On Matrix Product States for Periodic Boundary Conditions

TL;DR

This paper investigates the applicability of matrix product states to zero-energy eigenstates in quantum spin chains with periodic boundary conditions, revealing limitations compared to open boundary cases.

Contribution

It demonstrates that not all zero-energy eigenstates in periodic boundary conditions can be represented as matrix product states, providing a counter-example and contrasting with open boundary conditions.

Findings

Not all zero-energy eigenstates are matrix product states in periodic boundary conditions.

A counter-example shows limitations of the matrix product state representation.

Open boundary conditions allow all zero-energy states to be represented as matrix product states.

Abstract

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 {\em 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.