Antiferromagnetic Ising spin glass

Fyodorov, Ya V; Korenblit, I. Ya.; Shender, E. F.

doi:10.1088/0022-3719/20/12/011

Cited by 34 publications

(36 citation statements)

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“…In this section we present the results of several numerical studies and show that they are in good agreement. First, we numerically solved for the critical point of the RSB-1 free energy 13 We have glossed over one important point, which is that although the expressions for the free energies Eq. 3.5 and Eq.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Replica symmetry breaking in bipartite spin glasses and neural networks

2018

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Some interesting recent advances in the theoretical understanding of neural networks have been informed by results from the physics of disordered many-body systems. Motivated by these findings, this work uses the replica technique to study the mathematically tractable bipartite Sherrington-Kirkpatrick (SK) spin-glass model, which is formally similar to a restricted Boltzmann machine (RBM) neural network. The bipartite SK model has been previously studied assuming replica symmetry; here this assumption is relaxed and a replica symmetry breaking analysis is performed. The bipartite SK model is found to have many features in common with Parisi's solution of the original, unipartite SK model, including the existence of a multitude of pure states which are related in a hierarchical, ultrametric fashion. As an application of this analysis, the optimal cost for a graph partitioning problem is shown to be simply related to the ground state energy of the bipartite SK model. As a second application, empirical investigations reveal that the Gibbs sampled outputs of an RBM trained on the MNIST data set are more ultrametrically distributed than the input data themselves.

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Section: Numerical Resultsmentioning

confidence: 99%

“…2.42). 13 We numerically tested the replica prediction Eq. 3.7 by simulating many instances of random bipartite graphs and using a simple variant of the well-known Kernighan-Lin (KL) [41] algorithm.…”

Section: )mentioning

confidence: 99%

Replica symmetry breaking in bipartite spin glasses and neural networks

2018

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“…Most studies have concentrated on situations where the exchange distributions are either symmetric or with an additional ferromagnetic interaction. More recently a twosublattice version of the SK model was introduced [10][11][12][13] to allow for antiferromagnetic interactions between different sublattices. Such extension is quite natural in view of the existence of many experimental systems such as Fe x Mg 1−x Cl 2 [14][15][16] and Fe x Mn 1−x TiO 3 [17,18], which exhibit a transition from an Ising antiferromagnetic into an Ising spin glass state for a certain range of x values.…”

Section: Introductionmentioning

confidence: 99%

“…Such extension is quite natural in view of the existence of many experimental systems such as Fe x Mg 1−x Cl 2 [14][15][16] and Fe x Mn 1−x TiO 3 [17,18], which exhibit a transition from an Ising antiferromagnetic into an Ising spin glass state for a certain range of x values. In contrast to the standard SK model, in the two-sublattice SK model with antiferromagnetic intersublattice interactions, the ordered (antiferromagnetic) phase extends to finite fields and the de Almeida-Thouless instability line [5] has distinct branches in the paramagnetic and antiferromagnetic phases, which do not meet at a first-order transition [10][11][12][13]. Experimental determination of the field-temperature phase diagram in Fe x Mn 1−x TiO 3 , as well as the de AlmeidaThouless instability line [19], are in qualitative agreement with mean-field results [13].…”

Section: Introductionmentioning

confidence: 99%

“…In the previous studies of this model the stability of the RS solution against transversal fluctuations, i.e., outside the RS space, has already been investigated [10][11][12][13], and the stability against longitudinal fluctuations, i.e., inside the RS space, was also briefly considered [12]. The stability of the RS solution against transversal fluctuations is important to establish whether replica symmetry breaking is necessary.…”

Section: Introductionmentioning

confidence: 99%

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Stability of a two-sublattice spin-glass model

Yokoi¹,

Costa²

2003

Braz. J. Phys.

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We study the stability of the replica-symmetric solution of a two-sublattice infinite-range spin-glass model, which can describe the transition from an antiferromagnetic to a spin-glass state. The eigenvalues associated with replica-symmetric perturbations are in general complex. The natural generalization of the usual stability condition is to require the real part of these eigenvalues to be positive. The necessary and sufficient conditions for all the roots of the secular equation to have positive real parts is given by the Hurwitz criterion. The generalized stability condition allows a consistent analysis of the phase diagram within the replica-symmetric approximation.

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Construction and purpose of effective field theories for frustrated magnetic order

Oppermann

Schmidt

2007

Phys. Status Solidi (c)

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In many fields of physics the discovery of an effective field theory represents a milestone of theoretical progress and sometimes allows analytical answers where otherwise only the brutal power of computers can find structure-less numerical hints. Let us mention nonlinear sigma models where nonlinearly realized symmetries [1] play a central role. Applications to slow dynamics in magnets, to the localizing effects of disorder in Anderson localization and, more recently, as a model for non-equilibrium dynamics of spin glasses [2] are well-known examples.In the field of spin glasses, Talagrand [3] proved recently that Parisi replica symmetry breaking [4] represents, as advocated since three decades, the exact structure of the basic SK model solution. Explicit solutions, which are necessary for applications in solid state physics at low temperatures, were however not yet available except in the vicinity of the critical temperature.Numerical methods combined with the renormalization group of De Dominicis [5], aiming at a relation between Parisi-and droplet-picture [6], appears to support the Parisi picture even for finite range interaction, and including low temperatures.Phenomenological theories which study the interplay between frustrated magnetism with other degrees of freedom, for example with electronic transport, exist [7], but systematic approaches were limited to the region of the spin glass freezing temperature, or neglected the details of Parisi RSB. As the literature shows, most applications were, probably due to the complication of the Parisi solution and the way how to include it, concerned with precursor effects in the nonmagnetic region above the critical temperature [8]. In the case of quantum phase transitions (driven by a non-thermal parameter) the nonmagnetic region can extend down to T = 0. Since the specific features of the frustrated order are not involved (only * e-mail: opperman@physik.uni-wuerzburg. This article reviews recent years' progress in the low temperature analysis of standard models of spin glass order such as the Sherrington-Kirkpatrick (SK) model. Applications to CdTe/CdMnTe layered systems and explanation of glassy antiferromagnetic order at lowest temperatures stimulated us to study in detail the beautifully complex physical effects of replica symmetry breaking (RSB). We discuss analytical ideas based on highly precise numerical data which lead to the construction of relatively simple effective field theories for the SK model and help to understand the mysterious features of its exact solution. The goal is to find construction principles for the theory of interplay between frustrated magnetic order and various relevant physical degrees of freedom. The emphasis in this article is on the role of Parisi's RSB, which surprisingly creates critical phenomena in the low temperature limit despite the absence of a standard phase transition.

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Antiferromagnetic Ising spin glass

Cited by 34 publications

References 10 publications

Replica symmetry breaking in bipartite spin glasses and neural networks

Replica symmetry breaking in bipartite spin glasses and neural networks

Stability of a two-sublattice spin-glass model

Construction and purpose of effective field theories for frustrated magnetic order

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