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.
Highlights
AND GENERAL FRAMEWORKA central question in nonequilibrium quantum dynamics is whether reversible unitary dynamics in a closed quantum system can establish local thermal equilibrium
Much insight into quantum thermalization follows from the eigenstate thermalization hypothesis (ETH) [1,2,3,4,5], a strong version of which posits that every finite-temperature eigenstate of a thermalizing system reproduces thermal expectation values locally [6]
The parent Hamiltonian Hsym generally has a larger symmetry, so that its eigenstates can still be simultaneously diagonalized with HSG and the picture of tunnels to towers still applies, the states in the tower of scars need not have a definite eigenvalue under the Casimir Q2 and are not contained within a single irreducible representation of G = SU(2)
Summary
A central question in nonequilibrium quantum dynamics is whether reversible unitary dynamics in a closed quantum system can establish local thermal equilibrium. While such constructions have been very useful for explicitly deriving and unifying the presence of scars in specific “one-off” models, qualitative pictures of when and how scars may arise more generally are still largely missing It is still largely unclear where in the general space of operators and states we may expect to find a set {H, Q+, |ψ0 } such that the conditions in Eq (2) leading to (weak) ETH violations are met. We will discuss models where the scar tower does not transform as an irrep of the symmetry group G and/or where HSG is not a generator of the symmetry but still has a SGA with the raising operators This is possible when Hsym has an expanded symmetry, which allows Hsym to be simultaneously diagonalized with HSG and have tunnels of degenerate eigenstates that do not transform as an irrep. V, and present various technical details in a series of appendices
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