Approximate symmetries of Hamiltonians

• Published in: Journal of mathematical physics 2017-08, Vol.58 (8), p.1
• Main Authors: Chubb, Christopher T., Flammia, Steven T.
• Format: Article
• Language: English
• Subjects: Approximation; Commutation; Commutators; Commuting; Eigenvalues; Eigenvectors; Error analysis; Error correcting codes; Error correction; Excitation spectra; Hamiltonian functions; Hilbert space; Normality; Norms; Operators; Physics; Symmetry
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.4998921

We explore the relationship between approximate symmetries of a gapped Hamiltonian and the structure of its ground space. We start by considering approximate symmetry operators, defined as unitary operators whose commutators with the Hamiltonian have norms that are sufficiently small. We show that when approximate symmetry operators can be restricted to the ground space while approximately preserving certain mutual commutation relations. 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 under an added normality constraint, which may be of independent interest.