A probabilistic approach to the models of spin glasses (original) (raw)

Gauge models for spin-glasses

Physical Review B

A Landau-Ginzburg description of a spin-glass which incorporates naturally the concept of "frustration, " or the incompatibility of different local stable spin configurations in neighboring regions, is presented. For a planar spin, the effective Hamiltonian has a form analogous to that of the Landau-Ginzburg functional for a superconductor in a magnetic field, except that the role of the vector potential is taken by a quenched random variable Q(x) which represents the wave vector of the spin-density wave of minimum local free energy. The model is thus a simple transcription to a Landau-Ginzburg picture of the basic notion of a spinglass as a material whose properties are determined by competition between ferromagnetic and antiferromagnetic interactions. The probability distribution of Q(x) is chosen not to depend on Q(x) directly (in order not to favor any particular value of Q), but to be Gaussian in the curl of Q(x). The variance f of this distribution, the mean-square vorticity in Q(z), is a measure of the degree of frustration. [Any longitudinal part of Q(x) is gauged away by rotating the local spin axes appropriately. ] For a classical vector (Heisenberg) spin system, the analogous description is a Hamiltonian of O(3) Yang-Mills form, again with the gauge random variable. Two calculations are presented. The first tests the stability of the f = 0 theory (thermodynamically identical to an ordinary ferromagnet) against the introduction of a small amount of frustration. The result is that the f = 0 fixed point is unstable, and no new fixed point (of order 4d) appears. Thus the spin-glass transition does not appear to be related to any normal sort of critical point with a particular local-spin-density configuration as a "hidden" order parameter. The second is a mean-field analysis of a transition to a state characterized by an Edwards-Anderson order parameter; its qualitative features are similar to those of mean-field theories for other models for spin-glasses. The conditions for the thermodynamic stability of such a state remain unknown,

On a classical spin glass model

Zeitschrift f�r Physik B Condensed Matter, 1983

A simple, exactly soluble, model of a spin-glass with weakly correlated disorder is presented. It includes both randomness and frustration, but its solution can be obtained without replicas. As the temperature T is lowered, the spin-glass phase is reached via an equilibrium phase transition at T--T I. The spin-glass magnetization exhibits a distinct S-shape character, which is indicative of a field-induced transition to a state of higher magnetization above a certain threshold field. For suitable probability distributions of the exchange interactions. (a) A mixed phase is found where spin-glass and ferromagnetism coexist. (b) The zero-field susceptibility has a flat plateau for 0_<T_< T~ and a Curie-Weiss behaviour for T > T I. (c) At low temperatures the magnetic specific heat is linearly dependent on the temperature. The physical origin of the dependence upon the probability distributions is explained, and a careful analysis of the ground state structure is given.

Random free energies in spin glasses

1985

Abstract The free energies of the pure states in the spin glass phase are studied in the mean field theory. They are shown to be independent random variables with an exponential distribution. Physical implications concerning the fluctuations from sample to sample are worked out. The physical nature of the mean field theory is fully characterized. Résumé Les énergies libres des états purs sont étudiées dans la théorie de champ moyen.

Some comments on the Sherrington-Kirkpatrick model of spin glasses

Communications in Mathematical Physics, 1987

In this paper the high-temperature phase of general mean-field spin glass models, including the Sherrington-Kirkpatrick (SK) model, is analyzed. The free energy in zero magnetic field is calculated explicitly for the SK model, and uniform bounds on quenched susceptibilities are established. It is also shown that, at high temperatures, mean-field spin glasses are limits of shortrange spin glasses, as the range of the interactions tends to infinity.

Universality-class dependence of energy distributions in spin glasses

Physical Review B, 2005

We study the probability distribution function of the ground-state energies of the disordered onedimensional Ising spin chain with power-law interactions using a combination of parallel tempering Monte Carlo and branch, cut, and price algorithms. By tuning the exponent of the power-law interactions we are able to scan several universality classes. Our results suggest that mean-field models have a non-Gaussian limiting distribution of the ground-state energies, whereas non-meanfield models have a Gaussian limiting distribution. We compare the results of the disordered onedimensional Ising chain to results for a disordered two-leg ladder, for which large system sizes can be studied, and find a qualitative agreement between the disordered one-dimensional Ising chain in the short-range universality class and the disordered two-leg ladder. We show that the mean and the standard deviation of the ground-state energy distributions scale with a power of the system size. In the mean-field universality class the skewness does not follow a power-law behavior and converges to a nonzero constant value. The data for the Sherrington-Kirkpatrick model seem to be acceptably well fitted by a modified Gumbel distribution. Finally, we discuss the distribution of the internal energy of the Sherrington-Kirkpatrick model at finite temperatures and show that it behaves similar to the ground-state energy of the system if the temperature is smaller than the critical temperature.

Hierarchical Random Energy Model of a Spin Glass

Physical Review Letters, 2010

We introduce a Random Energy Model on a hierarchical lattice where the interaction strength between variables is a decreasing function of their mutual hierarchical distance, making it a nonmean field model. Through small coupling series expansion and a direct numerical solution of the model, we provide evidence for a spin glass condensation transition similar to the one occurring in the usual mean field Random Energy Model. At variance with mean field, the high temperature branch of the free-energy is non-analytic at the transition point. PACS numbers: 05.10.-a,05.50.+q,75.10.Nr

Some considerations of finite-dimensional spin glasses

Journal of Physics A: Mathematical and Theoretical, 2008

In talk I will review the theoretical results that have been obtained for spin glasses, paying a particular attention to finite dimensional spin glasses. I will concentrate my attention on the formulation of the mean field approach and on its numerical and experimental verifications. I will mainly considered equilibrium properties at zero magnetic field, where the situation is clear and it should be not controversial. I will present the various hypothesis at the basis of the theory and I will discuss their physical status.

Short-Range Spin Glasses: The Metastate Approach

Encyclopedia of Mathematical Physics, 2006

We discuss the metastate, a probability measure on thermodynamic states, and its usefulness in addressing difficult questions pertaining to the statistical mechanics of systems with quenched disorder, in particular short-range spin glasses. The possible low-temperature structures of realistic (i.e., short-range) spin glass models are described, and a number of fundamental open questions are presented.

Non-equilibrium Relations for Spin Glasses with Gauge Symmetry

2010

We study the applications of non-equilibrium relations such as the Jarzynski equality and fluctuation theorem to spin glasses with gauge symmetry. It is shown that the exponentiated free-energy difference appearing in the Jarzynski equality reduces to a simple analytic function written explicitly in terms of the initial and final temperatures if the temperature satisfies a certain condition related to gauge symmetry. This result is used to derive a lower bound on the work done during the non-equilibrium process of temperature change. We also prove identities relating equilibrium and non-equilibrium quantities. These identities suggest a method to evaluate equilibrium quantities from non-equilibrium computations, which may be useful to avoid the problem of slow relaxation in spin glasses.

Gauge Theory for Quantum Spin Glasses

Journal of the Physical Society of Japan, 2006

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.