The exact solution of the asymmetric exclusion problem with particles of asrbitrary size: matrix product ansatz

Abstract

The exact solution of the asymmetric exclusion problem and several of its generalizations is obtained by a matrix product ansatz. Due to the similarity of the master equation and the Schrödinger equation at imaginary times the solution of these problems reduces to the diagonalization of a one dimensional quantum Hamiltonian. Initially, we present the solution of the problem when an arbitrary mixture of molecules, each of then having an arbitrary size (s = 0, 1, 2, . . .) 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 N distinct classes of particles (c = 1, . . . , N ), with hierarchical order and arbitrary sizes, is also presented. Our matrix product 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 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 N > 2 classes, the associativity of this algebra implies the Yang-Baxter relations of the exact integrable model. I IntroductionThe representation of interacting stochastic particle dynamics in terms of quantum spin systems produced interesting and fruitful interchanges between the areas of equilibrium and nonequilibrium statistical mechanics. The connection between these areas follows from the similarity between the master equation describing the time-fluctuations on the nonequilibrium stochastic problem and the quantum fluctuations of the equilibrium quantum spin chains [1]- [20].Unlike the area of nonequilibrium interacting systems, where very few models are fully solvable, there exists a huge family of quantum chains appearing in equilibrium problems that are exactly integrable. The machinery that allows the exact solutions of these quantum chains comes from the Bethe ansatz in its several formulations (see [21]-[24] for reviews). The above mentioned mathematical connection between equilibrium and nonequilibrium revealed that some quantum chains related to interacting stochastic problems are exactly solvable through the Bethe ansatz. The simplest example is the problem of asymmetric diffusion of hard-core particles on the one dimensional lattice (see [16,17,20] for reviews). The time fluctuations of this last model are governed by a time evolution operator that coincides with the exact integrable anisotropic Heisenberg chain, or the so called, XXZ quantum chain, in its ferromagnetically ordered regime. A generalization of this stochastic problem where exact integrability is also known [25]-[27] is the case where there are N (N = 1, 2, ...) classes of particles hierarchically ordered and diffusing asymmetrically on the lattice. The quantum chain related to this problem is known in the literature as the anisotropic Sutherland model [28...

The Matrix Product Ansatz for integrable U(1)N models in Lunin-Maldacena backgrounds

“…The MPA we are going to use in this paper, in order to solve the U(1) N quantum spin chain, was introduced in [20][21][22][23]. This ansatz was applied with success in the evaluation of the spectra of several integrable quantum Hamiltonians [20][21][22], transfer matrices [24][25][26] and the time-evolution operator of stochastic systems [23]. According to this ansatz, the amplitudes of the eigenfunctions are given in terms of a product of matrices where the matrices obey appropriated algebraic relations.…”

“…We analyse the Yang-Baxter equation in the N = 3 sector and the consistence of the algebraic relations among the matrices defining the ansatz and find that the solutions are separated in two class. In the first (class A) we obtain the models presented in [28][29][30][31], in the context of one loop dilatation operator, as well as in condensed matter physics and stochastic models [19,21,23,[32][33][34]. In the second class (class B) we obtain the model presented in [19] for the stochastic problem of fully asymmetric diffusion of two kinds of particles.…”

Integrable inhomogeneous spin chains in generalized Lunin-Maldacena backgrounds

“…where ε n is the energy of the eigenfunction (2) (α = 2, 3) is composed of n = n 2 + n 3 spectral parameter dependent matrices [22,24,30]. A second class of solutions is obtained if matrices A (α)…”

“…The MPA we are going to use in this paper was introduced in [21][22][23][24]. This ansatz was applied with success in the evaluation of the spectra of several integrable quantum Hamiltonians [21][22][23], transfer matrices [25][26][27] and the time-evolution operator of stochastic systems [24]. According to this ansatz, the amplitudes of the eigenfunctions are given in terms of a product of matrices where the matrices obey appropriated algebraic relations.…”

Exactly solvable interacting vertex models

“…Some of these extended quantum chains appear in the description of the stochastic dynamics of in the asymmetric exclusion problem with particles with extended sizes [18,15], [19]- [23]. The exact solution of these quantum chains were not obtained by the QISM but directly by the coordinate Bethe ansatz [24] or by the matrix product ansatz [25]- [28]. Consequently, the two dimensional vertex model generating these quantum chains are not known, and we cannot explain their integrability, up to now.…”