Quantum groups and vertex models

Nella tesi vengono presentate alcune relazioni fra gruppi quantici e modelli reticolari. In particolare si associa un modello vertex a una rappresentazione di un'algebra inviluppante quantizzata affine e si mostra che, specializzando il parametro quantistico ad una radice dell'unità, si manifestano speciali simmetrie.

On the construction of correlation functions for the integrable supersymmetric fermion models

We review the recent progress on the construction of the determinant representations of the correlation functions for the integrable supersymmetric fermion models. The factorizing F-matrices (or the so-called F-basis) play an important role in the construction. In the F-basis, the creation (and the annihilation) operators and the Bethe states of the integrable models are given in completely symmetric forms. This leads to the determinant representations of the scalar products of the Bethe states for the models. Based on the scalar products, the determinant representations of the correlation functions may be obtained. As an example, in this review, we give the determinant representations of the two-point correlation function for the U-q(gl(2 vertical bar 1)) (i.e. q-deformed) supersymmetric t-J model. The determinant representations are useful for analyzing physical properties of the integrable models in the thermodynamical limit.

Generalized Perk-Schultz models: solutions of the Yang-Baxter equation associated with quantized orthosymplectic superalgebras

The Perk-Schultz model may be expressed in terms of the solution of the Yang-Baxter equation associated with the fundamental representation of the untwisted affine extension of the general linear quantum superalgebra U-q (gl(m/n)], with a multiparametric coproduct action as given by Reshetikhin. Here, we present analogous explicit expressions for solutions of the Yang-Baxter equation associated with the fundamental representations of the twisted and untwisted affine extensions of the orthosymplectic quantum superalgebras U-q[osp(m/n)]. In this manner, we obtain generalizations of the Perk-Schultz model.

Dynamics of the two-dimensional gonihedric spin model

In this paper, we study dynamical aspects of the two-dimensional (2D) gonihedric spin model using both numerical and analytical methods. This spin model has vanishing microscopic surface tension and it actually describes an ensemble of loops living on a 2D surface. The self-avoidance of loops is parametrized by a parameter ¿. The ¿=0 model can be mapped to one of the six-vertex models discussed by Baxter, and it does not have critical behavior. We have found that allowing for ¿¿0 does not lead to critical behavior either. Finite-size effects are rather severe, and in order to understand these effects, a finite-volume calculation for non-self-avoiding loops is presented. This model, like his 3D counterpart, exhibits very slow dynamics, but a careful analysis of dynamical observables reveals nonglassy evolution (unlike its 3D counterpart). We find, also in this ¿=0 case, the law that governs the long-time, low-temperature evolution of the system, through a dual description in terms of defects. A power, rather than logarithmic, law for the approach to equilibrium has been found.

A conformal invariant growth model

We present a one-parameter extension of the raise and peel one-dimensional growth model. The model is defined in the configuration space of Dyck (RSOS) paths. Tiles from a rarefied gas hit the interface and change its shape. The adsorption rates are local but the desorption rates are non-local; they depend not only on the cluster hit by the tile but also on the total number of peaks (local maxima) belonging to all the clusters of the configuration. The domain of the parameter is determined by the condition that the rates are non-negative. In the finite-size scaling limit, the model is conformal invariant in the whole open domain. The parameter appears in the sound velocity only. At the boundary of the domain, the stationary state is an adsorbing state and conformal invariance is lost. The model allows us to check the universality of non-local observables in the raise and peel model. An example is given.

Nonlocal asymmetric exclusion process on a ring and conformal invariance

We present a one-dimensional nonlocal hopping model with exclusion on a ring. The model is related to the Raise and Peel growth model. A nonnegative parameter u controls the ratio of the local backwards and nonlocal forwards hopping rates. The phase diagram, and consequently the values of the current, depend on u and the density of particles. In the special case of half-lling and u = 1 the system is conformal invariant and an exact value of the current for any size L of the system is conjectured and checked for large lattice sizes in Monte Carlo simulations. For u > 1 the current has a non-analytic dependence on the density when the latter approaches the half-lling value.

