Approximate symmetries of Hamiltonians

TL;DR

This paper investigates how approximate symmetries of gapped Hamiltonians relate to their ground space structure, generalizing classical theorems and applying to topological phases and error correction.

Contribution

It extends the Stone-von Neumann theorem to approximate relations and shows how approximate symmetry operators can certify ground state degeneracy.

Findings

Approximate symmetry operators can be restricted to the ground space with low distortion.

Generalization of the Stone-von Neumann theorem to approximate canonical commutation relations.

Provides exponential bounds on shared approximate eigenvectors of approximately commuting operators.

Abstract

We explore the relationship between approximate symmetries of a gapped Hamiltonian and the structure of its ground space. We start by showing that approximate symmetry operators---unitary operators whose commutators with the Hamiltonian have norms that are sufficiently small---which possess certain mutual commutation relations can be restricted to the ground space with low distortion. We generalize the Stone-von Neumann theorem to matrices that approximately satisfy the canonical (Heisenberg-Weyl-type) commutation relations, and use this to show that approximate symmetry operators can certify the degeneracy of the ground space even though they only approximately form a group. Importantly, the notions of "approximate" and "small" are all independent of the dimension of the ambient Hilbert space, and depend only on the degeneracy in the ground space. Our analysis additionally holds for any gapped band of sufficiently small width in the excited spectrum of the Hamiltonian, and we discuss applications of these ideas to topological quantum phases of matter and topological quantum error correcting codes. Finally, in our analysis we also provide an exponential improvement upon bounds concerning the existence of shared approximate eigenvectors of approximately commuting operators which may be of independent interest.