Asymptotic Quantum Many-Body Scars

• Asymptotic scars are nonexact many-body states that exhibit vanishing energy variance as system size increases, resulting in anomalously slow thermalization.
• They arise in diverse models like the spin-1 XY model, random graph ensembles, and holographic setups, providing clear diagnostics via energy variance scaling and fidelity decay.
• Experimental and numerical studies, including trapped-ion simulations, validate that these scar states maintain nonthermal dynamics even under perturbations and in open quantum systems.

Asymptotic scars are scar-like states or structures whose anomalous signatures sharpen only in an asymptotic limit. In the many-body setting, the canonical notion is that of asymptotic quantum many-body scars: families of low-entanglement states that are not exact eigenstates at finite size, but whose energy variance vanishes as the system size increases, so that their relaxation time diverges in the thermodynamic limit (Gotta et al., 2023). Related usages now span gapless-excitation constructions in two-local qubit models, rare weakly entangled states in random-graph ensembles, thermodynamic-limit scar sectors in gauge, dimer, and Majorana systems, open-system analogues under Lindblad evolution, and scar-like sectors in holography; a separate literature uses “scars” for large-domain filamentary patterns in random waves rather than ETH-violating many-body states (Logarić et al., 14 Apr 2026, Jeevanesan, 2023, Biswas et al., 2022, Sun et al., 2022, Gotta, 22 Sep 2025, Liu et al., 4 May 2026, Gass et al., 5 Jun 2026).

1. Definition and diagnostic criteria

In a nonintegrable many-body system, the Eigenstate Thermalization Hypothesis states that almost every energy eigenstate in the bulk of the spectrum behaves thermally, with local observables and entanglement entropies matching those of a Gibbs ensemble at the same energy. Quantum many-body scars are isolated atypical eigenstates that violate ETH, typically showing anomalously low entanglement, such as $\sim \log L$ rather than $\sim L$, and supporting long-lived revivals. Asymptotic quantum many-body scars generalize this notion to states that are not exact eigenstates at finite $L$, but satisfy $\mathrm{Var}(H)\to 0$ as $L\to\infty$; for short times, $$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$, so $\tau\sim 1/\Delta H$ diverges when $\Delta H\to 0$. A central consequence is that scar-like nonthermal dynamics can occur even for states orthogonal to all exact scars and embedded in what is otherwise regarded as the thermal bulk of the spectrum (Gotta et al., 2023).

A closely related finite-size diagnostic uses the energy variance

$$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$

together with the Mandelstam–Tamm bound

$$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$

This makes explicit the distinction from conventional scars: exact finite-$\sim L$0 scar eigenstates generate slow dynamics that is $\sim L$1-independent, whereas asymptotic scars become progressively more stable as $\sim L$2 increases. A common misconception is therefore that scarring requires isolated exact eigenstates at every finite size; the asymptotic constructions show instead that vanishing variance is already sufficient to produce parametrically slow thermalization (Logarić et al., 14 Apr 2026).

2. Canonical construction in the spin-1 XY model

The prototypical closed-system construction starts from the spin-1 XY model with an exact scar tower

$\sim L$3

where

$\sim L$4

From the exact-scar state $\sim L$5, one constructs

$\sim L$6

For $\sim L$7, these states are orthogonal to the exact tower, with $\sim L$8, yet their average energy remains

$\sim L$9

independent of $L$0. Under periodic boundary conditions, the energy variance is

$L$1

Choosing

$L$2

gives

$L$3

so $L$4 although $L$5 is not an eigenstate at any finite $L$6. Exact overlap plots show support only on the bulk ETH-like eigenstates, not on the exact-scar eigenstates, which demonstrates that nonthermal slow modes can be hidden in the nominally thermal part of the spectrum. In numerics with $L$7 and $L$8, TEBD real-time evolution up to $L$9 shows that $\mathrm{Var}(H)\to 0$0 stays near its initial value for $\mathrm{Var}(H)\to 0$1, while the fidelity collapses as a function of $\mathrm{Var}(H)\to 0$2, confirming $\mathrm{Var}(H)\to 0$3. By contrast, generic product states or $\mathrm{Var}(H)\to 0$4 deformed states have $\mathrm{Var}(H)\to 0$5 variance and $\mathrm{Var}(H)\to 0$6-independent relaxation. A weak perturbation

$\mathrm{Var}(H)\to 0$7

with $\mathrm{Var}(H)\to 0$8 destroys the exact QMBS, yet the original exact-scar state becomes an asymptotic QMBS of $\mathrm{Var}(H)\to 0$9 because $L\to\infty$0; TEBD then yields fidelity plateaux with $L\to\infty$1. The model thus shows both that asymptotic scars can be orthogonal to all exact scars and that they can survive perturbations that eliminate exact finite-size scar eigenstates (Gotta et al., 2023).

