On spin and matrix models in the complex plane

We describe various aspects of statistical mechanics defined in the complex temperature or coupling-constant plane. Using exactly solvable models, we analyse such aspects as renormalization group flows in the complex plane, the distribution of partition function zeros, and the question of new coupling-constant symmetries of complex-plane spin models. The double-scaling form of matrix models is shown to be exactly equivalent to finite-size scaling of two-dimensional spin systems. This is used to show that the string susceptibility exponents derived from matrix models can be obtained numerically with very high accuracy from the scaling of finite-N partition function zeros in the complex plane.     

Lee-Yang zeros of periodic and quasiperiodic anisotropic XY chains in a transverse field

The partition function zeros of the anisotropic XY chain in a complex transverse field are studied analytically and numerically. It is found that the partition function zeros of the periodic and quasiperiodic quantum Ising chain lie on the circle at zero temperature and the radius equal to the values of the critical field. For the periodic and quasiperiodic anisotropic XY chains, the closed curves are split to two or three closed curves as the anisotropic parameter gamma decreases at a given ratio of two kinds of exchange interactions. For the isotropic XX case, the partition function zeros lie on the straight segments which are parallel to the real axis and the segments move towards the real axis as the temperature goes to zero.     

Partition Function Zeros at First-Order Phase Transitions: Pirogov—Sinai Theory

This paper is a continuation of our previous analysis(2) of partition functions zeros in models with first-order phase transitions and periodic boundary conditions. Here it is shown that the assumptions under which the results of ref. 2 were established are satisfied by a large class of lattice models. These models are characterized by two basic properties: The existence of only a finite number of ground states and the availability of an appropriate contour representation. This setting includes, for instance, the Ising, Potts, and Blume–Capel models at low temperatures. The combined results of ref. 2 and the present paper provide complete control of the zeros of the partition function with periodic boundary conditions for all models in the above class.     

Theory and calculations for a spin glass

We obtain the properties of a mean-field spin-glass (in which the bonds connecting each spin to every other spin are “frozen-in” with random signs), by locating the zeros of the partition function in the complex T plane. For N = 5 and 9 spins, we obtain the relevant polynomials and zeros explicitly, and the resulting thermodynamic properties (free energy, specific heat, magnetic susceptibility, etc.). We then analyze the properties of such a system in the thermodynamic limit N → ∞, where it is impossible to obtain the polynomials directly but where the presumed location of the zeros can be usefully construed. In this limit, the thermodynamic functions are obtainable as functions of the distribution functions of monopoles, quadrupoles, and possibly higher-order poles.     

Completeness of the classical 2D Ising model and universal quantum computation

We prove that the 2D Ising model is complete in the sense that the partition function of any classical q-state spin model (on an arbitrary graph) can be expressed as a special instance of the partition function of a 2D Ising model with complex inhomogeneous couplings and external fields. In the case where the original model is an Ising or Potts-type model, we find that the corresponding 2D square lattice requires only polynomially more spins with respect to the original one, and we give a constructive method to map such models to the 2D Ising model. For more general models the overhead in system size may be exponential. The results are established by connecting classical spin models with measurement-based quantum computation and invoking the universality of the 2D cluster states.     

Lee–Yang Theorems and the Complexity of Computing Averages

We study the complexity of computing average quantities related to spin systems, such as the mean magnetization and susceptibility in the ferromagnetic Ising model, and the average dimer count (or average size of a matching) in the monomer-dimer model. By establishing connections between the complexity of computing these averages and the location of the complex zeros of the partition function, we show that these averages are #P-hard to compute, and hence, under standard assumptions, computationally intractable. In the case of the Ising model, our approach requires us to prove an extension of the famous Lee–Yang Theorem from the 1950s.     

