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Journal de physique

Exact properties of spin glasses. - I. 2D supersymmetry and Nishimori's result

Antoine Georges, David Hansel, Pierre Le Doussal

Published Jan 1, 1985• 8 citations
Publisher

Abstract

We introduce an effective theory of interacting fermions and bosons in order to express the quenched internal energy of the 2D Ising spin glass. We show that an exact result derived by Nishimori appears, in this formulation, as a dimensional reduction due to the apparition of a supersymmetry. For a general Ising spin glass, this suggests new insights into the physical meaning of this exact result.

DOI
10.1051/jphys:019850046080130900

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We introduce an effective theory of interacting fermions and bosons in order to express the quenched internal energy of the 2D Ising spin glass. We show that an exact result derived by Nishimori appears, in this formulation, as a dimensional reduction due to the apparition of a supersymmetry. For a general Ising spin glass, this suggests new insights into the physical meaning of this exact result.
We introduce an Ising spin-glass model with correlated disorder which continuously interpolates between the pure ferromagnetic Ising model and the Edwards-Anderson model with symmetric disorder. For this model, we prove that a Nishimori line (NL) can be defined, analogously to the Edwards-Anderson model, on which physical quantities can be expressed exactly in terms of those of the pure Ising model at a well-defined effective temperature on any lattice in any dimension. For example, the energy on the NL is equal to the energy of the pure Ising model at the effective temperature up to a constant and a trivial factor. More remarkably, the specific heat on the NL equals the energy, not the specific heat, of the pure Ising model at the effective temperature, again up to a constant and a trivial factor. Gauge-noninvariant quantities such as the magnetization and correlation functions are exactly equal to the corresponding quantities of the pure Ising model at the effective temperature. These exact relations imply that the leading critical behavior at that multicritical point for the disorder-correlated model is pure-Ising-like, in contrast to the conventional multicritical universality class of the standard Edwards-Anderson model. Our results motivate further investigations of the relatively unexplored topic of correlations in disorder in spin glasses and related problems.
We show that the effective action of the quantum spherical spin glass is invariant under a generalized form of the Becchi-Rouet-Stora-Tyutin(BRST) supersymmetry. The Ward identities associated to this invariance indicate that the spin glass order parameter must vanish, and as a result the annealed average is exact in this model. We present new results for for the free energy, entropy and specific heat. Due to quantum effects the entropy remains finite and the specific het vanishes at zero temperature. results for the phase diagram coincide with those obtained by different formalisms. At zero temperature we derive the scaling behaviour with frequency of the dynamical susceptibility.
We show that the effective action of the quantum spherical spin glass is invariant under a generalized form of Becchi-Rouet-Stora-Tyutin supersymmetry. The Ward identities associated with this invariance indicate that the spin glass order parameter must vanish, and as a result the annealed average is exact in this model. We present results for the free energy, entropy, and specific heat. Due to quantum effects, the entropy remains finite and the specific heat vanishes at zero temperature. Results for the phase diagram coincide with those obtained by different formalisms. At zero temperature we derive the scaling behavior with frequency of the dynamical susceptibility.
The gauge theory for random spin systems is extended to quantum spin glasses to derive a number of exact and/or rigorous results. The transverse Ising model and the quantum gauge glass are shown to be gauge invariant. For these models, an identity is proved that the expectation value of the gauge invariant operator in the ferromagnetic limit is equal to the one in the classical equilibrium state on the Nishimori line. As a result, a set of inequalities for the correlation function are proved, which restrict the location of the ordered phase. It is also proved that there is no long-range order in the two-dimensional quantum gauge glass in the ground state. The phase diagram for the quantum XY Mattis model is determined.
The multicritical behavior at the Nishimori point of two-dimensional Ising spin glasses is investigated by using numerical transfer-matrix methods to calculate probability distributions P(C)P(C)P(C) and associated moments of spin-spin correlation-functions C on strips. The angular dependence of the shape of correlation function distributions P(C)P(C)P(C) provides a stringent test of how well they obey predictions of conformal invariance; and an even symmetry of $(1\ensuremath{-}C)P(C)$ reflects the consequences of the Ising spin-glass gauge (Nishimori) symmetry. We show that conformal invariance is obeyed in its strictest form, and the associated scaling of the moments of the distribution is examined, in order to assess the validity of a recent conjecture on the exact localization of the Nishimori point. Power-law divergences of P(C)P(C)P(C) are observed near C=1C=1C=1 and C=0,C=0,C=0, in partial accord with a simple scaling scheme which preserves the gauge symmetry.
Abstract In recent years scale invariant scattering theory provided the first exact access to the magnetic critical properties of two-dimensional statistical systems with quenched disorder. We show how the theory extends to the overlap variables entering the characterization of spin glass properties. The resulting exact fixed point equations yield both the magnetic and, for the first time, the spin glass renormalization group fixed points. For the case of the random bond Ising model, on which we focus, the spin glass subspace of solutions is found to contain a line of fixed points. We discuss the implications of the results for Ising spin glass criticality and compare with the available numerical results.