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Generalized Shiraishi--Mori construction is exhaustive for ferromagnetic quantum many-body scars

Keita Omiya

TL;DR

This paper analyzes quantum many-body scars (QMBS) that take a ferromagnetic, totally symmetric form and shows that any local Hamiltonian hosting such scar states must decompose into an annihilator built from local projectors and a Zeeman term acting on the scar manifold. The annihilator itself further factorizes into strictly local projector components, rendering a generalized Shiraishi–Mori construction essentially exhaustive for this scar class. The authors leverage Schur–Weyl duality and the bicommutant theorem to connect the scar subspace with symmetric-group representations, providing a universal structural framework for QMBS. They also discuss how, in the weight-non-preserving sector, DM-like interactions can appear while preserving locality, and outline open questions about locality of the non-preserving part and extensions to subspaces beyond the full symmetric sector. Overall, the work clarifies the operator architecture underlying ferromagnetic QMBS and offers a unifying lens for constructing and analyzing scarred Hamiltonians.

Abstract

Quantum many-body scars (QMBS) constitute a subtle violation of ergodicity through a set of non-thermal eigenstates, referred to as scar states, which are embedded in an otherwise thermal spectrum. In a broad class of known examples, these scar states admit a simple interpretation: they are magnon excitations of fixed momentum on top of a ferromagnetic background. In this paper we prove that any Hamiltonian hosting such ``ferromagnetic scar states'' necessarily admits a structural decomposition into a Zeeman term and an ``annihilator'' that annihilates the entire scar manifold. Moreover, we show that this annihilator must itself decompose into a sum of terms built from local projectors that locally annihilate the scar states. This architecture is closely related to the Shiraishi--Mori construction, and our main theorem establishes that an appropriate generalization of that construction is in fact essentially exhaustive for this class of scar states.

Generalized Shiraishi--Mori construction is exhaustive for ferromagnetic quantum many-body scars

TL;DR

This paper analyzes quantum many-body scars (QMBS) that take a ferromagnetic, totally symmetric form and shows that any local Hamiltonian hosting such scar states must decompose into an annihilator built from local projectors and a Zeeman term acting on the scar manifold. The annihilator itself further factorizes into strictly local projector components, rendering a generalized Shiraishi–Mori construction essentially exhaustive for this scar class. The authors leverage Schur–Weyl duality and the bicommutant theorem to connect the scar subspace with symmetric-group representations, providing a universal structural framework for QMBS. They also discuss how, in the weight-non-preserving sector, DM-like interactions can appear while preserving locality, and outline open questions about locality of the non-preserving part and extensions to subspaces beyond the full symmetric sector. Overall, the work clarifies the operator architecture underlying ferromagnetic QMBS and offers a unifying lens for constructing and analyzing scarred Hamiltonians.

Abstract

Quantum many-body scars (QMBS) constitute a subtle violation of ergodicity through a set of non-thermal eigenstates, referred to as scar states, which are embedded in an otherwise thermal spectrum. In a broad class of known examples, these scar states admit a simple interpretation: they are magnon excitations of fixed momentum on top of a ferromagnetic background. In this paper we prove that any Hamiltonian hosting such ``ferromagnetic scar states'' necessarily admits a structural decomposition into a Zeeman term and an ``annihilator'' that annihilates the entire scar manifold. Moreover, we show that this annihilator must itself decompose into a sum of terms built from local projectors that locally annihilate the scar states. This architecture is closely related to the Shiraishi--Mori construction, and our main theorem establishes that an appropriate generalization of that construction is in fact essentially exhaustive for this class of scar states.
Paper Structure (16 sections, 15 theorems, 109 equations, 2 figures)

This paper contains 16 sections, 15 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation: the spin-1 XY model
  3. Setup
  4. Preliminaries
  5. Representation theory of the symmetric group
  6. Bicommutant theorem
  7. Annihilation implies local projectors
  8. Locality implies a Zeeman term
  9. Sufficient conditions for local annihilators
  10. Discussions
  11. Proofs
  12. Proof of the orthogonality in different tableaux
  13. Proof of Theorem \ref{['thm:GL(V)']}
  14. Proof of Lemma \ref{['lemma:su(V)']}
  15. Proof of Lemma \ref{['lemma:local HA']}
  16. ...and 1 more sections

Key Result

Theorem 4.1

For any finite group $G$, the regular representation $\mathbb{C}[G]$ contains every irreps as a subrepresentation, where $\lambda$ labels irreps and $V_\lambda$ denotes the corresponding irrep.

Figures (2)

  • Figure 1: (Left) a partition of $5$ into $(3,2)$, which correspond to an irrep of $\mathfrak{S}_5$. (Right) the construction of the Young symmetrizer corresponding to $(3,2)$. The subgroup $\mathfrak{P}_{(3,2)}$, which preserves each row, is isomorphic to $\mathfrak{S}_3\times\mathfrak{S}_2$, permuting within $\{1,3,4\}$ and $\{2,5\}$. Similarly, the subgroup $\mathfrak{Q}_{(3,2)}$, which preserves each column, is isomorphic to $\mathfrak{S}_2\times\mathfrak{S}_2(\times\mathfrak{S}_1)$, permuting within $\{1,2\}$ and $\{3,5\}$.
  • Figure 2: The doubled loop connects the bottom left point and the top right point.

Theorems & Definitions (25)

  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 5.1
  • Remark
  • Remark
  • proof
  • Corollary 5.2
  • Lemma 6.1
  • Theorem 6.2
  • ...and 15 more