General theory of lee-yang zeros in models with first-order phase transitions

We present a general, rigorous theory of Lee-Yang zeros for models with first-order phase transitions that admit convergent contour expansions. We derive formulas for the positions and the density of the zeros. In particular, we show that, for models without symmetry, the curves on which the zeros lie are generically not circles, and can have topologically nontrivial features, such as bifurcation. Our results are illustrated in three models in a complex field: the low-temperature Ising and Blume-Capel models, and the q-state Potts model for large q.     

Zeros of the partition function using theorems of Ruelle

Theorems of Ruelle which provide a technique for finding regions of the relevant complex planes free of zeros of the partition function are used to study certain Ising spin systems. Of particular interest is the antiferromagnetic triangle lattice system with h≠0 and systems having three-body interactions.     

On Complex Zeros of the <Emphasis Type="Italic">q</Emphasis>-Potts Partition Function for a Self-dual Family of Graphs

This paper deals with the location of the complex zeros of q-Potts partition function for a class of self-dual graphs. For this class of graphs, as the form of the eigenvalues is known, the regions of the complex plane can be focused on the sets where there is only one dominant eigenvalue in particular containing the positive half plane. Thus, in these regions, the analyticity of the free energy per site can be derived easily. Next, some examples of graphs with their Tutte polynomial having few eigenvalues are given. The case of the cycle with an edge having a high order of multiplicity is presented in detail. In particular, we show that the well known conjecture of Chen et al. is false in the finite case. Furthermore we obtain a sequence of self-dual graphs for which the unit circle does not belong to the accumulation sets of the zeros.     

The Curie–Weiss model with Complex Temperature: Phase Transitions

We study the partition function of the Curie–Weiss model with complex temperature, and partially describe its phase transitions. As a consequence, we obtain information on the locations of zeros of the partition function (the Fisher zeros).     

Lee–Yang zeros in the Rydberg atoms

Lee–Yang (LY) zeros play a fundamental role in the formulation of statistical physics in terms of (grand) partition functions, and assume theoretical significance for the phenomenon of phase transitions. In this paper, motivated by recent progress in cold Rydberg atom experiments, we explore the LY zeros in classical Rydberg blockade models. We find that the distribution of zeros of partition functions for these models in one dimension (1d) can be obtained analytically. We prove that all the LY zeros are real and negative for such models with arbitrary blockade radii. Therefore, no phase transitions happen in 1d classical Rydberg chains. We investigate how the zeros redistribute as one interpolates between different blockade radii. We also discuss possible experimental measurements of these zeros.     

Region Graph Partition Function Expansion and Approximate Free Energy Landscapes: Theory and Some Numerical Results

Graphical models for finite-dimensional spin glasses and real-world combinatorial optimization and satisfaction problems usually have an abundant number of short loops. The cluster variation method and its extension, the region graph method, are theoretical approaches for treating the complicated short-loop-induced local correlations. For graphical models represented by non-redundant or redundant region graphs, approximate free energy landscapes are constructed in this paper through the mathematical framework of region graph partition function expansion. Several free energy functionals are obtained, each of which use a set of probability distribution functions or functionals as order parameters. These probability distribution function/functionals are required to satisfy the region graph belief-propagation equation or the region graph survey-propagation equation to ensure vanishing correction contributions of region subgraphs with dangling edges. As a simple application of the general theory, we perform region graph belief-propagation simulations on the square-lattice ferromagnetic Ising model and the Edwards-Anderson model. Considerable improvements over the conventional Bethe-Peierls approximation are achieved. Collective domains of different sizes in the disordered and frustrated square lattice are identified by the message-passing procedure. Such collective domains and the frustrations among them are responsible for the low-temperature glass-like dynamical behaviors of the system.     

