Exhaustive Characterization of Quantum Many-Body Scars using Commutant Algebras

TL;DR

The paper presents a universal algebraic framework for quantum many-body scars based on local and commutant (bond) algebras. By applying the Double Commutant Theorem, it constructs exhaustive algebras of all local Hamiltonians that possess a given QMBS content, revealing a deep link between scars, conventional symmetries, and Hilbert-space fragmentation. It unifies prior QMBS formalisms (Shiraishi-Mori, Group-Invariant, quasisymmetry) within this algebraic language and demonstrates with canonical towers (AKLT, spin-1 XY bimagnon, Hubbard η-pairing, and the spin-1/2 ferromagnet) how Type I and Type II symmetric Hamiltonians emerge. The framework leads to precise definitions of QMBS, predicts equal-spacing towers in spectra, and clarifies dynamics and ETH violation, providing a powerful tool to classify, construct, and analyze scars across models. These results not only consolidate existing approaches but also guide the search for new QMBS by its exhaustive, locality-aware algebraic characterizations.

Abstract

We study Quantum Many-Body Scars (QMBS) in the language of commutant algebras, which are defined as symmetry algebras of families of local Hamiltonians. This framework explains the origin of dynamically disconnected subspaces seen in models with exact QMBS, i.e., the large "thermal" subspace and the small "non-thermal" subspace, which are attributed to the existence of unconventional non-local conserved quantities in the commutant; hence this unifies the study of conventional symmetries and weak ergodicity breaking phenomena into a single framework. Furthermore, this language enables us to use the von Neumann Double Commutant Theorem (DCT) to formally write down the exhaustive algebra of all Hamiltonians with a desired set of QMBS, which demonstrates that QMBS survive under large classes of local perturbations. We illustrate this using several standard examples of QMBS, including the spin-1/2 ferromagnetic, AKLT, spin-1 XY $π$-bimagnon, and the electronic $η$-pairing towers of states; and in each of these cases we explicitly write down a set of generators for the full algebra of Hamiltonians with these QMBS.Understanding this hidden structure in QMBS Hamiltonians also allows us to recover results of previous "brute-force" numerical searches for such Hamiltonians. In addition, this language clearly demonstrates the equivalence of several unified formalisms for QMBS proposed in the literature, and also illustrates the connection between two apparently distinct classes of QMBS Hamiltonians -- those that are captured by the so-called Shiraishi-Mori construction, and those that lie beyond. Finally, we show that this framework motivates a precise definition for QMBS that automatically implies that they violate the conventional Eigenstate Thermalization Hypothesis (ETH), and we discuss its implications to dynamics.

Figures (1)

• Figure 1: Summary of the local and commutant algebras and associated block decompositions that appear enroute the exhaustive description of towers of QMBS. We show explicit algebras for the ferromagnetic states $\{\ket{\Psi_n}\}$ as QMBS, but similar block decompositions hold for other examples we study. Particular Hamiltonians that realize these block decompositions have been studied previously, but the algebra language provides an exhaustive characterization of such Hamiltonians. This framework is also much more general, treats symmetries and scars on similar footing, and captures examples such as the AKLT tower of QMBS that do not fit into previous frameworks. (a) Pre-bond algebra generated by a set of strictly local terms that have a degenerate set of common eigenstates, which are the "target states." Such an algebra usually has a larger commutant which leads to other symmetry sectors in the block decomposition; this could be a conventional non-Abelian symmetry such as $SU(2)$, although not always. (b) An extensive local "lifting operator" added to the generators of the pre-bond algebra lifts degeneracies between the target states, while still preserving some symmetry sectors. In the ferromagnetic example, this corresponds to a dynamical $SU(2)$ symmetry, and the states are split into an equally spaced tower in any local Hamiltonian from this algebra. (c) Additional symmetries of the pre-bond algebra can be broken by the inclusion of terms that preserve the degenerate target states while mixing the remaining symmetry sectors into a single large thermal block; the target states are now examples of degenerate scars. (d) The lifting operator and the terms that break other symmetries of the pre-bond algebra can be added to obtain the typical decomposition in the case of QMBS systems, into a thermal block and a scar block composed of non-degenerate scar states. In the ferromagnetic example, we conjecture that the scar states appear as an equally spaced tower in any local Hamiltonian in this algebra. This is the algebra that ultimately exhaustively characterizes the QMBS, while the algebras in (a)-(c) are simply motivating steps.

Theorems & Definitions (22)

• Theorem 2.1: DCT
• Definition
• Definition
• Conjecture 3.1
• Conjecture 3.2
• Lemma 4.1
• Lemma 6.1
• Lemma A.1
• proof
• Lemma B.1