Thermodynamics of some layered Ising models

Thermodynamic properties of some inhomogeneous Ising models with layered distribution of couplings are studied. In particular, the specific heat is investigated in detail, both analytically and numerically. It is shown that different ordering mechanisms, namely ordering of finite regions and global ordering of infinite range, can occur in different temperature ranges. This results in remarkable structures of the specific heat curves. In addition we investigate the case where the random distribution of couplings extends over an infinite distance in one space direction. The ordinary Ising singularity then changes to one of infinite order provided the transition temperature remains finite.     

Thermodynamics of the quantum and classical Ising models with skew magnetic field

The thermodynamics of the unitary (normalized spin) quantum and classical Ising models with skew magnetic field, for |J|β?0.9, is derived for the ferromagnetic and antiferromagnetic cases. The high-temperature expansion (β-expansion) of the Helmholtz free energy is calculated up to order β⁷ for the quantum version (spin S≥1/2) and up to order β¹⁹ for the classical version. In contrast to the S=1/2 case, the thermodynamics of the transverse Ising and that of the XY model for S>1/2 are not equivalent. Moreover, the critical line of the T=0 classical antiferromagnetic Ising model with skew magnetic field is absent from this classical model, at least in the temperature range of |J|β?0.9.     

Thermodynamics of layered strong coupling superconductors

Using the full Eliashberg equations valid for layered compounds, we calculate the effect of various simple anisotropy models on the critical temperature and thermodynamic properties. The changes introduced by the layered structure can be large, but become less significant as the coupling is increased.     

Critical temperatures of Ising models

A general and simple method for calculating critical points of two-dimensional Ising models is presented. As an example we derive Utiyama's critical relation for the generalized square lattice.     

Exact treatment of some phenomenological models of layered high-Tc superconductors

Existing treatments of phenomenological models of Bi and Tl based superconductors consisting of n layers of copper oxide are unsatisfactory because they ignore the important φ⁴ terms in the free energy. An exact treatment of these models is presented, and it is shown that the condition of minimum free energy automatically excludes (n − 1) of the n critical temperatures. The critical exponent β of the order parameter is found to be independent of n and equal to as usual in the Landau theory. The specific heat discontinuity Δc at the critical temperature is calculated as a function of n. A meaningful comparison with experimental measurements of Δc at the moment is hindered by the limited availability of experimental results. Reliable determinations of Δc of a given series of layered superconductors will enable a check of the validity of theoretical results.     

Universality classes of Ising models

We discuss a transformation of Ising spins which maps a d-dimentional Ising problem into a series of different problems in the same universality class.     

Correlations in inhomogeneous Ising models

We study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant either in horizontal or in diagonal direction, i.e. the models have a layered structure. By using transfer matrix techniques the spin-spin correlations are calculated parallel to the layering and are expressed as Toeplitz determinants. After working out the general methods we discuss two special examples in detail: the fully frustrated square lattice (FFS) and the chessboard model, both having no phase transition. At zero temperature correlations in the chessboard model decay exponentially, while in the FFS model one has algebraic decay with a critical index =1/2, i.e.T=0 is a critical point. At finite temperature we find exponential decay in both models with a correlation length determined by the excitation gap in the fermion spectrum. Due to frustration correlations may develop on oscillatory structure and spins separated by an odd diagonal distance are totally uncorrelated at all temperatures.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köla     

Correlations in inhomogeneous Ising models

Using general methods developed in a previous treatment we study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings can vary both in strength and sign such that the coupling distribution is translationally invariant in horizontal direction. We calculate correlations parallel to the layering in the horizontally layered model with periodv=2. If the model has a finite critical temperature,T _c>0, the order parameter in the frustrated case may become discontinuous forT0. Correlations atT=T _c decay algebraically with critical exponent =1/4 and exponentially forT>T _c. If the critical temperature vanishes,T _c=0, we always have exponential decay at finite temperatures, while atT=T _c=0 we encounter either long-range order or algebraic decay with critical index =1/2, i.e.T=0 is thus a critical point.Work performed within the research program of the Sonder forschungsbereich 125 Aachen-Jülich-Köln     

