Interaction Round A Face Research Articles

Interaction-round-a-face and consistency-around-a-face-centered-cube

There is a correspondence between integrable lattice models of statistical mechanics and discrete integrable equations that satisfy multidimensional consistency, where the latter may be found in a quasi-classical expansion of the former. This paper extends this correspondence to interaction-round-a-face (IRF) models, resulting in a new formulation of the consistency-around-a-cube integrability condition applicable to five-point equations in the square lattice. Multidimensional consistency for these equations is formulated as consistency-around-a-face-centered-cube (CAFCC), which, namely, involves satisfying an overdetermined system of 14 five-point lattice equations for eight unknown variables on the face-centered cubic unit cell. From the quasi-classical limit of IRF models, which are constructed from the continuous spin solutions of the star–triangle relations associated with the Adler–Bobenko–Suris list, 15 sets of equations are obtained, which satisfy CAFCC.

Stochastic PDE limit of the dynamic ASEP

We study a stochastic PDE limit of the height function of the dynamic asymmetric simple exclusion process (dynamic ASEP). A degeneration of the stochastic Interaction Round-a-Face (IRF) model of arXiv:1701.05239, dynamic ASEP has a jump parameter $q\in (0,1)$ and a dynamical parameter $\alpha>0$. It degenerates to the standard ASEP height function when $\alpha$ goes to $0$ or $\infty$. We consider very weakly asymmetric scaling, i.e., for $\varepsilon$ tending to zero we set $q=e^{-\varepsilon}$ and look at fluctuations, space and time in the scales $\varepsilon^{-1}$, $\varepsilon^{-2}$ and $\varepsilon^{-4}$. We show that under such scaling the height function of the dynamic ASEP converges to the solution of the space-time Ornstein-Uhlenbeck process. We also introduce the dynamic ASEP on a ring with generalized rate functions. Under the very weakly asymmetric scaling, we show that the dynamic ASEP (with generalized jump rates) on a ring also converges to the solution of the space-time Ornstein-Uhlenbeck process on $[0,1]$ with periodic boundary conditions.

Symmetric elliptic functions, IRF models, and dynamic exclusion processes

We introduce stochastic Interaction-Round-a-Face (IRF) models that are related to representations of the elliptic quantum group E_{\tau, \eta}(sl_2) . For stochastic IRF models in a quadrant, we evaluate averages for a broad family of observables that can be viewed as higher analogs of q -moments of the height function for the stochastic (higher spin) six vertex models. In a certain limit, the stochastic IRF models degenerate to (1+1)d interacting particle systems that we call dynamic ASEP and SSEP; their jump rates depend on local values of the height function. For the step initial condition, we evaluate averages of observables for them as well, and use those to investigate one-point asymptotics of the dynamic SSEP. The construction and proofs are based on remarkable properties (branching and Pieri rules, Cauchy identities) of a (seemingly new) family of symmetric elliptic functions that arise as matrix elements in an infinite volume limit of the algebraic Bethe ansatz for E_{\tau, \eta}(sl_2) .

Extended Z-Invariance for Integrable Vector and Face Models and Multi-component Integrable Quad Equations

In a previous paper, the author has established an extension of the Z-invariance property for integrable edge-interaction models of statistical mechanics, that satisfy the star-triangle relation (STR) form of the Yang-Baxter equation (YBE). In the present paper, an analogous extended Z-invariance property is shown to also hold for integrable vector models and interaction-round-a-face (IRF) models of statistical mechanics respectively. As for the previous case of the STR, the Z-invariance property is shown through the use of local cubic-type deformations of a 2-dimensional surface associated to the models, which allow an extension of the models onto a subset of next nearest neighbour vertices of $\mathbb{Z}^3$, while leaving the partition functions invariant. These deformations are permitted as a consequence of the respective YBE's satisfied by the models. The quasi-classical limit is also considered, and it is shown that an analogous Z-invariance property holds for the variational formulation of classical discrete Laplace equations which arise in this limit. From this limit, new integrable 3D-consistent multi-component quad equations are proposed, which are constructed from a degeneration of the equations of motion for IRF Boltzmann weights.

Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices

In this paper we propose versions of the associative Yang-Baxter equation and higher order R-matrix identities which can be applied to quantum dynamical R-matrices. As is known quantum non-dynamical R-matrices of Baxter-Belavin type satisfy this equation. Together with unitarity condition and skew-symmetry it provides the quantum Yang-Baxter equation and a set of identities useful for different applications in integrable systems. The dynamical R-matrices satisfy the Gervais-Neveu-Felder (or dynamical Yang-Baxter) equation. Relation between the dynamical and non-dynamical cases is described by the IRF (interaction-round-a-face)-Vertex transformation. An alternative approach to quantum (semi-)dynamical R-matrices and related quantum algebras was suggested by Arutyunov, Chekhov, and Frolov (ACF) in their study of the quantum Ruijsenaars-Schneider model. The purpose of this paper is twofold. First, we prove that the ACF elliptic R-matrix satisfies the associative Yang-Baxter equation with shifted spectral parameters. Second, we directly prove a simple relation of the IRF-Vertex type between the Baxter-Belavin and the ACF elliptic R-matrices predicted previously by Avan and Rollet. It provides the higher order R-matrix identities and an explanation of the obtained equations through those for non-dynamical R-matrices. As a by-product we also get an interpretation of the intertwining transformation as matrix extension of scalar theta function likewise R-matrix is interpreted as matrix extension of the Kronecker function. Relations to the Gervais-Neveu-Felder equation and identities for the Felder’s elliptic R-matrix are also discussed.

