Enforced Gaplessness from States with Exponentially Decaying Correlations

It is well known that an exponentially localized Hamiltonian must be gapless if its ground state has algebraic correlations. We show that even certain exponentially decaying correlations can imply gaplessness. This is exemplified by the deformed toric code $\propto \exp(\beta \sum_{\ell} Z_{\ell}) |\mathsf{TC}\rangle$, where $|\mathsf{TC}\rangle$ is a fixed-point toric code wavefunction. Although it has a confined regime for $\beta > \beta_c$, recent work has drawn attention to its perimeter law loop correlations. Here, we show that these unusual loop correlations -- namely, perimeter law coexisting with a 1-form symmetry whose disorder operator has long-range order -- imply that any local parent Hamiltonian must either be gapless or have a degeneracy scaling with system size. Moreover, we construct a variational low-energy state for arbitrary local frustration-free Hamiltonians, upper bounding the finite-size gap by $O(1/L^3)$ on periodic boundary conditions. Strikingly, these variational states look like loop waves -- non-quasiparticle analogs of spin waves -- generated from the ground state by non-local loop operators. Our findings have implications for identifying the subset of Hilbert space to which gapped ground states belong, and the techniques have wide applicability. For instance, a corollary of our first result is that Glauber dynamics for the ordered phase of the two-dimensional classical Ising model on the torus must have a gapless Markov transition matrix, with our second result bounding its gap.