Locality forces equal energy spacing of quantum many-body scar towers

TL;DR

This paper proves that local Hamiltonians hosting quantum many-body scars, such as Dicke states, necessarily produce equally spaced energy levels, revealing a fundamental link between locality and scar structure.

Contribution

It establishes that local Hamiltonians with scar eigenstates like Dicke states must have equally spaced energies, extending to various graph structures and interaction types.

Findings

Dicke states' energies are necessarily equally spaced in local Hamiltonians.

Equal spacing extends to arbitrary bounded-degree graphs and multi-site quasiparticles.

States within the scar manifold exhibit frozen entanglement dynamics.

Abstract

Quantum many-body scars are non-thermal eigenstates embedded in the spectra of otherwise non-integrable Hamiltonians. Paradigmatic examples often appear as quasiparticle towers of states, such as the maximally ferromagnetic spin-1/2 states, also known as Dicke states. A distinguishing feature of quantum many-body scars is that they admit multiple local "parent" Hamiltonians for which they are exact eigenstates. In this work, we show that the locality of such parent Hamiltonians strongly constrains the relative placement of these states within the energy spectrum. In particular, we prove that if the full set of Dicke states are exact eigenstates of an extensive local Hamiltonian, then their energies must necessarily be equally spaced. Our proof builds on recent results concerning parent Hamiltonians of the $W$ state, together with general algebraic structures underlying such quasiparticle towers. We further demonstrate that this equal spacing property extends to local Hamiltonians defined on arbitrary bounded-degree graphs, including regular lattices in any spatial dimension and expander graphs. Hamiltonians with $k$-local interactions and a bounded number of interaction terms per site are also encompassed by our proof. On the same classes of graphs, we additionally establish equal spacing for towers constructed from multi-site quasiparticles on top of product states. For the towers considered here, an immediate corollary of the equal spacing property is that any state initialized entirely within the quantum many-body scar manifold exhibits completely frozen entanglement dynamics under any local Hamiltonian for which those scars are exact eigenstates. Overall, our results reveal a stringent interplay between locality and the structure of quantum many-body scars.