Matrix Product Eigenstates for One-Dimensional Stochastic Models and Quantum Spin Chains

TL;DR

This paper demonstrates that zero energy eigenstates of certain quantum spin chains and stochastic models can be represented as matrix product states, with weights expressed as expectation values in a specific algebraic framework.

Contribution

It introduces a general method to construct matrix product eigenstates for a broad class of quantum and stochastic models using algebraic relations.

Findings

All zero energy eigenstates can be written as matrix product states.

Weights are expressed as expectation values in a Fock representation.

Applicable to arbitrary m-state quantum spin chains with nearest neighbor interactions.

Abstract

We show that all zero energy eigenstates of an arbitrary $m$--state quantum spin chain Hamiltonian with nearest neighbor interaction in the bulk and single site boundary terms, which can also describe the dynamics of stochastic models, can be written as matrix product states. This means that the weights in these states can be expressed as expectation values in a Fock representation of an algebra generated by $2m$ operators fulfilling $m^2$ quadratic relations which are defined by the Hamiltonian.