3. Mechanistic generalizations in lattice, graph, and group-theoretic models

A different microscopic route to asymptotic scarring arises from gapless excitations of a frustration-free reference Hamiltonian. In the two-local qubit construction based on the PVBS chain,

$L\to\infty$2

the critical point $L\to\infty$3 is gapless in the single-excitation sector, with eigenstates

$L\to\infty$4

For fixed $L\to\infty$5, $L\to\infty$6 as $L\to\infty$7, so these modes become zero-energy eigenstates of the reference Hamiltonian and asymptotic scars of the deformed nonintegrable model. Since $L\to\infty$8 at small $L\to\infty$9, one obtains $$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$0 and $$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$1. The same work also shows that these single-excitation AQMBS can be prepared on a fully connected processor in depth $$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$2 with $$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$3 entangling gates (Logarić et al., 14 Apr 2026).

In coupled random-graph Hamiltonians, asymptotic scarring appears in two distinct forms. One family consists of approximate product-state scars with

$$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$4

and an approximate rank-one Schmidt form $$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$5, implying $$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$6 as $$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$7. The second family consists of exact $$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$8 scars generated by special subgraphs, with reduced density matrix of rank $$F(t)=|\langle \Psi|e^{-iHt}|\Psi\rangle|^2\approx \exp[-\Delta H^2 t^2]$$9 and $\tau\sim 1/\Delta H$0. Generic mid-spectrum states obey a volume law $\tau\sim 1/\Delta H$1, while the bulk level statistics are Wigner–Dyson for $\tau\sim 1/\Delta H$2 and Poisson at the noninteracting point $\tau\sim 1/\Delta H$3. The paper identifies the large-$\tau\sim 1/\Delta H$4 regime of these rare area-law states in an otherwise ergodic spectrum as the regime of “asymptotic scars” (Jeevanesan, 2023).

The Rokhsar–Kivelson $\tau\sim 1/\Delta H$5 quantum link and quantum dimer models provide a contrasting thermodynamic-limit construction. At $\tau\sim 1/\Delta H$6, the Hamiltonian $\tau\sim 1/\Delta H$7 has an exponentially large zero-mode manifold protected by a chiral-reflection index theorem, $\tau\sim 1/\Delta H$8 and $\tau\sim 1/\Delta H$9, with $\Delta H\to 0$0. On an $\Delta H\to 0$1 torus, one can write explicit type-I “lego scars” as tensor products of localized three-row blocks,

$\Delta H\to 0$2

with entanglement cut scaling $\Delta H\to 0$3. These are exact thermodynamic-limit scar states rather than finite-size approximate ones, but they sharpen the same theme of subthermal structure persisting at arbitrarily large system size (Biswas et al., 2022).

A further large-$\Delta H\to 0$4 scar sector appears in lattice Majorana systems. There the Hilbert space decomposes under $\Delta H\to 0$5, and the scars are $\Delta H\to 0$6 singlets, organized into $\Delta H\to 0$7- and $\Delta H\to 0$8-families. For $\Delta H\to 0$9, explicit formulas yield

$$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$0

and more generally

$$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$1

The $$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$2-family dimension grows as $$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$3, while the scar fraction tends to zero because $$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$4. This suggests a group-theoretic route to thermodynamic scarring with parametrically subthermal entanglement and a vanishing density of scar states (Sun et al., 2022).

