Citation: Chubb, Christopher T., and Steven T. Flammia. "Approximate symmetries of Hamiltonians." Journal of Mathematical Physics 58.8 (2017).
Web: https://arxiv.org/abs/1608.02600
Tags: Approximate-structure
In this paper, the authors discuss approximate symmetries of Hamiltonians. That is, cases in which you have several unitary operators which approximately commute with a Hamiltonian. The point is that in general these unitaries don't imply that the Hamiltonian has a degenerate ground space - there could be a splitting in the degeneracy. However, if we know that the Hamiltonian has an exactly degenerate ground space with an energy gap then we can ask whether these approximate symmetries imply a degeneracy. The answers obtained are all positive (yes, approximate symmetries imply degeneracy) which is in line with what one would exact from Ulam stability.
It seems like the takeaway from this paper is that symmetry tends to be an Ulam-stable property, even one you go beyond considerations from standard Ulam stability and compact Lie groups. Maybe the point here is that they consider symmetry under finitely many operators, and imposed commutation relations on those operators why imply that the algebra they generate is approximately a group extension of finitely many copies of Z. Since extensions of amenable groups by amenable groups are amenable, this means that the algebra they approximately generate is amenable. Thus, we would expect to be able to apply Ulam stability results. Note that this isn't how they prove their results - they use more standard techniques from perturbation theory.