Weak instability of frustrated fermionic models

Abstract

We study the Almeida-Thouless instability of two fermionic models analogous to spin glasses that exhibit frustration and that were solved some time ago with a replica symmetric ansatz. In the first model (I) we consider only the anisotropic, Ising-like limit, while in the second model (II) we consider the isotropic, Heisenberg-like Hamiltonian. In both models the interactions are of the Sherrington-Kirkpatrick type and the spins are represented by bilinear combinations of fermionic fields. While model I is almost classical, exhibiting a negative entropy at low temperatures, we show in this paper that the eigenvalue ${\ensuremath{\lambda}}_{\mathrm{RS}}$ is positive at the critical temperature and becomes negative at a temperature below the transition point. Model II is more interesting because ${\ensuremath{\lambda}}_{\mathrm{RS}}$ is positive at the critical temperature ${T}_{\mathrm{SG}},$ vanishes at ${T}_{1}<{T}_{\mathrm{SG}},$ and becomes positive again at ${T}_{2}<{T}_{1}.$ Although the entropy remains positive all the way down to $T=0,$ it presents a break of monotonicity when ${\ensuremath{\lambda}}_{\mathrm{RS}}$ becomes negative, indicating a negative specific heat in part of the instability region ${T}_{2}<T<{T}_{1}.$ The two stability regions in the ordered phase for $T<{T}_{2}$ and ${T}_{1}<T<{T}_{\mathrm{SG}}$ are characterized by the correct sign of the entropy and specific heat. This seems to indicate that replica symmetry stability is enhanced in frustrated fermionic spin models.