Energy density of variational states

TL;DR

This paper investigates the conditions under which a trial wavefunction can approximate the energy density of a ground state from a different phase, revealing limitations based on symmetry and topological order.

Contribution

It demonstrates that low variational energy density can be achieved with trial states representing different phases, especially in low dimensions or with topological order.

Findings

Low energy density approximation is possible across different phases in certain cases.

Limitations exist for higher dimensions with symmetry-breaking phases and trivial phases with topological order.

The results clarify when phase differences prevent accurate energy density approximation.

Abstract

We show, in several important and general cases, that a low variational energy density of a trial state is possible even when the trial state represents a different phase from the ground state. Specifically, we ask whether the ground state energy density of a Hamiltonian whose ground state is in phase A can be approximated to arbitrary accuracy by a wavefunction which represents a different phase B. We show this is indeed the case when A has discrete symmetry breaking order in one dimension or topological order in two dimensions, while B is disordered. We argue that, if reasonable conditions of physicality are imposed upon the trial wavefunction, then this is not possible when A has discrete symmetry breaking in dimensions greater than one and B is symmetric, or when A is topologically trivial and B has topological order.