Asymptotic Quantum Many-Body Scars (AQMBS)

• AQMBS are non-eigenstates in finite quantum lattice systems that exhibit vanishing energy variance and diverging relaxation times as the system grows, illustrating weak ergodicity breaking.
• Their construction relies on momentum-resolved superpositions and parent Hamiltonians that ensure orthogonality to conventional scar towers and maintain subvolume-law entanglement.
• Numerical simulations and analytical techniques confirm persistent nonthermal dynamics and revival behavior, making AQMBS promising for quantum simulation and information storage.

Asymptotic Quantum Many-Body Scars (AQMBS) designate highly structured, low-entanglement families of states in nonintegrable quantum lattice systems that exhibit slow relaxation and long-lived coherent dynamics for system sizes approaching the thermodynamic limit, despite not being exact energy eigenstates. AQMBS are distinguished from conventional quantum many-body scars (QMBS) by being strictly non-eigenstates at any finite system size but displaying vanishing energy variance and diverging relaxation times as system size grows, with orthogonality to the scarred eigenstate manifold. These states reveal persistent non-ergodic dynamics in regions of the spectrum that otherwise conform to the eigenstate thermalization hypothesis (ETH), thus demonstrating weak ergodicity breaking beyond what is captured by exact eigenstate analysis (Gotta et al., 2023, Kunimi et al., 7 May 2025, Hashimoto et al., 8 Jan 2026).

1. Defining Asymptotic Quantum Many-Body Scars

Quantum many-body scars (QMBS) are isolated, nonthermal eigenstates of a nonintegrable Hamiltonian $H$ that weakly violate ETH, often forming tower structures through spectrum-generating algebras (SGA) and showing subvolume (typically $O(\log L)$) entanglement entropy even at high energy. The prototypical example is the tower of eigenstates $$|n,\pi\rangle = \frac{(J^+_\pi)^n}{\sqrt{N_{n,\pi}}}|\Downarrow\rangle$$ in the spin-1 XY chain defined by $$H = J\sum_j(S^x_jS^x_{j+1}+S^y_jS^y_{j+1}) + h\sum_j S^z_j + D\sum_j (S^z_j)^2 + J_3\sum_j (S^x_jS^x_{j+3}+S^y_jS^y_{j+3})$$, where $J^+_k = \frac{1}{2}\sum_{j=1}^L e^{ikj}(S^+_j)^2$.

By contrast, AQMBS are constructed as coherent superpositions of states that are not energy eigenstates for any finite $L$, but satisfy three central properties:

• Orthogonality to the exact QMBS tower for all finite $L$: For any $k \neq \pi$, $$\langle n,k|n',\pi\rangle = \delta_{n,n'}\delta_{k,\pi}$$.
• Finite, subvolume-law entanglement entropy $S_A \sim \log L + O(1)$.
• Asymptotically vanishing energy variance: For $k = \pi + \frac{2\pi}{L}m$, $\Delta H^2_k \sim O(L^{-2}) \rightarrow 0$ as $L \rightarrow \infty$.

The asymptotic suppression of energy uncertainty yields diverging relaxation times $\tau \sim 1/\Delta H \sim L$, such that these states exhibit scar-like revival and nonthermal behavior even when immersed in a spectral continuum that would otherwise be classified as thermal by ETH (Gotta et al., 2023).

2. Explicit Construction and Analytical Properties

A canonical construction uses momentum-resolved superpositions. For each $k$ in the Brillouin zone, define

$$|n,k\rangle = \frac{J^+_k (J^+_\pi)^{n-1}}{\sqrt{N_{n,k}}}|\Downarrow\rangle, \quad N_{n,k} = \langle\Downarrow|(J^-_\pi)^{n-1}(J^-_k J^+_k)(J^+_\pi)^{n-1}|\Downarrow\rangle$$

Real-space wavepackets may be built as $$|\psi_j\rangle = \frac{1}{\sqrt{L(L-n+1)}}\sum_{k} e^{-ikj}|n,k\rangle$$, providing spatially localized analogues.

Energy variance scaling is key to their identification: $$\Delta H^2_k = 4[J^2\cos^2(k/2) + J_3^2\cos^2(3k/2)].$$ For $k = \pi + 2\pi m/L$ and fixed $m$, $$\Delta H^2_k \sim [J^2 + 9J_3^2] (\pi m / L)^2 \to 0$$, resulting in slow dephasing and long-lived coherent oscillations under time evolution (Gotta et al., 2023).

Fidelity decay from an initial AQMBS state obeys a Gaussian law $F(t) \approx \exp[-(\Delta H)^2 t^2]$ and relaxation time $\tau \sim 1/\Delta H \sim L$. Numerically, time-evolving block decimation (TEBD) simulations confirm the $t/L$ scaling collapse for both fidelity and local magnetization for system sizes up to $L=60$, unambiguously establishing the asymptotic nature of the phenomenon.

