Frustration free gapless Hamiltonians for Matrix Product States

TL;DR

This paper introduces a method to construct frustration-free, gapless Hamiltonians ('uncle' Hamiltonians) for Matrix Product States, extending the known parent Hamiltonian framework to cases with degenerate or unique ground states.

Contribution

It presents a novel construction of gapless, frustration-free Hamiltonians for MPS by perturbing the matrices, applicable to both injective and non-injective cases.

Findings

Constructed gapless 'uncle' Hamiltonians for non-injective MPS.

Extended the construction to injective MPS in the thermodynamic limit.

Provided a spectrum that is the positive real line for the uncle Hamiltonians.

Abstract

For every Matrix Product State (MPS) one can always construct a so-called parent Hamiltonian. This is a local, frustration free, Hamiltonian which has the MPS as ground state and is gapped. Whenever that parent Hamiltonian has a degenerate ground state (the so-called non-injective case), we construct another 'uncle' Hamiltonian which is local and frustration free but gapless, and its spectrum is $\R^+$. The construction is obtained by linearly perturbing the matrices building up the state in a random direction, and then taking the limit where the perturbation goes to zero. For MPS where the parent Hamiltonian has a unique ground state (the so-called injective case) we also build such uncle Hamiltonian with the same properties in the thermodynamic limit.