Chiral Potts Model Research Articles

STAR-TRIANGLE EQUATIONS AND IDENTITIES IN HYPERGEOMETRIC SERIES

In this paper, we introduce the cyclic basic hypergeometric series p + 1Φp with q → ω where ωN = 1. This is a terminating series with N terms, whose summand has period N. We show how the Fourier transform of the weights of the integrable chiral Potts model are related to the 2Φ1, which is summable. We show that 3Φ2 satisfies certain transformation formulae. We then show that the Saalschützian 4Φ3 series is summable at argument z = ω. This then gives the simplest proof of the star-triangle relation in the chiral Potts model. Finally, we let N → ∞, where the star-triangle equation becomes a two-sided identity for the hypergeometric series.

ONSAGER'S ALGEBRA AND PARTIALLY ORTHOGONAL POLYNOMIALS

The energy eigenvalues of the superintegrable chiral Potts model are determined by the zeros of special polynomials which define finite representations of Onsager's algebra. The polynomials determining the low-sector eigenvalues have been given by Baxter in 1988. In the ℤ3-case they satisfy 4-term recursion relations and so cannot form orthogonal sequences. However, we show that they are closely related to Jacobi polynomials and satisfy a special "partial orthogonality" with respect to a Jacobi weight function.

Random matrix theory and higher genus integrability: the quantum chiral Potts model

We perform a random matrix theory (RMT) analysis of the quantum fourstate chiral Potts chain for different sizes of the chain up to size L = 8. Ou ra nalysis gives clear evidence of a Gaussian orthogonal ensemble (GOE) statistics, suggesting the existence of a generalized time-reversal invariance. Furthermore, a change from the (generic) GOE distribution to a Poisson distribution occurs when the integrability conditions are met. The chiral Potts model is known to correspond to a (star-triangle) integrability associated with curves of genus higher than zero or one. Therefore, the RMT analysis can also be seen as a detector of ‘higher genus integrability’.

Cluster Monte Carlo: Extending the range

Monte Carlo simulations with local updates tend to become time-consuming when large-scale correlations exist, such as in critical systems. For a limited, but increasing number of model systems, nonlocal ‘cluster’ algorithms are available that are orders of magnitude more efficient than algorithms with local updates. Cluster algorithms can be defined on the basis of the symmetry properties of the Hamiltonian; different symmetries can thus lead to different cluster algorithms. We review a number of existing cluster algorithms, and describe new ones for an Ising-like model with two- and three-spin interactions, and for the chiral Potts model. New simulation data for the Ising-like model allow an accurate determination of its specific-heat exponent; this result confirms existing ideas that the model belongs to the 4-state Potts universality class.

Automorphisms of the Yang-Baxter equations, for the chiral Potts model

For more than a decade now, the chiral Potts model in statistical mechanics has attracted much attention. A number of people have written quite extensively about it. The solutions give rise to a curve over . Au-Yang and Perk found a large subgroup of the automorphism group of this curve. In this letter, we compute the automorphism group precisely.

Algebraic geometry of the three-state chiral Potts model

For more than a decade now, the chiral Potts model in statistical mechanics has attracted much attention. A number of mathematical physicists have written quite extensively about it. The solutions give rise to a curve over ℂ, and much effort has gone into studying the curve and its Jacobian.

DYNAMICS OF CREMONA MAPS FROM PHYSICAL MODELS

A Cremona transformation X=f(x, y), Y=g(x, y) is a rational mapping (meaning that f and g are ratios of polynomials) with rational inverse x=F(X, Y), y=G(X, Y). Discrete dynamical systems defined by such transformations are well studied. They include symmetries of the Yang-Baxter equations and their generalizations. In this paper we comment on two types of dynamical systems based on Cremona transformations. The first is the P1 case of Bellon et al. which pertains to the inversion relation for the matrix of Boltzmann weights of the 4-state chiral Potts model. The resulting dynamical system decouples completely to one in a single variable. The sub case z=x corresponds to the symmetric Ashkin-Teller model. We solve this case explicitly giving orbits as closed formulas in the number n of iterations. The second type of system treated is an extension from the famous example due to McMillan of invariant curves of area preserving maps in two dimensions to the case of invariant curves and surfaces of three dimensional Cremona maps that preserve volume. The trace map of the renormalization of transmission through a Fibonacci chain, first introduced by Kohmoto, Kadanoff and Tang, is considered as an example of such a system.

Strange quark polarization of the nucleon: a parameter-independent prediction of the chiral potential model.

We perform a one-loop calculation of the strange quark polarization (Deltas) of the nucleon in an SU(3) chiral potential model. We find that if the intermediate quark excited states are summed over in a proper way, i.e., summed up to a given energy instead of given radial and orbital quantum numbers, Deltas turns out to be almost independent of all the model parameters: quark masses and potential strengths. The contribution from the quark-antiquark pair creation and annihilation " Z" diagrams is found to be significant. Our numerical results agree quite reasonably with experiments and lattice QCD calculations.

Tunneling between Luttinger liquids

We consider the problem of tunneling between spinless 1D Luttinger liquid chains. We show how to map the problem onto the 4-state chiral clock model together with a free boson for the total charge mode. We use a variety of results, some of them exact, from the integrable chiral Potts model and the study of commensurate/incommensurate transitions to deduce the physics of coupled Luttinger liquids. For those intrachain interaction strengths for which interchain tunneling is relevant, we find that it can lead to the formation of symmetric and antisymmetric bands with split Fermi surfaces, depending on the relative strengths of interchain tunneling and interchain interactions. With split Fermi surfaces, the tunneling is coherent. It is not possible to have two gapless Fermi surfaces with the same Fermi momentum when interchain tunneling is relevant. However, interchain interactions can drive the formation of a gap, in which case interchain transport will be incoherent. We comment on the possible relevance of our results to c-axis transport in high-T_c superconductors.

