Partition Function Of Lattice Model Research Articles

Correspondence between spanning trees and the Ising model on a square lattice.

An important problem in statistical physics concerns the fascinating connections between partition functions of lattice models studied in equilibrium statistical mechanics on the one hand and graph theoretical enumeration problems on the other hand. We investigate the nature of the relationship between the number of spanning trees and the partition function of the Ising model on the square lattice. The spanning tree generating function T(z) gives the spanning tree constant when evaluated at z=1, while giving the lattice green function when differentiated. It is known that for the infinite square lattice the partition function Z(K) of the Ising model evaluated at the critical temperature K=K_{c} is related to T(1). Here we show that this idea in fact generalizes to all real temperatures. We prove that [Z(K)sech2K]^{2}=kexp[T(k)], where k=2tanh(2K)sech(2K). The identical Mahler measure connects the two seemingly disparate quantities T(z) and Z(K). In turn, the Mahler measure is determined by the random walk structure function. Finally, we show that the the above correspondence does not generalize in a straightforward manner to nonplanar lattices.

Quiver gauge theories and integrable lattice models

We discuss connections between certain classes of supersymmetric quiver gauge theories and integrable lattice models from the point of view of topological quantum field theories (TQFTs). The relevant classes include 4d $\mathcal{N} = 1$ theories known as brane box and brane tilling models, 3d $\mathcal{N} = 2$ and 2d $\mathcal{N} = (2,2)$ theories obtained from them by compactification, and 2d $\mathcal{N} = (0,2)$ theories closely related to these theories. We argue that their supersymmetric indices carry structures of TQFTs equipped with line operators, and as a consequence, are equal to the partition functions of lattice models. The integrability of these models follows from the existence of extra dimension in the TQFTs, which emerges after the theories are embedded in M-theory. The Yang-Baxter equation expresses the invariance of supersymmetric indices under Seiberg duality and its lower-dimensional analogs.

Quantum algorithms for spin models and simulable gate sets for quantum computation

We present elementary mappings between classical lattice models and quantum circuits. These mappings provide a general framework to obtain efficiently simulable quantum gate sets from exactly solvable classical models. For example, we recover and generalize the simulability of Valiant's match-gates by invoking the solvability of the free-fermion eight-vertex model. Our mappings furthermore provide a systematic formalism to obtain simple quantum algorithms to approximate partition functions of lattice models in certain complex-parameter regimes. For example, we present an efficient quantum algorithm for the six-vertex model as well as a 2D Ising-type model. We finally show that simulating our quantum algorithms on a classical computer is as hard as simulating universal quantum computation (i.e. BQP-complete).

Distribution of Runs and Longest Runs

Exact distributions of run statistics are traditionally obtained using combinatorial methods, which, under certain situations, become very tedious. Run distributions of multiple object systems, although appearing frequently in applications from various fields, such as computational biology, are not commonly used, due in part to the lack of easy-to-use formulas. In this article, a method for evaluating partition functions of lattice models in the field of statistical mechanics is used to develop a systematic method to study various run statistics in multiple object systems. By using particular generating functions for the specified situation under study, many new distributions can be obtained in a unified and coherent way. The method makes it possible to manipulate formulas of run statistics by using binomial identities to obtain more general, yet simpler formulas. To illustrate the applications of the general method, the distributions of the total number of runs and the longest runs are investigated. Novel and general explicit formulas are derived for the distribution and moments of the total number of runs, and simple explicit formulas are derived for the distributions of the longest runs. In addition, some classical run statistics are recovered and generalized in the same unified way. As examples of applications to biological sequence analysis, the run statistics developed using the general method are applied to several protein sequences to examine their global and local features.

Recurrence relations for the three-dimensional Ising-like model in the external field

The method for calculation of the partition function of lattice model for the magnet in the external field near critical point (CP) is proposed. The recurrence relations and their explicit solution near the critical point are founded. It is shown that dependence on temperature of thermodynamic functions near CP, when the field value comes down to zero, is in good agreement with the previous results obtained using the collective variable method. The phase transition temperature (when h = 0) is calculated and the dependence on parameters of interaction potential is found.