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arXiv:cond-mat/9606033 · Journal of Physics A Mathematical and General

Site-disordered spin systems in the Gaussian variational approximation

David S. Dean, David Lancaster

Published Jan 7, 1997• 3 citations
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Abstract

We define a replica field theory describing finite dimensional site-disordered spin systems by introducing the notion of grand canonical disorder, where the number of spins in the system is random but quenched. A general analysis of this field theory is made using the Gaussian variational or Hartree - Fock method, and illustrated with several specific examples. Irrespective of the form of interaction between the spins this approximation predicts a spin-glass phase. We discuss the replica symmetric phase at length, explicitly identifying the correlator that diverges at the spin-glass transition. We also discuss the form of continuous replica symmetry breaking found just below the transition. Finally we show how an analysis of ferromagnetic ordering indicates a breakdown of the approximation.

DOI
10.1088/0305-4470/30/1/005

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•Jun 1, 2007•0 citations
We define a replica field theory describing finite dimensional site-disordered spin systems by introducing the notion of grand canonical disorder, where the number of spins in the system is random but quenched. A general analysis of this field theory is made using the Gaussian variational or Hartree - Fock method, and illustrated with several specific examples. Irrespective of the form of interaction between the spins this approximation predicts a spin-glass phase. We discuss the replica symmetric phase at length, explicitly identifying the correlator that diverges at the spin-glass transition. We also discuss the form of continuous replica symmetry breaking found just below the transition. Finally we show how an analysis of ferromagnetic ordering indicates a breakdown of the approximation.
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.
Spin glasses are the paradigm of complex systems. These materials present really slow dynamics. However, the nature of the spin glass phase in finite dimensional systems is still controversial. Different theories describing the low temperature phase have been proposed: droplet, replica symmetry breaking and chaotic pairs. We present analytical studies of critical properties of spin glasses, in particular, critical exponents at and below the phase transition, existence of a phase transition in a magnetic field, computation of the lower critical dimension (in presence/absence of a magnetic field). We also introduce some rigorous results based on the concept of metastate. Finally, we report some numerical results regarding the construction of the Aizenman-Wehr metastate, scaling of the correlation functions in the spin glass phase and existence of a phase transition in a field, confronting these results with the predictions of different theories.
A field-theory approach is used to investigate the 'spin-glass effects' on the critical behaviour of systems with weak temperature-like quenched disorder. The renormalization group (RG) analysis of the effective Hamiltonian of a model with replica symmetry breaking (RSB) potentials of a general type is carried out in the two-loop approximation. The fixed point (FP) stability, recently found within the one-step RSB RG treatment, is further explored in terms of replicon eigenvalues. We find that the traditional FPs, which are usually considered to describe the disorder-induced universal critical behaviour, remain stable when the continuous RSB modes are taken into account.
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.
A mean-field replica-type theory is presented for long-range Ising spin-glasses whose interactions can be written in the form J ij = J ij I J ij SK , where J SK is a quenched Gaussian interaction, as in the SK spin-glass, and J I is a deterministic matrix which defines the structure or inhomogeneity of the model and must have a finite rank in the thermodynamic limit. The spin-glass model proposed by Franz, Parisi and Virasoro ( Europhys. Lett. , 17 (1992) 5) can be written in this form, with J I having rank 2, and is used as a case study to illustrate our theory. In the context of replica theory we also consider the stable states of this model, which appear continuously as the temperature is lowered.
In the present work, a fermionic infinite-range Ising spin glass (SG) model with a transverse field Γ is studied. The model is analyzed in a one-lattice structure (quantum SG) and in a two-sublattice structure (SG/antiferromagnetism (AF) competition). In these two cases, the exchange interactions are random variables that follow Gaussian probability distributions (SK interactions). For the two-sublattice model, spins of distinct sublattices can couple with an SK-type interaction with an AF average. For a general case of this model, the disordered interactions can also occur with spins of same sublattices. The problem is formulated in a Grassmann path integral formalism within the static ansatz. The replica method is applied to treat the disorder, in which the replica symmetry (RS) and one-step replica symmetry breaking (1P-RSB) approximations are used. The one-lattice model results provide evidences that the 1S-RSB can be used to improve the physical description of the ordered quantum SG phase treated by the replica method. In the two-sublattice model, a parallel magnetic field H breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase (SG+AF) characterizing an asymmetric RSB region. In this region, intra-sublattice disordered interactions V increase the difference between the RSB solutions of each sublattice. The freezing temperature (Tf ) shows a higher increase with H when V enhances. A discontinuous phase transition from the replica symmetry (RS) solution to the RSB solution can appear with the presence of an intra-sublattice ferromagnetic average coupling. The Γ field introduces a quantum spin flip mechanism that suppresses the magnetic orders leading them to quantum critical points. Results suggest that the quantum effects are not able to restore the RS solution. Therefore, the presence of both fields, Γ and H, produces effects that are in competition on Tf for instance. However, in the asymmetric RSB region, Γ can produce a stable RS solution at any finite temperature for a particular sublattice while the other sublattice still presents RSB solution for the special case in which only the intra-sublattice spins couple with disordered interactions.