4. Experimental realization and dynamical probes

The first direct digital simulation of asymptotic scars was carried out on the Quantinuum H1-1 trapped-ion processor. The implemented Floquet unitary was

$$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$5

with

$$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$6

compiled into native single-qubit rotations and entangling

$$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$7

gates, exploiting all-to-all connectivity and up to five $$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$8 gates in parallel. Simulations reached $$\sigma_E^2(N)=\langle \mathcal A|\hat H^2|\mathcal A\rangle-\langle \mathcal A|\hat H|\mathcal A\rangle^2$$9 qubits and up to $$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$0 entangling $$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$1 gates over $$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$2 Floquet cycles. The monitored observable was the total magnetization

$$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$3

The product scar $$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$4 remained stationary, the non-scar $$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$5 rapidly thermalized to $$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$6, a boundary-QMBS probe $$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$7 showed slower decay as $$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$8, and the asymptotic-scar state $$\tau_{\rm relax}(N)\gtrsim \frac{\pi}{2\,\sigma_E(N)}.$$9 displayed decay that slowed as $\sim L$00 increased. The observed size-dependent suppression of thermalization is the central experimental signature distinguishing AQMBS from conventional scars (Logarić et al., 14 Apr 2026).

5. Open-system asymptotic scars

In open quantum systems, the scar concept can be formulated within the Lindblad equation

$\sim L$01

The relevant algebraic structure is the commutant $\sim L$02 of the bond algebra generated by the local Hamiltonian terms and Hermitian jump operators. The stationary-state manifold is the linear span of $\sim L$03; when this commutant isolates a one-dimensional singlet projector $\sim L$04, one obtains anomalous stationary scar states in addition to the typical infinite-temperature state on the orthogonal complement. Building on this structure, asymptotic open-system quantum many-body scars are defined as states that ultimately converge to the typical infinite-temperature state but do so on anomalously large time scales. A sharp criterion is

$\sim L$05

Two short-time scaling Ansätze were identified: $\sim L$06 corresponding respectively to $\sim L$07 and $\sim L$08. In the second case, the slow mode can be understood from a one-magnon energy $\sim L$09, so the smallest decay rate scales as $\sim L$10, giving

$\sim L$11

Numerical data collapse for $\sim L$12 confirms this universal slowdown, and the same framework predicts long-lived local observables, fidelity plateaux, and suppressed decoherence of off-diagonal coherences in dissipative scar sectors (Gotta, 22 Sep 2025).

6. Holographic and geometric extensions

A holographic realization of scar-like behavior is provided by asymptotically AdS mini-boson stars. In four bulk dimensions with $\sim L$13, $\sim L$14, and scalar mass $\sim L$15, the Einstein–Klein–Gordon system admits horizonless stationary solutions whose low-lying normal modes exhibit random-matrix behavior, while high-lying modes organize into asymptotically decoupled equispaced towers. Writing

$\sim L$16

the large-$\sim L$17 branches obey

$\sim L$18

For intermediate and large $\sim L$19, the average gap ratio $\sim L$20 approaches the GOE/GSE values $\sim L$21 or $\sim L$22, signaling strong level repulsion at low $\sim L$23, while the high-$\sim L$24 sector develops sharp spacing peaks near $\sim L$25. Entanglement diagnostics based on vacuum-subtracted Ryu–Takayanagi entropy show $\sim L$26 at the same mass, and Krylov complexity displays pronounced periodic revivals rather than black-hole-like monotonic growth. This combination of chaotic bulk statistics, embedded regular branches, subthermal entanglement, and revivals is the holographic analogue of scarring (Liu et al., 4 May 2026).

A distinct usage of “scars” appears in Berry’s random-wave model on $\sim L$27, where the term refers to filamentary visual patterns or “scarlets” rather than ETH-violating many-body states. For observables whose fluctuations are asymptotically fully correlated with their second Wiener-chaos projection, the large-domain limit belongs to a common universality class governed by a fractional Gaussian field with Hurst index

$\sim L$28

After rescaling by $\sim L$29, these observables converge in law to $\sim L$30, equivalently

$\sim L$31

with covariance kernel proportional to $\sim L$32. The same universality class includes the stationary Poisson line process, and suitable random-wave observables can be coupled so that their finite-dimensional marginals become arbitrarily close to those of a Poisson line field at the fluctuation scale. This is a geometric and distributional notion of asymptotic scar-like structure, not a statement about slow thermalization or weak ETH violation, but it shows that the scar vocabulary has broadened beyond the many-body context (Gass et al., 5 Jun 2026).