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arXiv:1301.1252 · Physical Review E

Numerically exact correlations and sampling in the two-dimensional Ising spin glass

Creighton K. Thomas, A. Alan Middleton

Published Apr 5, 2013• 11 citations
arXivHTMLPDF

Abstract

A powerful existing technique for evaluating statistical mechanical quantities in two-dimensional Ising models is based on constructing a matrix representing the nearest-neighbor spin couplings and then evaluating the Pfaffian of the matrix. Utilizing this technique and other more recent developments in evaluating elements of inverse matrices and exact sampling, a method and computer code for studying two-dimensional Ising models is developed. The formulation of this method is convenient and fast for computing the partition function and spin correlations. It is also useful for exact sampling, where configurations are directly generated with probability given by the Boltzmann distribution. These methods apply to Ising model samples with arbitrary nearest-neighbor couplings and can also be applied to general dimer models. Example results of computations are described, including comparisons with analytic results for the ferromagnetic Ising model, and timing information is provided.

DOI
10.1103/physreve.87.043303

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A powerful existing technique for evaluating statistical mechanical quantities in two-dimensional Ising models is based on constructing a matrix representing the nearest-neighbor spin couplings and then evaluating the Pfaffian of the matrix. Utilizing this technique and other more recent developments in evaluating elements of inverse matrices and exact sampling, a method and computer code for studying two-dimensional Ising models is developed. The formulation of this method is convenient and fast for computing the partition function and spin correlations. It is also useful for exact sampling, where configurations are directly generated with probability given by the Boltzmann distribution. These methods apply to Ising model samples with arbitrary nearest-neighbor couplings and can also be applied to general dimer models. Example results of computations are described, including comparisons with analytic results for the ferromagnetic Ising model, and timing information is provided.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.
In an effort to make statistical methods available for the treatment of cooperational phenomena, the Ising model of ferromagnetism is treated by rigorous Boltzmann statistics. A method is developed which yields the partition function as the largest eigenvalue of some finite matrix, as long as the manifold is only one dimensionally infinite. The method is carried out fully for the linear chain of spins which has no ferromagnetic properties. Then a sequence of finite matrices is found whose largest eigenvalue approaches the partition function of the two-dimensional square net as the matrix order gets large. It is shown that these matrices possess a symmetry property which permits location of the Curie temperature if it exists and is unique. It lies at JkTc=0.8814\frac{J}{k{T}_{c}}=0.8814kTc​J​=0.8814 if we denote by JJJ the coupling energy between neighboring spins. The symmetry relation also excludes certain forms of singularities at Tc{T}_{c}Tc​, as, e.g., a jump in the specific heat. However, the information thus gathered by rigorous analytic methods remains incomplete.
Two different methods of obtaining the partition function for a spin ½ open ended Ising chain with interactions between nearest and second nearest neighbour spins are presented. These are non-matrix methods, in that they do not require the diagonalization of a transfer matrix, and they provide a useful representation for studying the analytical structure of the partition function. The pair correlation functions are also derived, using the second of these methods. These correlation functions are utilized in a study of the ground state and the finite temperature spin configurations. An interesting application of these results to the problem of out-of-phase domain structure in binary alloys is discussed.
The partition functions of many lattice statistical problems, such as the Ising models, the dimer model, and the ferroelectric model, have been written in the following general form:[Formula: see text]For models that can be solved by the pfaffian technique, H(j) is a quadratic function of fermion operators, but otherwise H(j) also contains quartic functions of fermion operators, making the above expression intractable to exact evaluation. General field theoretical perturbation methods, such as the Green's function techniques and the diagonalization (by a Fourier transformation) of the Hamiltonian, Ho(j), are developed in relation to the above expression. It is shown that the partition function for unsolved problems can be expressed in terms of an "irreducible vertex part" for which low-order approximations can be obtained. This general formalism is applied to the next-nearest neighbor two-dimensional Ising lattice. A first-order approximation is obtained for the irreducible vertex part, and the critical properties are obtained. For the equivalent-neighbor model, the critical temperature obtained is in excellent agreement with the series expansion results, being only 0.4 % higher.
An algorithm to make exact numerical evaluations for finite-size two-dimensional Ising models with arbitrary bond configurations under the toroidal periodic boundary condition is proposed. The algorithm is based on a modified version of the Cholesky-like factorization (MCLF) method and columnwise matrix inversions of very sparse skew-symmetric matrices. This enables us to exactly evaluate the internal energy, the specific heat and the entropy without use of any numerical differentiation. It also provides a general method to get the numerical value of a Pfaffian with its sign. The amount of memory and the CPU time required to get a result is measured to be proportional to N1.5 ∼1.8 and N2.4 ∼2.8, respectively, where N is the number of the spins. As a demonstration we show temperature dependencies of the specific heat and the internal energy along the Nishimori line for ±J models on lattices with sizes 25 ×25, 50 ×50 and 100 ×100, and we know the performance achieved is very high. Size-dependent anomalous behaviors cannot be seen in the result for the specific heat.