Commuting transfer matrices for the four-state self-dual chiral Potts model with a genus-three uniformizing fermat curve

We consider the diagonal-to-diagonal transfer matrix of the four-state self-dual chiral Potts model on a square lattice. We show that there exists a family of models parametrized by a spectral variable u and an auxiliary variable b such that the commutation requirement [T(u,b), T(u′,b)] = 0 needed for the existence of an infinite number of commuting constants of the motion holds. This family is uniformized by the curve x⁴+y⁴ = 1 for all values of b except 0, 1 and ∞, where b = 1 corresponds to the critical self-dual model of Fateev and Zamolodchikov.     

Corner transfer matrices of the chiral Potts model

We present some symmetry and factorization relations satisfied by the corner transfer matrices (CTMs) of the chiral Potts model. We show how the single-spin expectation values can be expressed in terms of the CTMs, and in terms of the related boost operator. Low-temperature calculations lead naturally to the variables that uniformize the Boltzmann weights of the model.     

Corner transfer matrices of the chiral Potts model. II. The triangular lattice

We consider a two-dimensional edge-interaction model satisfying the star-triangle relations. For the triangular lattice, the corner transfer matrices are functions of three rapidities: we show that they possess various factorization properties and satisfy certain equations. We indicate how these equations can be solved for the Ising model. We then consider the three-state chiral Potts model and obtain low-temperature solutions to the equations. The conjectured formula for the order parameter (the spontaneous magnetization) is verified to one more order in a series expansion.     

Depinning and wetting in two-dimensional Potts and chiral clock models

The interplay of depinning and interfacial adsorption or wetting phenomena is studied for two-dimensional three-state Potts and chiral clock models where the variables on opposite boundaries are fixed in different states and the interactions near one of the surfaces are weakened compared to the ones in the bulk. Using a transfer matrix approach and Monte Carlo techniques a new interfacial multicritical point is found at which both interfacial properties become critical simultaneously. However, in general the two types of transitions are decoupled.     

Critical behaviour of random-bond Potts models: a transfer matrix study

We study the two-dimensional Potts model on the square lattice in the presence of quenched random-bond impurities. For q > 4 the first-order transitions of the pure model are softened due to the impurities, and we determine the resulting universality classes by combining transfer matrix data with conformal invariance. The magnetic exponent β/v varies continuously with q, assuming non-Ising values for q > 4, whereas the correlation length exponent ν is numerically consistent with unity. We present evidence for the correctness of a formerly proposed phase diagram, unifying pure, percolative and non-trivial random behaviour.     

Interfacial tension of the chiral Potts model

We calculate the interfacial tension of theN-state chiral Potts model by solving the functional relations for the transfer matrices of the model with skewed boundary conditions. Our result is valid for the general physical model (with positive Boltzmann weights) and at all subcritical temperatures. The interfacial tension has been calculated previously for the superintegrable chiral Potts model with skewed boundary conditions. UsingZ-invariance, Baxter has argued that the interfacial tension of this model should be the same as the interfacial tension of the general physical model. We show that this is indeed the case.     

Analyticity and integrability in the chiral Potts model

We study the perturbation theory for the general nonintegrable chiral Potts model depending on two chiral angles and a strength parameter and show how the analyticity of the ground-state energy and correlation functions dramatically increases when the angles and the strength parameter satisfy the integrability condition. We further specialize to the superintegrable case and verify that a sum rule is obeyed.     

Families of commuting transfer matrices and integrable models with disorder

A family of commuting transfer matrices is shown to be associated to each symmetry transformation of a given Yang-Baxter algebra. This applies in lattices models and field theory.The Yang-Baxter algebra remains unchanged when an arbitrary parameter μ_l is associated to each lattice site. We generate in this way integrable one-dimensional hamiltonians with long-range couplings and disorder given by the <{;μ₁<};. These operators are lattice versions of the non-local charges in sigma models. As a simple example we get a Dzialozhinski-Moriya interaction with an arbitrary coupling per site from the six-vertex model. A similar model with a disordered magnetic field follows too. Their exact solution by an algebraic Bethe ansatz is presented. We derive the excitations spectrum in terms of the density of parameters (μ).As another application, the total spin S² is computed for a XXZ Heisenberg chain (μ_l ≡ 0) as a function of the anisotropy Δ (− ∞ < Δ < + ∞).     

