From tunnels to towers: quantum scars from Lie Algebras and q-deformed Lie Algebras

TL;DR

This paper introduces a symmetry-based framework for constructing many-body Hamiltonians with quantum scarred eigenstates, extending the phenomenon to models with Lie algebra and q-deformed Lie algebra symmetries, including new models and generalizations.

Contribution

The authors develop a general method to generate Hamiltonians with scar states based on non-Abelian and q-deformed symmetries, broadening the class of models exhibiting quantum scars.

Findings

Identified how symmetry breaking preserves specific low-entanglement states as scars.

Constructed new models with scars transforming under SU(3) and q-deformed SU(2) symmetries.

Presented generalized AKLT models with scar states not forming irreducible representations.

Abstract

We present a general symmetry-based framework for obtaining many-body Hamiltonians with scarred eigenstates that do not obey the eigenstate thermalization hypothesis. Our models are derived from parent Hamiltonians with a non-Abelian (or q-deformed) symmetry, whose eigenspectra are organized as degenerate multiplets that transform as irreducible representations of the symmetry (`tunnels'). We show that large classes of perturbations break the symmetry, but in a manner that preserves a particular low-entanglement multiplet of states -- thereby giving generic, thermal spectra with a `shadow' of the broken symmetry in the form of scars. The generators of the Lie algebra furnish operators with `spectrum generating algebras' that can be used to lift the degeneracy of the scar states and promote them to equally spaced `towers'. Our framework applies to several known models with scars, but we also introduce new models with scars that transform as irreducible representations of symmetries such as SU(3) and $q$-deformed SU(2), significantly generalizing the types of systems known to harbor this phenomenon. Additionally, we present new examples of generalized AKLT models with scar states that do not transform in an irreducible representation of the relevant symmetry. These are derived from parent Hamiltonians with enhanced symmetries, and bring AKLT-like models into our framework.