The partition function zeros of the anisotropic Ising model with multisite interactions on a zigzag ladder

It is shown that the spin- anisotropic Ising model with multisite interactions on a zigzag ladder may be mapped into the one dimensional spin- Axial-Next-Nearest-Neighbor Ising (ANNNI) model with multisite interactions. The partition function zeros of the ANNNI model with multisite interactions are investigated. A comprehensive analysis of the partition function zeros of the ANNNI model with and without three-site interactions on a zigzag ladder is done using the transfer matrix method. Analytical equations for the distribution of the partition function zeros in the complex magnetic field (Yang-Lee zeros) and temperature (Fisher zeros) planes are derived. The Yang-Lee and Fisher zeros distributions are studied numerically for a variety of values of the model parameters. The densities of the Yang-Lee and Fisher zeros are studied and the corresponding edge singularity exponents are calculated. It is shown that the introduction of three-site interaction terms in the ANNNI model leads to a simpler distribution of the partition function zeros. For example, the Yang-Lee zeros tend to a circular distribution when increasing by modulus the three-site interactions term coefficient. It is found that the Yang-Lee and Fisher edge singularity exponents are universal and equal to each other, .     

Fisher zeros on the Husimi lattice

Fisher zeros for the partition function with respect to a temperature-dependent parameter are studied. The Ising approximation for Heisenberg model with two- and three-site exchange interactions on the Husimi lattice was used. This model approximates the third layer of ³He, absorbed on the surface of graphite (kagome lattice). Using dynamic approach, we have found an exact recursion relation for the partition function. The presence of a phase transition, both in the real and complex regions on the temperature plane was shown.     

Complex zeros of the partition function for two-dimensionalU(N) lattice gauge theories

The complex zeros of the partition function for the two-dimensionalU(N) lattice gauge model in the variable β or ξ=N/β are calculated analytically for large β and numerically. The zeros formN trajectories which forN→∞ fill a domain of the ξ plane densely. This domain touches the real ξ axis at ξ=1 with a kink that causes the Gross-Witten phase transition.     

Transfer Operators and Dynamical Zeta Functions for a Class of Lattice Spin Models

 We investigate the location of zeros and poles of a dynamical zeta function for a family of subshifts of finite type with an interaction function depending on the parameters . The system corresponds to the well known Kac-Baker lattice spin model in statistical mechanics. Its dynamical zeta function can be expressed in terms of the Fredholm determinants of two transfer operators and with the Ruelle operator acting in a Banach space of holomorphic functions, and an integral operator introduced originally by Kac, which acts in the space with a kernel which is symmetric and positive definite for positive β. By relating via the Segal-Bargmann transform to an operator closely related to the Kac operator we can prove equality of their spectra and hence reality, respectively positivity, for the eigenvalues of the operator for real, respectively positive, β. For a restricted range of parameters we can determine the asymptotic behavior of the eigenvalues of for large positive and negative values of β and deduce from this the existence of infinitely many non-trivial zeros and poles of the dynamical zeta functions on the real β line at least for generic . For the special choice , we find a family of eigenfunctions and eigenvalues of leading to an infinite sequence of equally spaced ``trivial' zeros and poles of the zeta function on a line parallel to the imaginary β-axis. Hence there seems to hold some generalized Riemann hypothesis also for this kind of dynamical zeta functions. Received: 14 March 2002 / Accepted: 24 June 2002 Published online: 14 November 2002     

Partition function zeros of the two-dimensional HP model for protein folding

The partition function of the two-dimensional lattice HP model for protein folding is computed by exact enumeration. For a protein-like sequence, the distribution of partiton function zeros shows roughly a two-ring pattern, while for a nonprotein-like sequence, the outer ring of zeros is ill-developed and cannot induce the folding transition. By tracing the peak or shoulder of the heat capacity in the complex plane, the phase boundaries can be determined along the positive real axis.     

Partition function zeros for the two-dimensional Ising model

The distribution of complex temperature zeros of the partition function of the two-dimensional Ising model in the absence of a magnetic field is investigated. For anisotropic square and triangular lattices the distribution function is two-dimensional and satisfies a partial differential equation derived from a generalized scaling theory. Corresponding results for the isotropic square, triangular and honeycomb lattices are also presented.