On the thermodynamics of the random one-dimensional Ising chain in a transverse field

For three simple one-dimensional disordered models: (a) the Ising chain with random magnetic moments in a transverse field, (b) the Ising chain with random coupling constants in a transverse field, and (c) the X-Y model with a special type of disorder, the asymptotic equivalence in the thermodynamic limit is proved and some of its consequences are discussed. The spectral density of the finite chain for the model (a) is calculated by Dean's method for several representative cases and the presence of the local modes is indicated. The expressions for the initial susceptibilities for the models (a) and (b) are reviewed and (in two cases) the derivations are simplified.     

Stabilization of a steady state in oscillators coupled by a digital delayed connection

We propose the mapping of polynomial of degree 2S constructed as a linear combination of powers of spin-S (for simplicity, we called as spin-S polynomial) onto spin-crossover state. The spin-S polynomial in general can be projected onto non-symmetric degenerated spin up (high-spin) and spin down (low-spin) momenta. The total number of mapping for each general spin-S is given by 2(2^2S ? 1). As an application of this mapping, we consider a general non-bilinear spin-S Ising model which can be transformed onto spin-crossover described by Wajnflasz model. Using a further transformation we obtain the partition function of the effective spin-1/2 Ising model, making a suitable mapping the non-symmetric contribution leads us to a spin-1/2 Ising model with a fixed external magnetic field, which in general cannot be solved exactly. However, for a particular case of non-bilinear spin-S Ising model could become equivalent to an exactly solvable Ising model. The transformed Ising model exhibits a residual entropy, then it should be understood also as a frustrated spin model, due to competing parameters coupling of the non-bilinear spin-S Ising model.     

Graphical Representations for Ising Systems in External Fields

A graphical representation based on duplication is developed that is suitable for the study of Ising systems in external fields. Two independent replicas of the Ising system in the same field are treated as a single four-state (Ashkin–Teller) model. Bonds in the graphical representation connect the Ashkin–Teller spins. For ferromagnetic systems it is proved that ordering is characterized by percolation in this representation. The representation leads immediately to cluster algorithms; some applications along these lines are discussed.     

Constant coupling approximation for antiferromagnets with mixed ferro- and antiferromagnetic interactions

The constant-coupling approximation is extended to an antiferromagnetic spin-$\text{1}\text{2}$ system with two distinct anisotropic exchange interactions. The thermodynamic properties such as the transition temperature, magnetization, susceptibility and specific heat are discussed for three special cases: (i) ferro- and antiferromagnetic Ising interactions, (ii) isotropic ferro- and antiferromagnetic Heisenberg interactions; and (iii) isotropic ferromagnetic Heisenberg interactions and antiferromagnetic Ising interactions, allowing in each case for two different nearest-neighbour coupling constants. Numerical calculations have been performed for a layer structure with z = 6 intraplanar and z' = 6 interplanar nearest neighbours and the results are compared with those obtained in other approximations. Applying the theory to FeCl₂, the exchange constants are evaluated. It is shown that the calculated magnitudes of the interactions strongly depend upon the exchange-interaction model assumed.     

Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line

We study the critical two-dimensional Ising model with a defect line (altered bond strength along a line) in the continuum limit. By folding the system at the defect line, the problem is mapped to a special case of the critical Ashkin-Teller model, the continuum limit of which is the Z₂ orbifold of the free boson, with a boundary. Possible boundary states on the Z₂ orbifold theory are explored, and a special case is applied to the Ising defect problem. We find the complete spectrum of boundary operators, exact two-point correlation functions and the universal term in the free energy of the defect line for arbitrary strength of the defect. We also find a new universality class of defect lines. It is conjectured that we have found all the possible universality classes of defect lines in the Ising model. Relative stabilities among the defect universality classes are discussed.     

Numerical investigation of logarithmic corrections in two-dimensional spin models

The analysis of correlation function data obtained by Monte Carlo simulations of the two-dimensional four-state Potts model, XY model, and self-dual disordered Ising model at criticality are presented. We study the logarithmic corrections to the algebraic decay exhibited in these models. A conformal mapping is used to relate the finite-geometry information to that of the infinite plane. Extraction of the leading singularity is altered by the expected logarithmic corrections, and we show numerically that both leading and correction terms are mutually consistent.     

Eight-vertex model in lattice statistics and one-dimensional anisotropic heisenberg chain. II. Equivalence to a generalized ice-type lattice model

We establish an equivalence between the zero-field eight-vertex model and an Ising model (with four-spin interaction) in which each spin has L possible values, labeled 1, …, L, and two adjacent spins must differ by one (to modulus L). Such an Ising model can also be thought of as a generalized ice-type model and we will later show that the eigenvectors of the transfer matrix can be obtained by a Bethe-type ansatz.     

Symmetric vertex models on planar random graphs

We solve a 4-(bond)-vertex model on an ensemble of 3-regular (Φ³) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent – a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models).The relations between the vertex weights and Ising model parameters in the 4-vertex model on Φ³ graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model.Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.     

Beyond Inverse Ising Model: Structure of the Analytical Solution

I consider the problem of deriving couplings of a statistical model from measured correlations, a task which generalizes the well-known inverse Ising problem. After reminding that such problem can be mapped on the one of expressing the entropy of a system as a function of its corresponding observables, I show the conditions under which this can be done without resorting to iterative algorithms. I find that inverse problems are local (the inverse Fisher information is sparse) whenever the corresponding models have a factorized form, and the entropy can be split in a sum of small cluster contributions. I illustrate these ideas through two examples (the Ising model on a tree and the one-dimensional periodic chain with arbitrary order interaction) and support the results with numerical simulations. The extension of these methods to more general scenarios is finally discussed.     