The dynamical Yang–Baxter relation and the minimal representation of the elliptic quantum group

In this article, we give the general forms of the minimal L matrix (the elements of the L-matrix are c numbers) associated with the Boltzmann weights of the An−11 interaction-round-a-face (IRF) model and the minimal representation of the An−1 series elliptic quantum group given by Felder and Varchenko. The explicit dependence of elements of L-matrices on spectral parameter z are given. They are of five different forms [A(1-4) and B]. The algebra for the coefficients (which do not depend on z) are given. The algebra of form A is proved to be trivial, while that of form B obeys Yang–Baxter equation (YBE). We also give the PBW base and the centers for the algebra of form B.

Interaction-round-a-face density-matrix renormalization-group method

We demonstrate the numerical superiority of the interaction-round-a-face (IRF) density-matrix renormalization-group (DMRG) method applied to SU(2) invariant quantum spin chains over the conventional DMRG. The ground state energy densities and the gap energies of both S=1 and S=2 spin chains can be calculated using the IRF-DMRG without extensive computations. We have also studied the effect of tuning boundary interaction J end at both ends of the chain from the IRF view point. It is clearly observed that the magnon distribution is uniform when the best J end is chosen.

On the Combinatorics of Row and Corner Transfer Matrices of the $A^{(1)}_{n-1}$ Restricted Face Models

We establish a weight-preserving bijection between the index sets of the spectral data of row-to-row and corner transfer matrices for [Formula: see text] restricted interaction round a face (IRF) models. The evaluation of momenta by adding Takahashi integers in the spin chain language is shown to directly correspond to the computation of the energy of a path on the weight lattice in the two-dimensional model. As a consequence we derive fermionic forms of polynomial analogs of branching functions for the cosets [Formula: see text], and establish a bosonic–fermionic polynomial identity.

Thermodynamics Properties of One-Dimensional Z3 Quantum Potts Chain

The method of Discretized Path Integral Representation (DPIR) is used to convert the one-dimensional quantum Z3 Potts chain into a two-dimensional classical Interaction-Round a Face (IRF) model. Then we use the Double-Chain approximation to obtain the free energy and thermodynamic properties of the model.

Surface free energies, interfacial tensions and correlation lengths of the ABF models

In a previous paper, we introduced reflection equations for interaction-round-a-face (IRF) models and used these to construct commuting double-row transfer matrices for solvable lattice spin models with fixed boundary conditions. In particular, for the Andrews-Baxter-Forrester (ABF) models, we derived special functional equations satisfied by the eigenvalues of the commuting double-row transfer matrices. Here we introduce a generalized inversion relation method to solve these functional equations for the surface free energies. Although the surface free energies depend on the boundary spins we find that the associated surface critical exponent $\alpha_s=(7-L)/4$ is independent of the choice of boundary.

Reflection equations and exactly solvable lattice spin models

Reflection equations are used to obtain families of commuting double-row transfer matrices for interaction-round-a-face (IRF) models with fixed and free boundary conditions. We illustrate our methods for the Andrews-Baxter-Forrester (ABF) models which areL-state models associated with the quantum groupU q (su(2)) at a root of unity. We construct elliptic solutions to the reflection equations for the ABF models by a procedure which uses fusion to build the solutions starting from a trivial solution. Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.

Density Matrix Renormalization Group Method for 2D Classical Models

The density matrix renormalization group (DMRG) method is applied to the interaction round a face (IRF) model. When the transfer matrix is asymmetric, singular-value decomposition of the density matrix is required. A trial numerical calculation is performed on the square lattice Ising model, which is a special case of the IRF model.

THE YANG-BAXTER EQUATION IN KNOT THEORY

The role played by the Yang-Baxter equation in generating knot invariants using the method of statistical mechanics is reexamined and elucidated. The formulation of knot invariants is made precise with the introduction of piecewise-linear lattices and enhanced vertex and interaction-round-a-face (IRF) models with strictly local weights. It is shown that the Yang-Baxter equation and the need of using its solution in the infinite rapidity limit arise naturally in the realization of invariances under Reidemeister moves III. It is also shown that it is essential for the vertex models be charge-conserving, and that the construction of knot invariants from IRF models follows directly from the vertex model formulation.

Colored Vertex Models, Colored IRF Models and Invariants of Trivalent Colored Graphs

We present formulas for the Clebsch-Gordan coefficients and the Racah coefficients for the root of unity representations ($N$-dimensional representations with $q^{2N}=1$) of $U_q(sl(2))$. We discuss colored vertex models and colored IRF (Interaction Round a Face) models from the color representations of $U_q(sl(2))$. We construct invariants of trivalent colored oriented framed graphs from color representations of $U_q(sl(2))$.

