The modified XXZ Heisenberg chain, conformal invariance, surface exponents of c<1 systems, and hidden symmetries of the finite chains

The spin-1/2 XXZ Heisenberg chain with two types of boundary terms is considered. For the first type, the Hamiltonian is hermitian but not for the second type which includes the U_{q}[SU(2)] symmetric case. It is shown that for a certain `tuning' between the anisotropy angle and the boundary terms the spectra present unexpected degeneracies. These degeneracies are related to the structure of the irreducible representations of the Virasoro algebras for c<1.Comment: 9 pages; an old preprint from the pre-arXiv (but not pre-LaTeX) era, published version not (yet?) electronically accessibl

The Pokrovski-Talapov Phase Transitions and Quantum Groups

We show that the XY quantum chain in a magnetic field is invariant under a two parameter deformation of the $SU(1/1)$ superalgebra. One is led to an extension of the braid group and the Hecke algebras which reduce to the known ones when the two parameter coincide. The physical significance of the two parameters is discussed. When both are equal to one, one gets a Pokrovski-Talapov phase transition. We also show that the representation theory of the quantum superalgebras indicates how to take the appropriate thermodynamical limits.Comment: 9 page

Nonlocal growth processes and conformal invariance

Up to now the raise and peel model was the single known example of a one-dimensional stochastic process where one can observe conformal invariance. The model has one-parameter. Depending on its value one has a gapped phase, a critical point where one has conformal invariance and a gapless phase with changing values of the dynamical critical exponent $z$. In this model, adsorption is local but desorption is not. The raise and strip model presented here in which desorption is also nonlocal, has the same phase diagram. The critical exponents are different as are some physical properties of the model. Our study suggest the possible existence of a whole class of stochastic models in which one can observe conformal invariance.Comment: 27 pages, 22 figure

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 to check the universality of nonlocal observables in the raise and peel model. An example is given.Comment: 11 pages and 8 figure

Density profiles in the raise and peel model with and without a wall. Physics and combinatorics

We consider the raise and peel model of a one-dimensional fluctuating interface in the presence of an attractive wall. The model can also describe a pair annihilation process in a disordered unquenched media with a source at one end of the system. For the stationary states, several density profiles are studied using Monte Carlo simulations. We point out a deep connection between some profiles seen in the presence of the wall and in its absence. Our results are discussed in the context of conformal invariance ($c = 0$ theory). We discover some unexpected values for the critical exponents, which were obtained using combinatorial methods. We have solved known (Pascal's hexagon) and new (split-hexagon) bilinear recurrence relations. The solutions of these equations are interesting on their own since they give information on certain classes of alternating sign matrices.Comment: 39 pages, 28 figure

Cyclic representations of the periodic Temperley Lieb algebra, complex Virasoro representations and stochastic processes

An $N$ ${L} \choose {L/2}$-dimensional representation of the periodic Temperley-Lieb algebra $TL_L(x)$ is presented. It is also a representation of the cyclic group $Z_N$. We choose $x = 1$ and define a Hamiltonian as a sum of the generators of the algebra acting in this representation. This Hamiltonian gives the time evolution operator of a stochastic process. In the finite-size scaling limit, the spectrum of the Hamiltonian contains representations of the Virasoro algebra with complex highest weights. The $N = 3$ case is discussed in detail. One discusses shortly the consequences of the existence of complex Virasoro representations on the physical properties of the systems.Comment: 5 pages, 6 figure

Stochastic processes with Z_N symmetry and complex Virasoro representations. The partition functions

In a previous Letter (J. Phys. A v.47 (2014) 212003) we have presented numerical evidence that a Hamiltonian expressed in terms of the generators of the periodic Temperley-Lieb algebra has, in the finite-size scaling limit, a spectrum given by representations of the Virasoro algebra with complex highest weights. This Hamiltonian defines a stochastic process with a Z_N symmetry. We give here analytical expressions for the partition functions for this system which confirm the numerics. For N even, the Hamiltonian has a symmetry which makes the spectrum doubly degenerate leading to two independent stochastic processes. The existence of a complex spectrum leads to an oscillating approach to the stationary state. This phenomenon is illustrated by an example.Comment: 8 pages, 4 figures, in a revised version few misprints corrected, one relevant reference adde