2,099 research outputs found
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 . 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
The Exact Solution of the Asymmetric Exclusion Problem With Particles of Arbitrary Size: Matrix Product Ansatz
The exact solution of the asymmetric exclusion problem and several of its
generalizations is obtained by a matrix product {\it ansatz}. Due to the
similarity of the master equation and the Schr\"odinger equation at imaginary
times the solution of these problems reduces to the diagonalization of a one
dimensional quantum Hamiltonian. We present initially the solution of the
problem when an arbitrary mixture of molecules, each of then having an
arbitrary size () in units of lattice spacing, diffuses
asymmetrically on the lattice. The solution of the more general problem where
we have | the diffusion of particles belonging to distinct class of
particles (), with hierarchical order, and arbitrary sizes is also
solved. Our matrix product {\it ansatz} asserts that the amplitudes of an
arbitrary eigenfunction of the associated quantum Hamiltonian can be expressed
by a product of matrices. The algebraic properties of the matrices defining the
{\it ansatz} depend on the particular associated Hamiltonian. The absence of
contradictions in the algebraic relations defining the algebra ensures the
exact integrability of the model. In the case of particles distributed in
classes, the associativity of the above algebra implies the Yang-Baxter
relations of the exact integrable model.Comment: 42 pages, 1 figur
Anomalous bulk behaviour in the free parafermion spin chain
We demonstrate using direct numerical diagonalization and extrapolation
methods that boundary conditions have a profound effect on the bulk properties
of a simple model for for which the model hamiltonian is
non-hermitian. For the model reduces to the well known quantum Ising
model in a transverse field. For open boundary conditions the model is
known to be solved exactly in terms of free parafermions. Once the ends of the
open chain are connected by considering the model on a ring, the bulk
properties, including the ground-state energy per site, are seen to differ
dramatically with increasing . Other properties, such as the leading
finite-size corrections to the ground-state energy, the mass gap exponent and
the specific heat exponent, are also seen to be dependent on the boundary
conditions. We speculate that this anomalous bulk behaviour is a topological
effect.Comment: 8 pages, 8 figures, minor change
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 ( 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 -dimensional representation of the periodic
Temperley-Lieb algebra is presented. It is also a representation of
the cyclic group . We choose 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 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
Exactly solvable interacting vertex models
We introduce and solvev a special family of integrable interacting vertex
models that generalizes the well known six-vertex model. In addition to the
usual nearest-neighbor interactions among the vertices, there exist extra
hard-core interactions among pair of vertices at larger distances.The
associated row-to-row transfer matrices are diagonalized by using the recently
introduced matrix product {\it ansatz}. Similarly as the relation of the
six-vertex model with the XXZ quantum chain, the row-to-row transfer matrices
of these new models are also the generating functions of an infinite set of
commuting conserved charges. Among these charges we identify the integrable
generalization of the XXZ chain that contains hard-core exclusion interactions
among the spins. These quantum chains already appeared in the literature. The
present paper explains their integrability.Comment: 20 pages, 3 figure
- …