Angelo Luciaanlucia

The problem of determining the existence of a spectral gap in a lattice quantum spin system was previously shown to be undecidable for one [J. Bausch et al., "Undecidability of the spectral gap in one dimension", Physical Review X 10 (2020)] or more dimensions [T. S. Cubitt et al., "Undecidability of the spectral gap", Nature 528 (2015)]. In these works, families of nearest-neighbor interactions are constructed whose spectral gap depends on the outcome of a Turing machine Halting problem, therefore making it impossible for an algorithm to predict its existence. While these models are translationally invariant, they are not invariant under the other symmetries of the lattice, a property which is commonly found in physically relevant cases, posing the question of whether the spectral gap is still an undecidable problem for Hamiltonians with stronger symmetry constraints. We give a positive answer to this question, in the case of models with 4-body (plaquette) interactions on the square lattice satisfying rotation, but not reflection, symmetry: rotational symmetry is not enough to make the problem decidable.

We prove that the critical finite-size gap scaling for frustration-free Hamiltonians is of inverse-square type. The novelty of this note is that the result is proved on general graphs and for general finite-range interactions. Therefore, the inverse-square critical gap scaling is a robust, universal property of finite-range frustration-free Hamiltonians. This places further limits on their ability to produce conformal field theories in the continuum limit. Our proof refines the divide-and-conquer strategy of Kastoryano and the second author through the refined Detectability Lemma of Gosset--Huang.

We consider the spectral gap question for AKLT models defined on decorated versions of simple, connected graphs G. This class of decorated graphs, which are defined by replacing all edges of $G$ with a chain of $n$ sites, in particular includes any decorated multi-dimensional lattice. Using the Tensor Network States (TNS) approach from a work by Abdul-Rahman et. al. 2020, we prove that if the decoration parameter is larger than a linear function of the maximal vertex degree, then the decorated model has a nonvanishing spectral gap above the ground state energy.