3. General Systematic Construction and Parent Hamiltonians

A systematic approach for constructing AQMBS leverages the decomposition of the Hamiltonian into subspaces where QMBS are embedded as ground states of a positive semi-definite "parent Hamiltonian", with AQMBS realized as gapless low-lying excitations. The procedure:

• Identify the restricted spectrum-generating algebra (RSGA): operators $H$, $Q^\dagger$, and a root state $|S_0\rangle$ such that $H|S_0\rangle=E_0|S_0\rangle$, $$[H,Q^\dagger]|S_0\rangle = \mathcal E Q^\dagger|S_0\rangle$$, etc.
• Decompose $H_A = H_0 + H'_p$ with $H_0 = \sum_j h_j$ local and an appropriately chosen projector $\mathcal P$ to a subspace $\mathcal H_P$ containing the original scar states.
• Construct the parent Hamiltonian $$H_\mathrm{parent} = \mathcal P_n H_0^2 \mathcal P_n$$, ensuring the original scar state is its zero mode.
• AQMBS are then the lowest-lying excitations in this sector, with energy eigenvalues $\varepsilon^{(\alpha)}_n \to 0$ as $L\to\infty$ (Kunimi et al., 7 May 2025, Hashimoto et al., 8 Jan 2026).

In SU($N$) Hubbard models, this construction leads to a mapping from the scar sector to a ferromagnetic SU($N$) Heisenberg chain, whose gapless magnons are explicit, analytic AQMBS with proven asymptotic properties (orthogonality to the scar tower, vanishing energy variance, subvolume-law entanglement) (Hashimoto et al., 8 Jan 2026).

Furthermore, the parent Hamiltonian formalism reveals an emergent $\mathcal N=2$ supersymmetry, with the QMBS wavefunction interpreted as the unbroken SUSY ground state, and the AQMBS forming SUSY partners (Goldstino-like gapless modes) (Kunimi et al., 7 May 2025).

4. Dynamical Signatures, Robustness, and Numerical Evidence

AQMBS exhibit slow dynamics and revival behavior captured by diverging relaxation times in the thermodynamic limit. In the spin-1 XY model, TEBD/MPS simulations show that fidelity $F(t)$ and local magnetizations remain near initial values up to $t \sim O(L)$. This behavior stands in stark contrast with typical thermal or product states, which rapidly decay on $O(1)$ or $O(\sqrt L)$ time scales.

Weak perturbations that remove the exact QMBS, such as $V = \frac{J_z}{L}\sum_j S^z_jS^z_{j+1}$ (with norm $O(1)$), only minimally affect AQMBS: the energy variance under $H' = H + V$ scales as $\Delta H'^2 \sim C/L \to 0$, and AQMBS persist with relaxation times $\tau \sim \sqrt{L}$ (Gotta et al., 2023).

Finite-$S$ scaling in models such as the truncated Schwinger model (TSM) shows exceptionally robust AQMBS for all $S$ and $L$, in contrast to the quantum link model (QLM) where the scarring signals diminish as $S\to\infty$ (Desaules et al., 2022).

5. Semiclassical and Classical-Quantum Correspondence

In systems allowing for a true classical (mean-field) limit, AQMBS emerge as quantum analogues of phase-space localization near unstable periodic orbits, consistent with Heller's scar criterion: $\sum_j \lambda_j T \lesssim 2\pi$, where $\lambda_j$ are positive Lyapunov exponents and $T$ the period. Staggered-dimer scars in the Bose-Hubbard model exemplify this, with tube (coherent) states localized about unstable orbits showing persistent oscillations over timescales scaling with Lyapunov exponents (Hummel et al., 2022).

Hybridization of scarred eigenstates manifests as irregular, nonexponential oscillations of local observables, distinctly different from regular tunneling between integrable "cat" states. Numerical and analytical studies for Bose-Hubbard clusters ($L=4$, $8$, $12$) confirm high inverse participation ratios well above ergodic backgrounds and persistent oscillations that do not decay up to $Jt \gg 1$.

6. Experimental Realization and Observational Implications

Cold atom and Rydberg array platforms are viable for observing AQMBS. In the TSM, one can prepare the extreme vacuum state $|0_{-}\rangle$ and quench to the TSM Hamiltonian with $S \sim 2-5$ and $L \lesssim 20$, observing fidelity revivals and slow entanglement growth. In the Bose-Hubbard context, initialized staggered-dimer states display persistent oscillations of on-site populations in optical lattices for experimentally relevant atom numbers and couplings (Desaules et al., 2022, Hummel et al., 2022).

Observable signatures include extended coherence in fidelity $F(t)$, slow decoherence of local order parameters, and suppressed entanglement entropy relative to thermal backgrounds. These phenomena distinguish AQMBS as robust nonthermal dynamical subspaces even in models meeting all criteria for nonintegrable chaos.

7. Relation to Non-Hermitian Scar Phases and Future Directions

In open or measurement-driven quantum systems, non-Hermitian dynamics can sharply stabilize quantum scar subspaces as true nonequilibrium steady states—a direct dynamical realization of AQMBS. This non-Hermitian QMBS phase undergoes first-order transitions, tunable by projective measurement strength or dissipative "scar-stabilizing" driving, with spectral and order-parameter signatures admitting classical Ising and matrix product operator (MPO) analogues (Omiya et al., 30 Jul 2025).

The mapping between parent-Hamiltonian constructions, restricted algebraic structures, and non-Hermitian stabilization underpins a unified framework for asymptotic scarring in quantum many-body systems, with analytic control in a growing class of nonintegrable models, potential for experimental quantum simulation, and relevance for quantum information storage in non-ergodic phases.