The spin-spin correlation function in the two-dimensional Ising model in a magnetic field at T = Tc

The form-factor bootstrap approach is used to compute the exact contributions in the large-distance expansion of the correlation function 〈σ(x)σ(0)〉 of the two-dimensional Ising modeliin a magnetic field at T = T_c The matrix elements of the magnetization operator σ,(x) present a rich analytic structure induced by the (multi-)) scattering processes of the eight massive particles of the model. The spectral representation series has a fast rate of convergence and perfectly agrees with the numerical determination of the correlation function.     

Superfluidity of 3He in aerogel at T=0 in a magnetic field

The transition of liquid ³He to the superfluid B phase in aerogel at T=0 is considered. It is shown that in a magnetic field, the quantum phase transition with respect to pressure is split in two. The amount of splitting δP is estimated. The components of the superfluid density tensor are calculated near the critical pressures. Zh. éksp. Teor. Fiz. 115, 754–762 (February 1999) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

The Ising model in a random magnetic field

The existence of a spontaneous magnetization in the three-dimensional Ising model in a weak random magnetic field (RFIM) is investgated. Following Imry and Ma, we consider the energy change, E, from the fully aligned ferromagnetic state caused by flipping all the spins inside a connected surface, . It is proved rigorously that with high probability, E is positive forall enclosing the origin. Under the unproven assumption that the expectation value of the spin at one site is weakly correlated with the random fields at far away sites (which is true if surfaces within surfaces can be ignored) it follows that the three-dimensional RFIM has a spontaneous magnetization at low temperatures. The proof works for all dimensions greater than two, providing support for the conjecture that two is the lower critical dimension.Work supported in part by NSF grant No. DMR 8100417.     

Metastable states and T=0 hysteresis in the random-field Ising model on random graphs

We study the ferromagnetic random-field Ising model on random graphs of fixed connectivity z (Bethe lattice) in the presence of an external magnetic field H. We compute the number of single-spin-flip stable configurations with a given magnetization m and study the connection between the distribution of these metastable states in the H-m plane (focusing on the region where the number is exponentially large) and the shape of the saturation hysteresis loop obtained by cycling the field between - and + at T=0. The annealed complexity A(m,H) is calculated for z=2,3,4 and the quenched complexity Q(m,H) for z=2. We prove explicitly for z=2 that the contour Q(m,H)=0 coincides with the saturation loop. On the other hand, we show that A(m,H) is irrelevant for describing, even qualitatively, the observable hysteresis properties of the system.     

Asymptotics of decay of correlations for lattice spin fields at high temperatures. I. The Ising model

We find the asymptotic decrease of correlations < _{A +y} , _B >,yZ ^v +1, |y|, in the Ising model at high temperatures. For the case when monomials _A and _B both are odd, using the saddle-point method, we find the asymptotics of the correlations for any dimension . For even monomials _A , _B we formulate a general hypothesis about the form of the asymptotics and confirm it in two cases: (1) =1 and the vectory has an arbitrary direction, (2)y is directed along a fixed axis and arbitrary . Here we use besides the saddle-point method, some arguments from scattering theory.     

Response in kinetic Ising model to oscillating magnetic fields

Ising models obeying Glauber dynamics in a temporally oscillating magnetic field are analyzed. In the context of stochastic resonance, the response in the magnetization is calculated by means of both a mean-field theory with linear-response approximation, and the time-dependent Ginzburg-Landau equation. Analytic results for the temperature and frequency dependent response, including the resonance temperature, compare favorably with simulation data.     

On the Ising model in the presence of magnetic field

We demonstrate that the partition function of the 2-dimensional Ising model in the presence of a magnetic field can be expressed in terms of the distributions of the prime numbers in closed form.     