SOLUTIONS OF THE STAR-TRIANGLE EQUATIONS IN (Nα,Nβ) POTTS MODELS

In this paper we present the solutions of the star-triangle equation for Boltzmann weights of the (Nα,Nβ) model. This model consists of two scalar Potts models (of Nα and Nβ states, respectively) coupled together by four-spin interaction terms and was introduced by Domany and Riedel to study the phase transitions in adsorbed thin films. If Nα,Nβ≠2, the star-triangle equations have nontrivial self-dual and non-self-dual solutions only for Nα=Nβ; whereas if Nα≠Nβ=2, we have other self-dual and non-self-dual solutions. After applying partial duality transformations several of our solutions can be identified with known solutions for Interaction-Round-a-Face (IRF) models. Our results also mean that our high-genus solutions found earlier are not the only solutions to the star-triangle equations in chiral Potts models with six or more states per site.

IRF MODELS ASSOCIATED WITH REPRESENTATIONS OF THE LIE SUPERALGEBRAS gl(m|n) AND sl(m|n)

We present two families of exactly solvable interaction round a face (IRF) models associated with representation of the Lie superalgebras gl (m|n) and sl (m|n). These IRF models are the generalizations of integrable spin chains with bosons and fermions. We also presents fusion models associated with higher representation of gl (m|n) and sl (m|n). We introduce restricted IRF models both for gl (m|n) and sl (m|n).

Restricted IRF Models with Complex Coupling Parameters

We present restricted IRF (Interaction Round a Face) models with discrete complex-valued crossing parameters. We discuss two types of restriction mechanisms of restricted IRF models and apply them to generalize the ABF models and A n (1) -cyclic RSOS models.

Scattering Amplitudes in Offcritical Models and RSOS Integrable Models

In the present paper we study scaling field theories related to the class of exactly solvable noncritical models of statistical mechanics which are known as RSOS (restricted solid-on-solid) or IRF (interaction round a face) models. These models are related to simple Lie algebras and they generalize the integrable version of RSOS model proposed by Andrews, Baxter and Forrester. 1> The models related to gl(n) were found by Jimbo, Miwa and Okado,2> the models related to other classical Lie algebras were obtained by the same authors in Ref. 3) and by Pasquier in Ref. 4) for Dn-case with a special coupling constant. Critical properties of JMO model were found in Ref. 5) where the authors obtain the explicit formulas for magnetization in the model and compute its critical behavior. It was established there that the critical behavior of these models was described by coset conformal field theories. 6>.7> Therefore it is natural to expect that corre­ sponding scaling field theories will be integrable perturbations of these conformal field theories. s> In this paper we describe the thermodynamics of the noncritical magnetic chain. Then we find the ground state, excitations over the ground state and the scattering amplitudes of excitations. By getting the scaling limit over the ground state of the model we compute the central charges of corresponding field theories, the dimensions of perturbations, the spectrum and scattering amplitudes of excitations. Under certain assumptions we generalize these results for any Lie algebra of A-D-E type. In certain simplest cases this gives known S-matrices like Es, E1, Es scalar S-matrices.9> In certain other models matrix elements of the S-matrix are given by Pasquier's solutions 4> of the Yang-Baxter equation. We do not discuss here the relation of RSOS models to integrable models of vertex type. 23>' 25> Partially the relation was explained in Refs. 4) and 26). The relation between the spectrum of row-to-row transfer matrices in RSOS and in vertex models will be given in a separate publication. For some of the scaling models the same answer was obtained by Bernard and

Relations for Clebsch–Gordan and Racah coefficients in suq(2) and Yang–Baxter equations

Relations are exploited among Clebsch–Gordan (CG) and Racah coefficients in the algebra suq(2), known as a deformation of su(2). These are used to show that the Yang–Baxter (YB) relation for the IRF (interaction round a face) model results from one of the symmetry relations for the 9-j symbol specific to suq(2), and that in an asymptotic limit this YB relation becomes the YB relation for the two-dimensional vertex model. The Racah coefficient, which has a particularly simple dependence on q, is efficiently used such that an asymptotic limit of the Racah coefficient is the CG coefficient and another limit gives the factorized S matrix of the vertex model.

Yang-Baxter Relations in Terms ofn-jSymbols ofsuq(2) Algebra

A trigonometric function parametrization of Yang-Baxter (Y-B) relations is described in terms of n - j symbols of su q (2), a deformation of su (2). The formalism starts with deduction of the Y-B relation for the IRF (interaction round a face) model of su q (2) from a general operator relation for the vertex model. The relation obtained for the IRF model is transformed into the Y-B relation for the vertex model using the property that the 3- j symbol is an asymptotic limit of the 6- j symbol. A class of very general solutions to Y-B equations is also presented. This solution of the IRF model with infinite rapidity parameters is linked to a symmetry of the 12- j symbol of the second kind.