Free energy of the solvable chiral Potts model

Very recently, it has been shown that there are chiralN-state Potts models in statistical mechanics that satisfy the star-triangle relation. Here it is shown that the relation implies that the free energy (and its derivatives) satisfies certain functional relations. These can be used to obtain the free energy: in particular, we expand about the critical case and find that the exponent is 1–2/N.     

(Z N×) n−1 generalization of the chiral Potts model

We show that theR-matrix which intertwines twon-by-N ^n–1 state cyclicL-operators related with a generalization ofU _q(sl(n)) algebra can be considered as a Boltzmann weight of four-spin box for a lattice model with two-spin interaction just as theR-matrix of the checkerboard chiral Potts model. The rapidity variables lie on the algebraic curve of the genusg=N ^{2(n–1)((n–1)N-n)+1} defined by 2n–3 independent moduli. This curve is a natural generalization of the curve which appeared in the chiral Potts model. Factorization properties of theL-operator and its connection to the SOS models are also discussed.     

Free energy of the chiral Potts model in the scaling region

We explicitly calculate the free energy of the general solvableN-state chiral Potts model in the scaling region, forT<T _c . We do this from both of the two available results for the free energy, and verify that they are mutually consistent. Ift=T _c –T, then we find that - _c /t has a Taylor expansion in powers oft ^2/N (together with higher-order non-scaling terms of ordert, ort logt).     

Derivation of the order parameter of the chiral Potts model

We derive the order parameter of the chiral Potts model, using the method of Jimbo et al. The result agrees with previous conjectures.     

Excitation spectrum and phase structure of the chiral Potts model

The spectrum of low lying excitations of the integrable chiral Potts model is computed. It is shown that there is a region in the parameter space where the excitation energies become negative thus indicating that a level crossing transition to a new incommensurate ground state has occurred.     

A characterization of “rapidity” curve in the chiral Potts model

We study the geometry of high genus curves of rapidity variables in chiral Potts model. In terms of symmetries, we characterize these Rieman surfaces and derive their period matrices. By the theory of prime forms, the temperature-like parameter is expressed by hyperelliptic theta functions.     

Self-dual renormalization group analysis of the Potts models

We study the Potts models P(N) in 1 + 1 dimensions using a self-dual renormalization-group (RG) approach. The method is very successful in calculating the critical index ν. For N > 4 a pseudocritical behaviour is obtained.     

Potts models with magnetic field: Arithmetic,geometry, and computation

We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the vanishing of the partition function is affected by changes in the magnetic field: elementary examples suffice to see non-polynomially countable cases that become polynomially countable after a perturbation of the magnetic field. The same recursive formula for the Grothendieck classes, under edge-doubling operations, holds as in the case without magnetic field, but the closed formulae for specific examples like banana graphs differ in the presence of magnetic field. We give examples of computation of the Euler characteristic with compact support, for the set of real zeros, and find a similar exponential growth with the size of the graph. This can be viewed as a measure of topological and algorithmic complexity. We also consider the computational complexity question for evaluations of the polynomial, and show both tractable and NP-hard examples, using dynamic programming.     

Statistical mechanics of one-dimensional Ising and Potts models with exponential interactions

This paper continues the authors' work on a new method for discussing one-dimensional systems in statistical mechanics with exponentially decreasing interactions. It is shown how in the case of the S-spin Ising and the N-state Potts model the results in the classic paper of Kac et al. for these models emerge also from our method. It is the aim of the present paper to compare these two mathematically completely different methods and prepare the extension of our method to two-dimensional systems.     

Solutions of the Einstein equations with zero coupling

A study is made of the algebraic properties and geometric structure of solutions of the Einstein equations the metric tensors of which differ by the product of two identical isotropic vectors. Proof is offered for a theorem which states that when the congruence of isotropic lines with a tangent vector field used for the coupling is geodesic, both spaces are algebraically special in the Petrov-Penrose sense. A noncoordinate transformation of this type can be used to find new exact solutions from known solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 10, pp. 32–37, October, 1971.The author thanks Professor V. I. Rodichev for interest in the study and for valuable discussions.