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arXiv:cond-mat/9910034 · Journal of Physics A Mathematical and General

Replica symmetry breaking solution for the fermionic Ising spin-glass and the Ghatak-Sherrington model

H Feldmann, R Oppermann

Published Feb 10, 2000• 15 citations
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Abstract

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.

DOI
10.1088/0305-4470/33/7/303

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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 investigate the Ghatak-Sherrington model with S ≥1, including S =∞, in a framework of the replica method, namely in terms of the replica symmetry (RS) solutions, the de Almeida and Thouless (AT) stability analysis and the one-step replica symmetry breaking (RSB) solutions. We find that there is a first order transition of van der Waals type between the paramagnetic phase and the spin glass phase and also a first order transition between the RS solution and the RSB solution in the model with integral S in some range of anisotropy parameter, while there is only a continuous transition in the model with half-integral S , including S =∞. We find that the model with integral S has a critical end point under the external magnetic field. We obtain phase diagrams as for the AT stability for the model under the external magnetic field.
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 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 ${\ensuremath{\lambda}}_{\mathrm{RS}}$ 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.
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).
We investigate the expansion near T_c of the free energy corresponding to the generalized Sherrington--Kirkpatrick model with arbitrary diagonal operators U^\hat{U}U^ standing instead of Ising spins. We focus on the case when U is an operator without reflection symmetry. The principal prescription for obtaining a full replica symmetry breaking solution is derived in general case in the neighbourhood of the transition temperature. An illustrating example is considered demonstrating the implicitly build solution.
This thesis is concerned with the mean field theory of spin glasses within the replica method approach. We have studied the Sherrington-Kirkpatrick model [33,4δ] both with Ising and m-component spins. It is shown that the replica symmetric solution gives an instability in the condensed phase which requires a breaking of replica symmetry in the spin glass phase. The various broken replica symmetry solutions proposed thus far (Bray and Moore [52], Blandin et al [53], Parisi [54]) are reviewed and particular attention is given to the one proposed by Parisi. The order parameter susceptibility and the stability of the solution are studied within this breaking of replica symmetry scheme which give results in agreement with the predictions of TAP [47] and Bray and Moore [52]. The breakdown of linear response theory in the spin glass phase is also discussed.
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.