Correlations in inhomogeneous Ising models

Using general methods developed in a previous treatment we study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings can vary both in strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate correlations parallel to the layering in the diagonally layered model with periodv=2, the so-called “general square lattice” model (GS). If the model has a finite critical temperature,T _c>0, we have a spontaneous magnetization belowT _c vanishing atT _c with the Ising exponent β=1/8. AtT _c correlations decay algebraically with critical exponnet η=1/4 and exponentially forT>T _c. In the frustrated case we have oscillatory behaviour superposed on the exponential decay where the wavevector of the oscillations changes at some “disorder temperature”T _D(>T _c) from commensurate to temperature-dependent in commensurate periods. If the critical temperature vanishes,T _c=0 we always have exponential decay at finite temperatures, while atT=T _c=0 we encounter either long-range order or algebraic decay with critical index η=1/2, i.e.T=0 is thus a critical point.     

Ising models on hyperbolic graphs

We consider Ising models on a hyperbolic graph which, loosely speaking, is a discretization of the hyperbolic planeH ² in the same sense asZ ^d is a discretization ofR ^d . We prove that the models exhibit multiple phase transitions. Analogous results for Potts models can be obtained in the same way.     

Self-similar potentials and Ising models

A new link between soliton solutions of integrable nonlinear equations and one-dimensional Ising models is established. Translational invariance of the spin lattice associated with the KdV equation is related to self-similar potentials of the Schrödinger equation. This gives antiferromagnets with exponentially decaying interaction between the spins. The partition function is calculated exactly for a uniform magnetic field and two discrete values of the temperature.     

Applications of the interface approach to some frustrated Ising models on a simple cubic lattice

Three 3-dimensional frustrated Ising models are studied by the interface method. We calculate the interface free energies by Monte Carlo simulation, and estimate the critical temperatures with a size-dependent analysis of the interface free energies.     

Some properties of random Ising models

We consider an Ising model with random magnetic fieldh _i and random nearest-neighbor couplingsJ _ij . The random variablesh _i andJ _ij are independent and identically distributed with a nice enough distribution, e.g., Gaussian. We will prove that (i) at high temperature, infinite volume correlation functions are independent on the boundary conditions and decay exponentially fast with probability 1 and (ii) for any temperature with sufficiently strong magnetic field the correlation functions are again independent on the boundary conditions and decay exponentially fast with probability 1. We also prove that the averaged magnetization of the ground state configuration of the one-dimensional Ising model with random magnetic field is zero, no matter how small is the variance of theh _i .     

Thermodynamics of the Ising Model Encoded in Restricted Boltzmann Machines

The restricted Boltzmann machine (RBM) is a two-layer energy-based model that uses its hidden–visible connections to learn the underlying distribution of visible units, whose interactions are often complicated by high-order correlations. Previous studies on the Ising model of small system sizes have shown that RBMs are able to accurately learn the Boltzmann distribution and reconstruct thermal quantities at temperatures away from the critical point Tc. How the RBM encodes the Boltzmann distribution and captures the phase transition are, however, not well explained. In this work, we perform RBM learning of the 2d and 3d Ising model and carefully examine how the RBM extracts useful probabilistic and physical information from Ising configurations. We find several indicators derived from the weight matrix that could characterize the Ising phase transition. We verify that the hidden encoding of a visible state tends to have an equal number of positive and negative units, whose sequence is randomly assigned during training and can be inferred by analyzing the weight matrix. We also explore the physical meaning of the visible energy and loss function (pseudo-likelihood) of the RBM and show that they could be harnessed to predict the critical point or estimate physical quantities such as entropy.     

Thermodynamics of Ising rare-earth magnet in the static fluctuation approximation

The authors obtain the closed self-consistent system of equations for calculation of the desired thermodynamic values of the Ising magnet with arbitrary spin and in presence of the single-ion anisotropy in the frame of the original static fluctuation approach (SFA). The essence of the SFA is based on accurate calculations of fluctuations of a molecular field. As an example the complete analysis of the critical behavior of the Ising magnet with spin S = 1 (Blume-Capiel model) is shown. The expressions for the phase transitions lines of the first and the second order are found. In addition, the coordinates of the tricritical point for some types of cubic lattices are found also.     

Elastic anomalies in compressible Ising models

The renormalization-group-study indications of critical elastic anomalies (softening) inducing first order transitions in compressible Ising systems are examined here and are compared with the results obtained by employing other methods. The indications are concluded to be spurious.