Unraveling additional quantum many-body scars of the spin-$1$ $XY$ model with Fock-space cages and commutant algebras

TL;DR

This work addresses weak ergodicity breaking in a nonintegrable spin-$1$ XY chain by identifying an extensive zero-energy manifold protected by $U(1)$ magnetization and chiral symmetry, enabling exact scar eigenstates embedded in a thermal spectrum. It introduces Fock-space cage (FSC) states, whose destructive interference confines their support to sparse subgraphs, yielding subextensive entanglement and coherent revivals under transverse fields, and it develops a commutant-algebra framework to classify scars as simultaneous eigenstates of noncommuting local operators. The authors uncover new scar families beyond FSCs, including a tower of volume-entangled states and a set of mirror-dimer states with unconstrained center spins, all robust to carefully chosen perturbations. By combining geometric interference with algebraic structure, the paper provides systematic routes to identify, classify, and engineer QMBS in generic many-body quantum systems and highlights potential implications for long-lived coherent dynamics and quantum information processing.

Abstract

Quantum many-body scars (QMBS) represent a mechanism for weak ergodicity breaking, characterized by the coexistence of atypical non-thermal eigenstates within an otherwise thermalizing many-body spectrum. In this work, we revisit the spin-$1$ $XY$ model on a periodic chain and construct several new families of exact scar eigenstates embedded within its extensively degenerate manifolds that owe their origins to an interplay of $U(1)$ magnetization conservation and chiral symmetries. We go beyond previously studied towers of states and first identify a novel set of interference-protected eigenstates resembling Fock space cage states, where destructive interference confines the wave function to sparse subgraphs of the Fock space. These states exhibit subextensive entanglement entropy, and when subjected to a transverse magnetic field, form equally spaced ladders whose coherent superpositions display long-lived fidelity oscillations. We further reveal a simpler organizing principle behind these nonthermal states by using the commutant algebra framework, in particular by showing that they are simultaneous eigenstates of non-commuting local operators. Moreover, in doing so, we uncover two more novel families of exact scars: a tower of volume-entangled states, and a set of mirror-dimer states with some free local degrees of freedom. Our results illustrate the power and interplay of interference-based and algebraic mechanisms of non-ergodicity, offering systematic routes to identifying and classifying QMBS in generic many-body quantum systems.

Figures (13)

• Figure 1: Many-body density of states $\rho(E)$ as a function of energy $E$, in units of the coupling strength $J$, for the $XY$ Hamiltonian $H_{XY}$ [see Eq. \ref{['eq: full_Hamiltonian']}] for $2L{=}8$ sites. The sharp central peak at $E{=}0$ reflects the extensive degeneracy of exact zero-energy eigenstates of $H_{XY}$.
• Figure 2: Fock-space graph of $H_{XY}$ [see Eq. \ref{['eq: full_Hamiltonian']}] for $2L{=}4$ sites. Basis states are colored by their $\hat{\mathcal{C}}$-parity [see Eq. \ref{['eq: chiral_symmetry_operator']} for definition of the operator $\hat{\mathcal{C}}$]: orange for sublattice $A$ (even) and blue for sublattice $B$ (odd). Edges indicate nonzero matrix elements of $H_{XY}$, connecting only opposite sublattices, revealing the bipartite nature of its Fock-space graph.
• Figure 3: Schematic illustration of the Fock-space graph connectivity for the Fock-space cage state $|\Omega_2\rangle$ in a system with $2L{=}12$ sites in the magnetization sector $M{=}{-}10$. The orange nodes represent the basis states $|F_{2}^{i}\rangle$ [see Eq. \ref{['eq: FSC_states']}] that constitute the state, and their amplitudes, ${\pm}1$, are indicated by just the sign. The blue nodes are direct neighbors of the orange nodes under the Hamiltonian $H_{XY}$ [see Eq. \ref{['eq: full_Hamiltonian']}], and green edges show these connections. Destructive interference ensures that all blue nodes receive zero net amplitude from the orange nodes. Dimmed nodes represent the remaining Fock space states, which are decoupled from the cage.
• Figure 4: Fidelity $\mathcal{F}(t){=}|\langle\psi(t)|\psi(0)\rangle|^2$ for a spin-$1$ chain of size $2L{=}12$ sites, shown for initial states $|\psi_{r}^{\rm init}\rangle$ [see Eq. \ref{['Eq: Initial_states_for_revivals']}] (with $r{=}1,2,{\cdots},6$) and $|\psi_{\rm vol}^{\rm init}\rangle$ [see Eq. \ref{['eq: EAP_state']}] constructed as a coherent superposition over the scar eigenstates of the towers $|r,n\rangle$, and the volume-entangled tower $|\mathbb{V}_n\rangle$, respectively, and evolved under $H_{0}$ [see Eq. \ref{['eq: full_Hamiltonian']}]. All cases exhibit persistent fidelity revivals with period $\pi/h$, consistent with the uniform energy spacing $2h$, except for $|\psi_{r{=}L}^{\rm init}\rangle$, which oscillates with period $\pi/(2h)$ as discussed in the main text.
• Figure 5: Bipartite entanglement entropy (EE) of all scar states constructed in this work for a spin-$1$ chain of size $2L{=}12$ sites. The plot shows EE of the tower states $\{|r,n\rangle\}$ [see Eq. \ref{['eq: new_towers']}] for $r{=}1,2,{\cdots},6$, and the special volume-entangled tower states $\{|\mathbb{V}_{n}\}\rangle$ [see Eq. \ref{['eq: volume_scar_tower']}], both for the standard bipartition with a cut at $L$, and the fine-tuned antipodal bipartition, discussed in the text. For comparison, the EE of all eigenstates of a non-integrable Hamiltonian $H_{\rm comm}^{(2)}$ [see Eq. \ref{['eq: V_comm_def']}] with $J{=}h{=}1$, $\epsilon_{i}{\in}[0,0.2]$ and periodicity $r{=}2$, is shown in the background as grey dots with color-coded by the density of states with a given EE (darker=higher density). The black dotted line marks the Page value, the expected EE of a random state. Although the scar states lie in different magnetization sectors and are exact eigenstates of the spin-$1$$XY$ model with different integrability-breaking perturbations, this plot provides a standard comparative representation of their subthermal EE, highlighting their nonthermal nature.