Restricted Solid-on-solid Models Research Articles

Factorization of density matrices in the critical RSOS models

We study reduced density matrices of the integrable critical restricted solid-on-solid (RSOS) model in a particular topological sector containing the ground state. Similar as in the spin- Heisenberg model it has been observed that correlation functions of this model on short segments can be ‘factorized’: they are completely determined by a single nearest-neighbour two-point function ω and a set of structure functions. While ω captures the dependence on the system size and the state of the system the structure functions can be expressed in terms of the possible operators on the segment, in the present case representations of the Temperley–Lieb algebra , and are independent of the model parameters. We present explicit results for the function ω in the infinite system ground state of the model and compute multi-point local height probabilities for up to four adjacent sites for the RSOS model and the related three-point correlation functions of non-Abelian anyons.

Lattice models, deformed Virasoro algebra and reduction equation

We study the fused currents of the deformed Virasoro algebra. By constructing a homotopy operator we show that for special values of the parameter of the algebra fused currents pairwise coincide on the cohomologies of the Felder resolution. Within the algebraic approach to lattice models these currents are known to describe neutral excitations of the solid-on-solid (SOS) models in the transfer-matrix picture. It allows us to prove the closeness of the system of excitations for a special nonunitary series of restricted SOS models. Though the results of the algebraic approach to lattice models were consistent with the results of other methods, the lack of such proof had been an essential gap in its construction.

RSOS quantum chains associated with off-critical minimal models and parafermions

We consider the φ1,3 off-critical perturbation of the general non-unitary minimal models where 2 ⩽ m ⩽ m′ and m, m′ are coprime and t measures the departure from criticality corresponding to the φ1,3 integrable perturbation. We view these models as the continuum scaling limit in the ferromagnetic Regime III of the Forrester–Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. We also consider the RSOS models in the antiferromagnetic Regime II related in the continuum scaling limit to parfermions with n = m′ − 2. Using an elliptic Yang–Baxter algebra of planar tiles encoding the allowed face configurations, we obtain the Hamiltonians of the associated quantum chains defined as the logarithmic derivative of the transfer matrices with periodic boundary conditions. The transfer matrices and Hamiltonians act on a vector space of paths on the Am′−1 Dynkin diagram whose dimension is counted by generalized Fibonacci numbers.

Off-critical logarithmic minimal models

We consider the integrable minimal models ℳ(m,m′;t), corresponding to the φ1,3 perturbation off-criticality, in the logarithmic limit m,m′ → ∞, m/m′ → p/p′ where p,p′ are coprime and the limit is taken through coprime values of m,m′. We view these off-critical minimal models ℳ(m,m′;t) as the continuum scaling limit of the Forrester–Baxter restricted solid-on-solid (RSOS) models on the square lattice. Applying corner transfer matrices to the Forrester–Baxter RSOS models in regime III, we argue that taking first the thermodynamic limit and second the logarithmic limit yields off-critical logarithmic minimal models ℒℳ(p,p′;t) corresponding to the φ1,3 perturbation of the critical logarithmic minimal models ℒℳ(p,p′). Specifically, in accord with the Kyoto correspondence principle, we show that the logarithmic limit of the one-dimensional configurational sums yields finitized quasi-rational characters of the Kac representations of the critical logarithmic minimal models ℒℳ(p,p′). We also calculate the logarithmic limit of certain off-critical observables \U0001d4aar,s related to one-point functions and show that the associated critical exponents produce all conformal dimensions in the infinitely extended Kac table. The corresponding Kac labels (r,s) satisfy (ps−p′r)2 < 8p(p′ − p). The exponent 2 − α = p′/2(p′ − p) is obtained from the logarithmic limit of the free energy giving the conformal dimension for the perturbing field t. As befits a non-unitary theory, some observables \U0001d4aar,s diverge at criticality.

