Approximate symmetries of Hamiltonians

Abstract

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.

Observation of gauge invariance in a 71-site Bose–Hubbard quantum simulator

“…If all gauge-violating many-body states experience such energy penalties, a deformed symmetry emerges that is perturbatively close to the ideal original one [40], and which indefinitely suppresses gauge invarianceviolating processes [18]. In the present case, the leading gauge-violating processes are energetically penalized, but violations at distant sites may in principle energetically compensate each other.…”

Reliability of Lattice Gauge Theories

“…This regime is expected once the gauge-invariant sector we start in is energetically well separated from other sectors, as can be shown in degenerate perturbation theory and made mathematically rigorous by adapting the results of Ref. [22] (see the Supplemental Material [21]). For any unitary symmetry that is broken on a scale ∼λ, the opening of a gap generates an emergent symmetry that is perturbatively in ðλ=VÞ 2 close to the original one.…”

Approximate Noether symmetries and collineations for regular perturbative Lagrangians

“…In this work we are interested in the application of Noether's theorem for approximate symmetries on some regular perturbative Lagrangians. Approximate Noether symmetries [14,15] provide approximate first integrals, functions which can be used as conservation laws until a specific step in numerical integrations. This kind of approximate conservation laws have played an important role for the study of chaotic systems -for an extended discussion we refer the reader to an application in Galactic dynamics [16,17,18].Whilst recently, the advent of automated software algorithms has made light work of calculating symmetries [19].…”