Exact solution of the XXZ Gaudin model with generic open boundaries

The XXZ Gaudin model with generic integrable boundaries specified by generic non-diagonal K-matrices is studied. The commuting families of Gaudin operators are diagonalized by the algebraic Bethe ansatz method. The eigenvalues and the corresponding Bethe ansatz equations are obtained. (C) 2004 Elsevier B.V. All rights reserved.

A(n-1) Gaudin model with open boundaries

The A(n-1) Gaudin model with integrable boundaries specified by non-diagonal K-matrices is studied. The commuting families of Gaudin operators are diagonalized by the algebraic Bethe ansatz method. The eigenvalues and the corresponding Bethe ansatz equations are obtained. (c) 2005 Elsevier B.V. All rights reserved.

Determinant representations of correlation functions for the supersymmetric t-J model

Working in the F-basis provided by the factorizing F-matrix, the scalar products of Bethe states for the supersymmetric t-J model are represented by determinants. By means of these results, we obtain determinant representations of correlation functions for the model.

Truncation identities for the small polaron fusion hierarchy

We study a one-dimensional lattice model of interacting spinless fermions. This model is integrable for both periodic and open boundary conditions; the latter case includes the presence of Grassmann valued non-diagonal boundary fields breaking the bulk U(1) symmetry of the model. Starting from the embedding of this model into a graded Yang-Baxter algebra, an infinite hierarchy of commuting transfer matrices is constructed by means of a fusion procedure. For certain values of the coupling constant related to anisotropies of the underlying vertex model taken at roots of unity, this hierarchy is shown to truncate giving a finite set of functional equations for the spectrum of the transfer matrices. For generic coupling constants, the spectral problem is formulated in terms of a functional (or TQ-)equation which can be solved by Bethe ansatz methods for periodic and diagonal open boundary conditions. Possible approaches for the solution of the model with generic non-diagonal boundary fields are discussed.

Exact solution of the A(n-1)((1)) trigonometric vertex model with non-diagonal open boundaries

The A(n-1)((1)) trigonometric vertex model with generic non-diagonal boundaries is studied. The double-row transfer matrix of the model is diagonalized by algebraic Bethe ansatz method in terms of the intertwiner and the corresponding face-vertex relation. The eigenvalues and the corresponding Bethe ansatz equations are obtained.

Non-diagonal solutions of the reflection equation for the trigonometric A((1)) (n-1) vertex model

We obtain a class of non-diagonal solutions of the reflection equation for the trigonometric A(n-1)((1)) vertex model. The solutions can be expressed in terms of intertwinner matrix and its inverse, which intertwine two trigonometric R-matrices. In addition to a discrete (positive integer) parameter l, 1 less than or equal to l less than or equal to n, the solution contains n + 2 continuous boundary parameters.

Fault tolerance of Mobius cubes under two forbidden fault set models

An n-dimensional Mobius cube, 0MQ(n) or 1MQ(n), is a variation of n-dimensional cube Q(n) which possesses many attractive properties such as significantly smaller communication delay and stronger graph-embedding capabilities. In some practical situations, the fault tolerance of a distributed memory multiprocessor system can be measured more precisely by the connectivity of the underlying graph under forbidden fault set models. This article addresses the connectivity of 0MQ(n)/1MQ(n), under two typical forbidden fault set models. We first prove that the connectivity of 0MQ(n)/1MQ(n) is 2n - 2 when the fault set does not contain the neighborhood of any vertex as a subset. We then prove that the connectivity of 0MQ(n)/1MQ(n) is 3n - 5 provided that the neighborhood of any vertex as well as that of any edge cannot fail simultaneously These results demonstrate that 0MQ(n)/1MQ(n) has the same connectivity as Q(n) under either of the previous assumptions.