Surface scaling properties of decagonal quasicrystals under nonequilibrium vertical growth

Restricted solid-on-solid (RSOS) growth models are studied on two different decagonal quasicrystal lattices, namely the Penrose tiling lattice and the random tiling lattice. There exist two types of growth blocks-fat and skinny tiles-which may have different sticking probabilities. We found that the RSOS growths on both lattices belong to the Kardar-Parisi-Zhang universality class when they have the same sticking probabilities in spite of the lack of periodicity in the substrates. However, when they have tile-type dependent sticking probabilities, the RSOS models on two lattices may produce different scaling behaviors. Growth on Penrose tiling shows that the roughness exponent is around 0.4 while that on random tiling is around 0.49. Our observation may provide an effective way to investigate the bulk structures of decagonal quasicrystals.

Critical RSOS and minimal models: fermionic paths, Virasoro algebra and fields

A framework is presented to extend the finitized characters and recursion methods of (off-critical) corner transfer matrices (CTMs), in a self-consistent fashion, to the calculation of CFT characters and conformal partition functions. More specifically, in this paper we consider sℓ(2) minimal conformal field theories on a cylinder from a lattice perspective. We argue that a general energy-preserving bijection exists between the one-dimensional configuration paths of the A L restricted solid-on-solid (RSOS) lattice models and the eigenstates of their double row transfer matrices and exhibit this bijection for the critical and tricritical Ising models in the vacuum sector. To each allowed one-dimensional configuration path we associate a physical state and a monomial in a finite fermionic algebra. The orthonormal states produced by the action of these monomials on the primary states | h〉 generate finite Virasoro modules with dimensions given by the finitized Virasoro characters χ ( N) h ( q). These finitized characters are the generating functions for the double row transfer matrix spectra of the critical RSOS models. We also propose a general level-by-level algorithm to build matrix representations of the Virasoro generators and chiral vertex operators (CVOs). The algorithm employs a distinguished basis which we call the L 1-basis. Our results extend to Z L−1 parafermion models by duality.

Universality class of the restricted solid-on-solid model with hopping.

We study the restricted solid-on-solid (RSOS) model with finite hopping distance l(0), using both analytical and numerical methods. Analytically, we use the hard-core bosonic field theory developed by the authors [Phys. Rev. E 62, 7642 (2000)] and derive the Villain-Lai-Das Sarma (VLD) equation for the l(0)=infinity case, which corresponds to the conserved RSOS (CRSOS) model and the Kardar-Parisi-Zhang (KPZ) equation for all finite values of l(0). Consequently, we find that the CRSOS model belongs to the VLD universality class and that the RSOS models with any finite hopping distance belong to the KPZ universality class. There is no phase transition at a certain finite hopping distance contrary to the previous result. We confirm the analytic results using the Monte Carlo simulations for several values of the finite hopping distance.

Two-dimensional wetting transition in a corrugated potential

A simple model for the wetting or depinning transition of a two-dimensional solid-on-solid (SOS) interface in a short-range periodic pinning potential which alternates between attraction and infinite repulsion is analysed exactly. The interface is specified by transverse displacement variables which vary continuously in the interval , and the stretching energy is proportional to . Both the semi-infinite and infinite geometries , are considered. For the most part the wetting transition in the continuum model is similar to the transition in restricted SOS models with corrugated potentials, in which the are restricted to integers and to , but there are some qualitative differences in the phase diagrams involving re-entrant behaviour.

Numerical Examination of the Universality of the Equilibrium Crystal Shape in Some SOS Models

By means of the Monte Carlo twist method, we examine the universality of the following critical properties of the equilibrium crystal shape in the discrete Gaussian (DG) model and restricted solid-on-solid (RSOS) model: c1) a finite curvature jump at the roughening temperature T R , c2) the profile of a facet with a singularlity at the facet edge for T < T R , and c3) the radius of the facet near T R proportional to \(\exp(-a/ \sqrt{1-T/T_{R}})\). We obtain the following results with the help of some results obtained analytically. The DG and RSOS models have T R =1.44 ±0.02 and 1.60 ±0.05, respectively. c2) is universal for these models. For the DG model c1) well agrees with the exact result of the body-centered SOS model, whereas for the RSOS model it is almost equal to twice the latter. Finite-size-scaling for c3) is excellent for the DG model, while it is consistent but inconclusive for the RSOS model.