GPU-based single-cluster algorithm for the simulation of the Ising model

We present the GPU calculation with the common unified device architecture (CUDA) for the Wolff single-cluster algorithm of the Ising model. Proposing an algorithm for a quasi-block synchronization, we realize the Wolff single-cluster Monte Carlo simulation with CUDA. We perform parallel computations for the newly added spins in the growing cluster. As a result, the GPU calculation speed for the two-dimensional Ising model at the critical temperature with the linear size L = 4096 is 5.60 times as fast as the calculation speed on a current CPU core. For the three-dimensional Ising model with the linear size L = 256, the GPU calculation speed is 7.90 times as fast as the CPU calculation speed. The idea of quasi-block synchronization can be used not only in the cluster algorithm but also in many fields where the synchronization of all threads is required.     

On clusters and critical dimensions in spin glasses

We discuss the problem of critical dimensions in the Sherrington-Kirkpatrick model of a spin glass, using the presence of clusters above T_c. It is argued that these clusters are ramified. This fact accounts for the different (upper) critical dimensions of the spherical and Ising models. The special role of dimension four is pointed out.     

A reformulation of the Feynman chessboard model

The chessboard model is reviewed and reformulated as a four-state process. In this formulation both the Dirac propagator of the chessboard model and the partition function of the associated Ising chain are observed to be projections of a single matrix of partition functions onto two orthogonal eigenspaces. This helps clarify the role played by the phase associated with Feynman paths in this model.     

Spectral self-similarity,one-dimensional Ising chains and random matrices

Partition functions of one-dimensional Ising chains with specific long distance exchange between N spins are connected to the N-soliton τ-functions of the Korteweg-de Vries (KdV) and B-type Kadomtsev-Petviashvili (BKP) integrable equations. The condition of translational invariance of the spin lattice selects infinite-soliton solutions with soliton amplitudes forming a number of geometric progressions. The KdV equation generates a spin chain with exponentially decaying antiferromagnetic exchange. The BKP case is richer. It comprises both ferromagnets and anti ferromagnets and, as a special case, includes an exchange decaying as 1/(i − j)² for large |i − j|. The corresponding partition functions are calculated exactly for a homogeneous magnetic field and some fixed values of the temperature. The connection between these Ising chains and random matrix models is considered as well. A short account of the basic ideas underlying the present work has been published in JETP Lett. 66 (1997) 789.     

A comparative study of the dynamic critical behavior of the four-state Potts like models

We investigate the short-time critical dynamics of the Baxter-Wu (BW) and n=3 Turban (3TU) models to estimate their global persistence exponent θ_g. We conclude that this new dynamical exponent can be useful in detecting differences between the critical behavior of these models which are very difficult to obtain in usual simulations. In addition, we estimate again the dynamical exponents of the four-state Potts (FSP) model in order to compare them with results previously obtained for the BW and 3TU models and to decide between two sets of estimates presented in the current literature. We also revisit the short-time dynamics of the 3TU model in order to check if, as already found for the FSP model, the anomalous dimension of the initial magnetization x₀ could be equal to zero.     

Decomposition of Ising spin of spin-S Ising model

We propose a general method by which an Ising spin of a spin-S Ising model is decomposed into Ising spins less than S. Some exactly solvable spin-S Ising models with S greater than 1/2 are obtained without using Horiguchi's condition or its extended conditions. These systems are reduced to the Ising model of spin ± 1 or an Ashkin-Teller model.     

Granger causality and the inverse Ising problem

The inference of the couplings of an Ising model with given means and correlations is called the inverse Ising problem. This approach has received a lot of attention as a tool to analyze neural data. We show that autoregressive methods may be used to learn the couplings of an Ising model, also in the case of asymmetric connections and for multispin interactions. We find that, for each link, the linear Granger causality is two times the corresponding transfer entropy (i.e., the information flow on that link) in the weak coupling limit. For sparse connections and a low number of samples, the ?₁ regularized least squares method is used to detect the interacting pairs of spins. Nonlinear Granger causality is related to multispin interactions.     

Two-dimensional decorated Ising model with classical vector spins and Ising spins of magnitude s

A two-dimensional decorated Ising model with ν-dimensional vector spins and Ising spins of magnitude s is considered. The partition function, magnetization and correlation functions are expressed in terms of the average of functions of the spins of the Ising model with effective exchange constants. These results, although derived for a two-dimensional lattice, are valid for a lattice of arbitrary dimensionality. The phase diagram is obtained exactly in the zero external field and two-dimensional lattice for arbitrary values of s and ν, and, as expected, three transition temperatures are obtained for some values of the parameters. It is also shown that for $\text{|}\text{S}\text{|=1, s>}\text{1}\text{2}$ there is an additional ordered phase (up-down/up-down), and for $\text{|}\text{S}\text{|=ν}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ this additional phase can be either up-down/up-down or up-up/down-down depending on the values of ν and s.     

Generalized transformation for decorated spin models

The paper discusses the transformation of decorated Ising models into an effective undecorated spin model, using the most general Hamiltonian for interacting Ising models including a long range and high order interactions. The inverse of a Vandermonde matrix with equidistant nodes [−s,s] is used to obtain an analytical expression of the transformation. This kind of transformation is very useful to obtain the partition function of decorated systems. The method presented by Fisher is also extended, in order to obtain the correlation functions of the decorated Ising models transforming into an effective undecorated Ising model. We apply this transformation to a particular mixed spin-(1/2, 1) and (1/2, 2) square lattice with only nearest site interaction. This model could be transformed into an effective uniform spin-S square lattice with nearest and next-nearest interaction, furthermore the effective Hamiltonian also includes combinations of three-body and four-body interactions; in particular we considered spin 1 and 2.