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On the ubiquity of matrix-product states in one-dimensional stochastic processes with boundary interactions

Kai Klauck, Andreas Schadschneider

Published Jan 1, 1999• 31 citations
OpenAlex

Abstract

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.

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The possibility of a matrix product representation for eigenstates with energy and momentum zero of a general m -state quantum spin Hamiltonian with nearest-neighbour interaction and periodic boundary condition is considered. For this representation we use the quadratic algebra defined in Krebs and Sandow ( 1997 J. Phys. A: Math. Gen. 30 3165 ), which is generated by 2 m operators which fulfil m 2 quadratic relations, endowed with a trace-like function reflecting periodic boundary conditions. It is shown that not every eigenstate with energy and momentum zero can be written in this way. An explicit counter-example is given. This is in contrast to the case of open boundary conditions where every zero-energy eigenstate can be written as a matrix product state using a Fock-like representation of the same quadratic algebra.

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We consider the asymmetric random average process which is a one-dimensional stochastic lattice model with nearest neighbour interaction but continuous and unbounded state variables. First, the explicit functional representations, so-called beta densities, of all local interactions leading to steady states of product measure form are rigorously derived. This also completes an outstanding proof given in a previous publication. Then, we present an alternative solution for the processes with factorized stationary states by using a matrix product ansatz. Due to continuous state variables we obtain a matrix algebra in form of a functional equation which can be solved exactly.

arXiv:cond-mat/0211472•Journal of Physics A Mathematical and General•Mar 19, 2003•22 citations
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.
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.
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.
The possibility of a matrix product representation for eigenstates with energy and momentum zero of a general m -state quantum spin Hamiltonian with nearest-neighbour interaction and periodic boundary condition is considered. For this representation we use the quadratic algebra defined in Krebs and Sandow ( 1997 J. Phys. A: Math. Gen. 30 3165 ), which is generated by 2 m operators which fulfil m 2 quadratic relations, endowed with a trace-like function reflecting periodic boundary conditions. It is shown that not every eigenstate with energy and momentum zero can be written in this way. An explicit counter-example is given. This is in contrast to the case of open boundary conditions where every zero-energy eigenstate can be written as a matrix product state using a Fock-like representation of the same quadratic algebra.
The possibility of a matrix product representation for eigenstates with energy and momentum zero of a general m–state quantum spin Hamiltonian with nearest neighbour interaction and periodic boundary condition is considered. The quadratic algebra used for this representation is generated by 2m operators which fulfil m 2 quadratic relations and is endowed with a trace. It is shown that not every eigenstate with energy and momentum zero can be written as matrix product state. An explicit counter–example is given. This is in contrast to the case of open boundary conditions where every zero energy eigenstate can be written as a matrix product state using a Fock–like representation of the same quadratic algebra.
We consider the asymmetric random average process which is a one-dimensional stochastic lattice model with nearest neighbour interaction but continuous and unbounded state variables. First, the explicit functional representations, so-called beta densities, of all local interactions leading to steady states of product measure form are rigorously derived. This also completes an outstanding proof given in a previous publication. Then, we present an alternative solution for the processes with factorized stationary states by using a matrix product ansatz. Due to continuous state variables we obtain a matrix algebra in form of a functional equation which can be solved exactly.