Solvable lattice models labelled by Dynkin diagrams

An equivalence between generalised restricted solid-on-solid (RSOS) models, associated with sets of graphs, and multi-colour loop models is established. As an application we consider solvable loop models and in this way obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply-laced Lie algebras. For one of the RSOS models, labelled by the Lie algebra pair (A$_L$,A$_L$) and related to the C$_2^{(1)}$ vertex model, we give an off-critical extension, which breaks the Z$_2$ symmetry of the Dynkin diagrams.

New construction of solvable lattice models including an Ising model in a field.

In this Letter we report a new construction to obtain restricted solid-on-solid (RSOS) models out of loop models. The method is a generalization of ideas developed by Owczarek and Baxter, and by Pasquier. In particular we consider a solvable O(n) model and point out that some of the RSOS models thus obtained admit an off-critical extension. Among these models we find a spin-1 Ising model, which is solvable not only at the critical point, but also in a fieldlike deviation away from it. We calculate the critical exponent \ensuremath{\delta}=15 directly from the relation between the free energy and the field. This is the first determination of this exponent without the use of scaling relations.

Ferro- and antiferro-magnetizations in RSOS models

We study a class of restricted solid-on-solid (RSOS) models related to the quantum group U q (sl( n)) with q a general primitive root of unity. Physical regimes of the models enjoy intriguing equivalence properties under the interchange of the rank n and the relevant level of the affine Lie algebra A n−1 (1) (level-rank-duality) and also the q- and q −-deformations ( q ±1 duality). By using these we give a classification scheme for ferromagnetic and antiferromagnetic orderings at low temperature. We also propose that the level-rank duality is inherited in rational Toda field theories with imaginary coupling constants. This explains the deficiency structure of the conserved currents in a series of deformed non-unitary conformal field theories.

Surface phase transitions: Roughening, preroughening, orientational roughening, and reconstruction in an isotropic frustrated Ising model.

We study the roughening, the preroughening, and the orientational roughening transitions for interfaces between phases in the isotropic competing Ising Hamiltonian H=-J${\mathrm{\ensuremath{\Sigma}}}_{\mathit{nm}}$${\mathit{s}}_{\mathit{i}}$${\mathit{s}}_{\mathit{j}}$-2M ${\mathrm{\ensuremath{\Sigma}}}_{\mathit{lnnn}}$${\mathit{s}}_{\mathit{i}}$${\mathit{s}}_{\mathit{j}}$. The symbols, NN, DNN, and LNNN represent nearest-neighbor, diagonal-nearest-neighbor, and linear-next-nearest-neighbor pairs, respectively, with Jg0 and -\ensuremath{\infty}M\ensuremath{\infty}. By performing low-temperature expansions in the three-dimensional cubic lattice, it is found that the roughening transition curve passes through the region in which paramagnetic, ferromagnetic, and modulated phases merge, and is parallel to the phase boundary of ferromagnetic and modulated phases. It is also found that crossover behavior occurs in the roughening transition from the surface tension dominant regime to the bending-energy dominant regime. We examine the preroughening transition for the case of the restricted solid-on-solid (RSOS) model. The preroughening transition line is found to be distinct from the roughening transition line. This confirms the recent prediction by den Nijs [Phys. Rev. Lett. 64, 435 (1990)]. Within the RSOS model, we find a multistate point for reconstructed surface structures. This is analogous to the multistate point in the bulk phase diagram. The orientational roughening transition is also examined. However, it was not possible to obtain the orientational roughening transition line at a temperature higher than that of the translational roughening transition, presumably due to the limited order of the low-temperature series. The results deduced from the low-temperature expansions are compared to the sine-Gordon renormalization-group calculations based on the SOS and RSOS models. Finally we discuss the implications of the surface structure, and the connections with bulk structure.

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