113 research outputs found
Integrable approach to simple exclusion processes with boundaries. Review and progress
We study the matrix ansatz in the quantum group framework, applying
integrable systems techniques to statistical physics models. We start by
reviewing the two approaches, and then show how one can use the former to get
new insight on the latter. We illustrate our method by solving a model of
reaction-diffusion. An eigenvector for the transfer matrix for the XXZ spin
chain with non-diagonal boundary is also obtained using a matrix ansatz.Comment: 44 page
3-state Hamiltonians associated to solvable 33-vertex models
Using the nested coordinate Bethe ansatz, we study 33-vertex models, where
only one global charge with degenerate eigenvalues exists and each site
possesses three internal degrees of freedom. In the context of Markovian
processes, they correspond to diffusing particles with two possible internal
states which may be exchanged during the diffusion (transmutation). The first
step of the nested coordinate Bethe ansatz is performed providing the
eigenvalues in terms of rapidities. We give the constraints ensuring the
consistency of the computations. These rapidities also satisfy Bethe equations
involving R-matrices, solutions of the Yang--Baxter equation which
implies new constraints on the models. We solve them allowing us to list all
the solvable 33-vertex models.Comment: 14 pages; title changed according to referee request; an appendix
added to describe explicitely the Hamiltonia
Inhomogeneous discrete-time exclusion processes
We study discrete time Markov processes with periodic or open boundary
conditions and with inhomogeneous rates in the bulk. The Markov matrices are
given by the inhomogeneous transfer matrices introduced previously to prove the
integrability of quantum spin chains. We show that these processes have a
simple graphical interpretation and correspond to a sequential update. We
compute their stationary state using a matrix ansatz and express their
normalization factors as Schur polynomials. A connection between Bethe roots
and Lee-Yang zeros is also pointed out.Comment: 30 pages, 10 figures; a short paragraph at the end to justify the
form of the sequential update has been added; the justification of the
transfer matrix degree is detaile
Open two-species exclusion processes with integrable boundaries
We give a complete classification of integrable Markovian boundary conditions
for the asymmetric simple exclusion process with two species (or classes) of
particles. Some of these boundary conditions lead to non-vanishing particle
currents for each species. We explain how the stationary state of all these
models can be expressed in a matrix product form, starting from two key
components, the Zamolodchikov-Faddeev and Ghoshal-Zamolodchikov relations. This
statement is illustrated by studying in detail a specific example, for which
the matrix Ansatz (involving 9 generators) is explicitly constructed and
physical observables (such as currents, densities) calculated.Comment: 19 pages; typos corrected, more details on the Matrix Ansatz algebr
Integrable boundary conditions for multi-species ASEP
The first result of the present paper is to provide classes of explicit
solutions for integrable boundary matrices for the multi-species ASEP with an
arbitrary number of species.
All the solutions we have obtained can be seen as representations of a new
algebra that contains the boundary Hecke algebra. The boundary Hecke algebra is
not sufficient to build these solutions. This is the second result of our
paper.Comment: 20 page
Integrable dissipative exclusion process: Correlation functions and physical properties
We study a one-parameter generalization of the symmetric simple exclusion
process on a one dimensional lattice. In addition to the usual dynamics (where
particles can hop with equal rates to the left or to the right with an
exclusion constraint), annihilation and creation of pairs can occur. The system
is driven out of equilibrium by two reservoirs at the boundaries. In this
setting the model is still integrable: it is related to the open XXZ spin chain
through a gauge transformation. This allows us to compute the full spectrum of
the Markov matrix using Bethe equations. Then, we derive the spectral gap in
the thermodynamical limit. We also show that the stationary state can be
expressed in a matrix product form permitting to compute the multi-points
correlation functions as well as the mean value of the lattice current and of
the creation-annihilation current. Finally the variance of the lattice current
is exactly computed for a finite size system. In the thermodynamical limit, it
matches perfectly the value obtained from the associated macroscopic
fluctuation theory. It provides a confirmation of the macroscopic fluctuation
theory for dissipative system from a microscopic point of view.Comment: 31 pages, 7 figures ; introduction expanded, typos corrected and
title change
Matrix product solution to a 2-species TASEP with open integrable boundaries
We present an explicit representation for the matrix product ansatz for some
two-species TASEP with open boundary conditions. The construction relies on the
integrability of the models, a property that constrains the possible rates at
the boundaries. The realisation is built on a tensor product of copies of the
DEHP algebras. Using this explicit construction, we are able to calculate the
partition function of the models. The densities and currents in the stationary
state are also computed. It leads to the phase diagram of the models. Depending
on the values of the boundary rates, we obtain for each species shock waves,
maximal current, or low/high densities phases.Comment: 23 page
Relaxation rate of the reverse biased asymmetric exclusion process
We compute the exact relaxation rate of the partially asymmetric exclusion
process with open boundaries, with boundary rates opposing the preferred
direction of flow in the bulk. This reverse bias introduces a length scale in
the system, at which we find a crossover between exponential and algebraic
relaxation on the coexistence line. Our results follow from a careful analysis
of the Bethe ansatz root structure.Comment: 22 pages, 12 figure
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