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arXiv:1707.06123 · Journal of Physics A Mathematical and Theoretical

Matrix product states for interacting particles without hardcore constraints

Amit Kumar Chatterjee, P. K. Mohanty

Published Sep 27, 2017• 3 citations
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Abstract

Abstract We construct matrix product steady states for a class of interacting particle systems where particles do not obey hardcore exclusion, meaning each site can occupy any number of particles subjected to the global conservation of the total number of particles in the system. To represent the arbitrary occupancy of the sites, the matrix product ansatz here requires an infinite set of matrices which in turn leads to an algebra involving an infinite number of matrix equations. We show that these matrix equations, in fact, can be reduced to a single functional relation when the matrices are parametric functions of the representative occupation number. We demonstrate this matrix formulation in a class of stochastic particle hopping processes on a one dimensional periodic lattice where hop rates depend on the occupation numbers of the departure site and its neighbors within a finite range; this includes some well known stochastic processes like, the totally or partially asymmetric zero range process, misanthrope process and finite range process.

DOI
10.1088/1751-8121/aa8f58

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Abstract We construct matrix product steady states for a class of interacting particle systems where particles do not obey hardcore exclusion, meaning each site can occupy any number of particles subjected to the global conservation of the total number of particles in the system. To represent the arbitrary occupancy of the sites, the matrix product ansatz here requires an infinite set of matrices which in turn leads to an algebra involving an infinite number of matrix equations. We show that these matrix equations, in fact, can be reduced to a single functional relation when the matrices are parametric functions of the representative occupation number. We demonstrate this matrix formulation in a class of stochastic particle hopping processes on a one dimensional periodic lattice where hop rates depend on the occupation numbers of the departure site and its neighbors within a finite range; this includes some well known stochastic processes like, the totally or partially asymmetric zero range process, misanthrope process and finite range process.
Abstract Matrix product ansatz (MPA) is a powerful framework for constructing exact steady state weights of one dimensional non-equilibrium stochastic processes; but its generalization to higher dimensions is limited. Here, we introduce the MPA formalism for two dimensions (2D). As a concrete application, we introduce and exactly solve a non-conserved assisted exclusion model (NAEM) in one and two dimensions with constrained hopping and local birth–death dynamics: a particle can hop to a neighbouring site only when exactly one of its neighbouring sites is vacant, while creation and annihilation occur exclusively at sites whose neighbours are all occupied. The MPA yields exact steady-state weights and provides a systematic method to compute observables such as density moments and particle currents. In the particle-conserving limit, the system undergoes an absorbing phase transition at the critical density ρ c =1/2 with order-parameter exponent β=3. We further show that the steady state of the NAEM maps exactly onto the well-studied hard-square lattice gas with nearest-neighbour exclusion, thereby providing a nonequilibrium dynamical route to realizing equilibrium states of constrained lattice gases. Our work generalizes matrix-product methods beyond one dimension, establishing a systematic approach to exact solutions of interacting stochastic systems in 2D.
Abstract Many one-dimensional lattice particle models with open boundaries, like the paradigmatic asymmetric simple exclusion process (ASEP), have their stationary states represented in the form of a matrix product, with matrices that do not explicitly depend on the lattice site. In contrast, the stationary state of the open 1D zero-range process (ZRP) takes an inhomogeneous factorized form, with site-dependent probability weights. We show that in spite of the absence of correlations, the stationary state of the open ZRP can also be represented in a matrix product form, where the matrices are site-independent, non-commuting and determined from algebraic relations resulting from the master equation. We recover the known distribution of the open ZRP in two different ways: first, using an explicit representation of the matrices and boundary vectors; second, from the sole knowledge of the algebraic relations satisfied by these matrices and vectors. Finally, an interpretation of the relation between the matrix product form and the inhomogeneous factorized form is proposed within the framework of hidden Markov chains.
We show that the steady state weights of all one-dimensional exclusion models which are mapped to the zero-range process (ZRP) can be written in a matrix product form, where the required matrices depend only on the steady state weights of the ZRP. An infinite-dimensional representation of these matrices which works for generic systems has also been provided. This is in contrast to the case for the usual matrix product ansatz which does not always guarantee a solution for the dynamics dependent algebra that the matrices need to satisfy. The formulation helps us study the spatial correlations of these exclusion processes which are unfeasibly difficult to obtain directly from their ZRP correspondence. To illustrate this method we reproduce certain known results, and then investigate unexplored correlations in some other model systems.
We address the problem of exactly describing stochastic nonequilibrium systems that are widely used to model one-dimensional transport, in biology, traffic flow and others. We review the matrix product states ansatz to interacting multiparticle systems and its extension to a tridiagonal (generalized Onsager) algebra approach. The stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra defined by the dynamics of the process. The states involved in the matrix dements are determined by the boundary conditions. This reflects the intriguing feature of open systems that the bulk behaviour in the steady state strongly depends on the boundary rates. The importance of the boundary conditions manifests itself in the fact that the boundary operators are generators of a tridiagonal algebra whose irreducible modules are the Askey-Wilson polynomials. The matrices of the matrix product ansatz obey the tridiagonal algebraic relations for particular values of the structure, constants. Previously known representations, both infinite dimensional and finite dimensional ones, are recovered within the tridiagonal framework. The boundary Askey-Wilson and. tridiagonal symmetry is the deep algebraic property of driven diffusive systems allowing for the exact solvability in the steady state and the exact description of the stochastic dynamics.
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 Schrodinger 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.
Recently, it has been shown that the zero-energy eigenstate – corresponding to the stationary state – of a stochastic Hamiltonian with nearest-neighbour interaction in the bulk and single-site boundary terms, can generically be written in the form of a so-called matrix-product state. We generalize this result to stochastic Hamiltonians with arbitrary, but finite, interaction range. As an application two different particle-hopping models with three-site bulk interaction are studied. For these models which can be interpreted as cellular automata for traffic flow, we present exact solutions for periodic boundary conditions and some suitably chosen boundary interactions.