Baxter's Solution for the Free Energy of the Chiral Potts Model

In honor of Baxter's sixtieth birthday, we would like to review some of his work on the free energy of the chiral Potts model. In spite of the enormous complexity and difficulty of the problem, Baxter, using functional relations was able to calculate not only the free energy, but also the interfacial tension. We here show that the integral for the free energy simplifies in the superintegrable case and is identical to his earlier results using entirely different approaches. His calculations are extended to include other regions. We also attempt to clarify some of his reasoning as several steps may be mysterious at first glance.

Equivalence of the Two Results for the Free Energy of the Chiral Potts Model

The free energy of the chiral Potts model has been obtained in two ways. The first used only the star-triangle relation, symmetries, and invariances, and led to a system of equations that implicitly define the free energy, and from which the critical behavior can be obtained The second used the functional relations derived by Bazhanov and Stroganov, solving them to obtain the free energy explicitly as a double integral. Here we obtain, for the first time, a direct verification that the two results are identical at all temperatures.

INTEGRABLE MODELS IN STATISTICAL MECHANICS: THE HIDDEN FIELD WITH UNSOLVED PROBLEMS

In the past 30 years there have been extensive discoveries in the theory of integrable statistical mechanical models including the discovery of nonlinear differential equations for Ising model correlation functions, the theory of random impurities, level crossing transitions in the chiral Potts model and the use of Rogers–Ramanujan identities to generalize our concepts of Bose/Fermi statistics. Each of these advances has led to the further discovery of major unsolved problems of great mathematical and physical interest. I will here discuss the mathematical advances, the physical insights and extraordinary lack of visibility of this field of physics.

Comment on “Glassy Potts model: A disordered Potts model without a ferromagnetic phase”

We report the equivalence of the ``glassy Potts model,'' recently introduced by Marinari et al. and the ``chiral Potts model'' investigated by Nishimori and Stephen. Both models do not exhibit any spontaneous magnetization at low temperature, differently from the ordinary glass Potts model. The phase transition of the glassy Potts model is easily interpreted as the spin-glass transition of the ordinary random Potts model.

The large- N limits of the chiral Potts model

In this paper we study the large- N limits of the integrable N-state chiral Potts model. Three chiral solutions of the star-triangle equations are derived, with states taken from all integers, or from a finite or infinite real interval. These solutions are expected to be chiral-field lattice deformations of parafermionic conformal field theories. A new two-sided hypergeometric identity is derived as a corollary.

Functional relations and nested Bethe ansatz forsl(3) chiral Potts model at

We obtain the functional relations for the eigenvalues of the transfer matrix of the sl(3) chiral Potts model for q^2=-1. For the homogeneous model in both directions a solution of these functional relations can be written in terms of roots of Bethe ansatz-like equations. In addition, a direct nested Bethe ansatz has also been developed for this case.

Bethe ansatz for the three-layer Zamolodchikov model

This paper is a continuation of our previous work (solv-int/9903001). We obtain two more functional relations for the eigenvalues of the transfer matrices for the $sl(3)$ chiral Potts model at $q^2=-1$. This model, up to a modification of boundary conditions, is equivalent to the three-layer three-dimensional Zamolodchikov model. From these relations we derive the Bethe ansatz equations.

Proofs of some hyperelliptic function identities

We give simple direct proofs of some hyperelliptic function identities conjectured by R.J. Baxter which arose in his study of the chiral Potts model.

Functional relations for the order parameters of the chiral Potts model: low-temperature expansions

This is the third in a series of papers in which we set up and discuss the functional relations for the “split rapidity line” correlation function in the N-state chiral Potts model. The order parameters of the model can be obtained from this function. Here we consider the case N=3 and write the equations explicitly in terms of the hyperelliptic functions parametrization. We also present four-term low-temperature series expansions, which we hope will cast light on the analyticity properties needed to solve the relations. The problem remains unsolved, but we hope that this will prove to be a step in the right direction.

A Lie algebra for closed strings, spin chains, and gauge theories

We consider quantum dynamical systems whose degrees of freedom are described by N×N matrices, in the planar limit N→∞. Examples are gauge theories and the M(atrix)-theory of strings. States invariant under U(N) are “closed strings,” modeled by traces of products of matrices. We have discovered that the U(N)-invariant operators acting on both open and closed string states form a remarkable new Lie algebra which we will call the heterix algebra. (The simplest special case, with one degree of freedom, is an extension of the Virasoro algebra by the infinite-dimensional general linear algebra.) Furthermore, these operators acting on closed string states only form a quotient algebra of the heterix algebra. We will call this quotient algebra the cyclix algebra. We express the Hamiltonian of some gauge field theories (like those with adjoint matter fields and dimensionally reduced pure QCD models) as elements of this Lie algebra. Finally, we apply this cyclix algebra to establish an isomorphism between certain planar matrix models and quantum spin chain systems. Thus we obtain some matrix models solvable in the planar limit; e.g., matrix models associated with the Ising model, the XYZ model, models satisfying the Dolan–Grady condition and the chiral Potts model. Thus our cyclix Lie algebra describes the dynamical symmetries of quantum spin chain systems, large-N gauge field theories, and the M(atrix)-theory of strings.

Some hyperelliptic function identities that occur in the chiral Potts model

We present a number of hyperelliptic function identities that occur in the chiral Potts model. Some of these are relevant to the outstanding problem of solving the functional relations for the order parameter.