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Physical review. B, Condensed matter

Weak instability of frustrated fermionic models

Alba Theumann

Published Sep 1, 1997• 3 citations
Publisher

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.

DOI
10.1103/physrevb.56.5500

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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 \ensuremathλRS{\ensuremath{\lambda}}_{\mathrm{RS}}\ensuremathλRS​ is positive at the critical temperature and becomes negative at a temperature below the transition point. Model II is more interesting because \ensuremathλRS{\ensuremath{\lambda}}_{\mathrm{RS}}\ensuremathλRS​ is positive at the critical temperature TSG,{T}_{\mathrm{SG}},TSG​, vanishes at T1<TSG,{T}_{1}<{T}_{\mathrm{SG}},T1​<TSG​, and becomes positive again at T2<T1.{T}_{2}<{T}_{1}.T2​<T1​. Although the entropy remains positive all the way down to T=0,T=0,T=0, it presents a break of monotonicity when \ensuremathλRS{\ensuremath{\lambda}}_{\mathrm{RS}}\ensuremathλRS​ becomes negative, indicating a negative specific heat in part of the instability region T2<T<T1.{T}_{2}<T<{T}_{1}.T2​<T<T1​. The two stability regions in the ordered phase for T<T2T<{T}_{2}T<T2​ and T1<T<TSG{T}_{1}<T<{T}_{\mathrm{SG}}T1​<T<TSG​ 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.
We solve the fermionic version of the Ising spin glass for arbitrary filling µ and temperature T taking into account replica symmetry breaking. Using a simple exact mapping from µ to the anisotropy parameter D , we also obtain the solution of the S = 1 Sherrington-Kirkpatrick model. An analytic expression for T = 0 gives an improved critical value for the first-order phase transition. We revisit the question of stability against replica-diagonal fluctuations and find that the appearance of complex eigenvalues of the Almeida-Thouless matrix is not an artefact of the replica-symmetric approximation.
We present details of the phase diagrams of fermionic systems with random and frustrated interactions, emphasizing the important role of the chemical potential. The insulating fermionic Ising spin glass model is shown to reveal different entangled magnetic instabilities and phase transitions. We review tricritical phenomena related to the strong correspondence between charge and spin fluctuations, being controlled by quantum statistics. We compare the spin density diluted Sherrington-Kirkpatrick spin glass with classical spin 1 models such as the BEG model. We analyse in detail the infinite range model and show that spin glass order must decay discontinuously as the chemical potential exceeds a critical value, provided the temperature is below the tricritical one, and that the T=0 transition is of classical type. Parisi replica permutation symmetry breaking (RPSB) governs the thermal spin glass transitions and fermionic modifications of the SK-models AT-line emerge. RPSB takes place everywhere within the fermionic spin glass phase. Although the critical field theory of the quantum paramagnet to spin glass transition in metallic systems remains replica--symmetric at T=0, with only small corrections at low T from RPSB, the phase diagram is affected at O(T^0) by RPSB. Generalizing our results for the fermionic Ising spin glass we consider aspects of models with additional spin and charge quantum--dynamics such as metallic spin glasses.
I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).
In the present paper we present a detailed analysis of quantum effects in the spin-glass transition as described by a quantum Heisenberg analogue of the Sherrington-Kirkpatrick model. The spin operators are represented in terms of two fermion fields and the problem is reduced to that of n fermion levels at one site in a random time-dependent field and with an interaction delocalized in time. It is shown that within a Hartree-Fock approximation in a replica-symmetric theory one obtains a mean-field description of the transition with satisfactory zero-temperature properties. The transition is described by two order parameters: the static magnetic susceptibility and the spin-glass order parameter. The saddle-point equations obtained in this work are analogous to those of Sommers for a theory of classical spins.
We analyze the infinite range Ising spin glass in a transverse-field $\ensuremath{\Gamma}$ below the critical temperature by a one step replica symmetry breaking theory. The set of nnn replicas is divided in rrr blocks of mmm replicas each. We present results for different values of the block-size parameter mmm. The spin operators are represented by bilinear combinations of fermionic fields and we compare the results of two models: In the four-states (4-S) model the diagonal Siz{S}_{i}^{z}Siz​ operator has two unphysical vanishing eigenvalues, that are suppressed by a restraint in the two-states (2-S) model. In the static approximation we obtain qualitatively similar results for both models. They both exhibit a critical temperature ${T}_{c}(\ensuremath{\Gamma})$ that decreases when $\ensuremath{\Gamma}$ increases, until it reaches a quantum critical point at the same value of ${\ensuremath{\Gamma}}_{c}$ and they are both unstable under replica symmetry breaking in the whole spin glass phase. Below the critical temperature we present results for the order parameters and free energy.
Within the replica approach to mean-field spin-glasses the transition from ergodic high-temperature behaviour to the glassy low-temperature phase is marked by the instability of the replica-symmetric saddle-point. For general spin-glass models with non-Gaussian field distributions the corresponding Hessian is a 2n×2n2^n\times 2^n2n×2n matrix with the number nnn of replicas tending to zero eventually. We block-diagonalize this Hessian matrix using representation theory of the permutation group and identify the blocks related to the spin-glass susceptibility. Performing the limit n→0n\to 0n→0 within these blocks we derive expressions for the de~Almeida-Thouless line of general spin-glass models. Specifying these expressions to the cases of the Sherrington-Kirkpatrick, Viana-Bray, and the Lévy spin glass respectively we obtain results in agreement with previous findings using the cavity approach.