Double criticality of the Sherrington-Kirkpatrick model at T=0

Numerical results up to the 42nd order of replica-symmetry breaking (RSB) are used to predict the singular structure of the Sherrington-Kirkpatrick spin glass at T=0. We confirm predominant single parameter scaling and derive corrections for the T=0 order function q(a), related to a Langevin equation with pseudotime 1/a. a=0 and a=infinity are shown to be two critical points for infinity-RSB, associated with two discrete spectra of Parisi block size ratios, attached to a continuous spectrum. Finite-RSB-size scaling, associated exponents, and T=0-energy are obtained with unprecedented accuracy.     

The relativistic quantum fields of the D = 2 Ising model

We identify the correct field operators which correspond to the order and disorder variables of the D = 2 Ising model. After infinite resummations we obtain equality between our correlation functions and the correlation functions of McCoy, Tracy and Wu [4].     

Random field Ising model and community structure in complex networks

We propose a method to determine the community structure of a complex network. In this method the ground state problem of a ferromagnetic random field Ising model is considered on the network with the magnetic field B_s = +∞, B_t = -∞, and B_i≠s,t=0 for a node pair s and t. The ground state problem is equivalent to the so-called maximum flow problem, which can be solved exactly numerically with the help of a combinatorial optimization algorithm. The community structure is then identified from the ground state Ising spin domains for all pairs of s and t. Our method provides a criterion for the existence of the community structure, and is applicable equally well to unweighted and weighted networks. We demonstrate the performance of the method by applying it to the Barabási-Albert network, Zachary karate club network, the scientific collaboration network, and the stock price correlation network. (Ising, Potts, etc.)     

Form factors in the Bullough-Dodd related models: The Ising model in a magnetic field

A particular modification of the free-field representation of the form factors in the Bullough-Dodd model is considered. The two-particles minimal form factors are excluded from the construction. As a consequence, a convenient representation for the multiparticle form factors has been obtained, recurrence relations between them have been established, and their properties have been studied. The proposed construction is used to obtain the free-field representation of the lightest particles form factors in the Φ_{1, 2} perturbed minimal models. The Ising model in a magnetic field is considered as a significant example. The results obtained in the framework of the proposed free-field representation are in agreement with the corresponding results obtained by solving the bootstrap equations.     

Random field effects on the anisotropic quantum Heisenberg model with Gaussian random magnetic field distribution

The effect of a zero-centered Gaussian random magnetic field distribution on the phase transition properties of the anisotropic quantum Heisenberg model has been investigated on a honeycomb lattice within the framework of effective field theory (EFT) for a two-spin cluster (which is abbreviated as EFT-2). Particular attention has been devoted to investigation of the effect of the anisotropy in the exchange interaction on a system with Gaussian random magnetic field distribution. The variation of the critical temperature with the randomness parameter (i.e., the width of the distribution) has been obtained for several anisotropy parameters. Critical Gaussian distribution width values, which make the critical temperature zero, have been obtained. Moreover, it has been concluded that all critical temperatures are of second order, and that reentrant behavior does not exist in the phase diagrams.     

Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model

An approach to the definition of infinite-volume Gibbs states for the (quenched) random-field Ising model is considered in the case of a Curie-Weiss ferromagnet. It turns out that these states are random quasi-free measures. They are random convex linear combinations of the free product-measures shifted by the corresponding effective mean fields. The conditional self-averaging property of the magnetization related to this randomness is also discussed.This paper is dedicated to Robert A. Minlos on the occasion of his 60th birthday.     

Ising model for the alloys according to the effective field theory

FeAl alloys in their disordered structural phase have been investigated through an Ising model where besides exchange interactions between nearest-neighbors Fe atoms, a superexchange interaction mediated by Al atoms is also taken into account. The model has been approximately treated according to the effective field theory. Although the phase diagram, as a function of Al concentration, is similar to the one previously obtained from Bogoliubov variational approach for the free energy, a different behavior for the superexchange interaction is achieved, which can also be physically accepted for this system.     

Variational principle for the distribution function of the effective field for the random Ising model in the Bethe approximation

We study the random Ising model in the pair approximation of the cluster variation method. We show that the distribution function of the effective field is determined either by a reducibility condition of the distribution function of two sites to that of one site or by a stationarity condition of the averaged free energy.