Undecidability of the spectral gap in rotationally symmetric Hamiltonians

TL;DR

This paper proves that the spectral gap problem remains undecidable even for rotationally symmetric Hamiltonians on a square lattice, extending previous results to models with specific symmetry constraints.

Contribution

It demonstrates that rotational symmetry alone does not make the spectral gap problem decidable in quantum lattice models.

Findings

Spectral gap undecidability extends to rotationally symmetric Hamiltonians.

Rotational symmetry does not simplify the spectral gap problem.

The result applies to 4-body interactions on the square lattice.

Abstract

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. This poses the question of whether the spectral gap problem could be decidable for Hamiltonians with stronger symmetry constraints. We give a negative 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.