symmetric theory (SK solution) and for broken replica permutation symmetry (Parisi solution). For model I f we also calculate the entropy, a modified Almeida-Thouless line depending on q(T c ) , and the dependence on chemical potential and/or filling. Finally we define an itinerant spin glass model RHI f, which consists of model I f and an additional random hopping hamiltonian. The self-consistency equations for q, q , and for the electron Green function are determined. Above T c, the spin glass susceptibility is derived by expanding the action to second order in the order parameter fluctuation fields. The critical temperature for the onset of metallic spinglass order is found to be T c = J q(T c ) with q(T c ) = max( 4 π 2 − 1 2πJτ , 0) , where J denotes the the variance of the spin-spin coupling and τ refers to the elastic scattering time. This result leads to a criterion for the suppression of spin glass order which is reminiscent of the Stoner criterion for itinerant ferromagnetism.