---
title: Asymptotic Quantum Many-Body Scars
url: https://www.emergentmind.com/topics/asymptotic-scars
type: topic
---

# Asymptotic Quantum Many-Body Scars

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 [2303.05407]. 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 [2604.12296] [2305.08123] [2202.03451] [2212.11914] [2509.18023] [2605.02446] [2606.07842].

## 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 [2303.05407].

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-\(N\) scar eigenstates generate slow dynamics that is \(N\)-independent, whereas asymptotic scars become progressively more stable as \(N\) 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 [2604.12296].

## 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
\[
|n,\pi\rangle \propto (J^+_{\pi})^n |\downarrow\downarrow\ldots\downarrow\rangle,\qquad
H|n,\pi\rangle=(-Lh+2nh+LD)|n,\pi\rangle,
\]
where
\[
J^+_{k}\coloneqq \tfrac12 \sum_{j=1}^L e^{ikj}(S^+_j)^2.
\]
From the exact-scar state \(|n-1,\pi\rangle\), one constructs
\[
|n,k\rangle \coloneqq [N_{n,k}]^{-1/2}J^+_{k}(J^+_{\pi})^{n-1}|\downarrow\downarrow\ldots\downarrow\rangle.
\]
For \(k\neq \pi\), these states are orthogonal to the exact tower, with \(\langle n,k|n',\pi\rangle=\delta_{n,n'}\delta_{k,\pi}\), yet their average energy remains
\[
\langle n,k|H|n,k\rangle=-Lh+2nh+LD,
\]
independent of \(k\). Under periodic boundary conditions, the energy variance is
\[
\mathrm{Var}(H)\equiv \Delta H^2
=4\,[J^2\cos^2(k/2)+J_3^2\cos^2(3k/2)].
\]
Choosing
\[
k=\pi+\frac{2\pi m}{L},\qquad m\in\mathbb Z,\qquad m=O(1),
\]
gives
\[
\Delta H^2\sim (J^2+9J_3^2)\left(\frac{2\pi m}{L}\right)^2=O(L^{-2}),
\]
so \(\Delta H\to 0\) although \(|n,k\rangle\) is not an eigenstate at any finite \(L\). 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 \(n=L/2\) and \(k=\pi-2\pi/L\), TEBD real-time evolution up to \(L=60\) shows that \(S^{z2}(t)=L^{-1}\sum_j \langle (S_j^z)^2\rangle\) stays near its initial value for \(t\lesssim O(L)\), while the fidelity collapses as a function of \(t/L\), confirming \(\tau\propto L\). By contrast, generic product states or \(k=0\) deformed states have \(O(1)\) variance and \(L\)-independent relaxation. A weak perturbation
\[
V=\frac{J_z}{L}\sum_j S^z_jS^z_{j+1}
\]
with \(\|V\|=O(1)\) destroys the exact QMBS, yet the original exact-scar state becomes an asymptotic QMBS of \(H'=H+V\) because \(\mathrm{Var}(V)\sim O(1/L)\); TEBD then yields fidelity plateaux with \(\tau\propto L^{1/2}\). 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 [2303.05407].

## 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,
\[
\hat H_+^{(g)}=\sum_{n=0}^{N-2}\hat P_{n,n+1}^{(g)},
\]
the critical point \(|g|=1\) is gapless in the single-excitation sector, with eigenstates
\[
|\mathcal A_k\rangle
=\frac{1}{\mathcal N_k}\sum_{n=0}^{N-1}e^{in\phi}
\cos\!\Bigl[\Bigl(N-n-\tfrac12\Bigr)\frac{k\pi}{N}\Bigr]|n\rangle,
\qquad
\varepsilon_k=1-\cos\!\Bigl(\frac{k\pi}{N}\Bigr).
\]
For fixed \(k>0\), \(\varepsilon_k\to 0\) as \(N\to\infty\), so these modes become zero-energy eigenstates of the reference Hamiltonian and asymptotic scars of the deformed nonintegrable model. Since \(\varepsilon_k\approx (k\pi)^2/(2N^2)\) at small \(k\), one obtains \(\sigma_E(N)\sim 1/N^2\) and \(\tau_{\rm relax}(N)\sim N^2\). The same work also shows that these single-excitation AQMBS can be prepared on a fully connected processor in depth \(D(N)\sim \lceil \log_2 N\rceil\) with \(N-1\) entangling gates [2604.12296].

In coupled random-graph Hamiltonians, asymptotic scarring appears in two distinct forms. One family consists of approximate product-state scars with
\[
\max_{i,j}|\langle \psi\,|\,\varphi_i\otimes \varphi_j\rangle|^2
\sim 1-\mathcal O(1/\sqrt{\mathcal D}),
\]
and an approximate rank-one Schmidt form \(\psi_{ij}\approx \lambda_1\varphi_i\varphi_j\), implying \(S_{\rm ent}\to 0\) as \(\mathcal D\to\infty\). The second family consists of exact \(\ln 2\) scars generated by special subgraphs, with reduced density matrix of rank \(2\) and \(S_{\rm ent}=\ln 2\). Generic mid-spectrum states obey a volume law \(S_{\rm ent}\sim \ln(\dim \mathcal H_L)\propto L\), while the bulk level statistics are Wigner–Dyson for \(\mathcal D'<\mathcal D\) and Poisson at the noninteracting point \(\mathcal D'=\mathcal D\). The paper identifies the large-\(L\) regime of these rare area-law states in an otherwise ergodic spectrum as the regime of “asymptotic scars” [2305.08123].

The Rokhsar–Kivelson \(U(1)\) quantum link and quantum dimer models provide a contrasting thermodynamic-limit construction. At \(\lambda=0\), the Hamiltonian \(H_{\rm RK}=O_{\rm kin}+\lambda O_{\rm pot}\) has an exponentially large zero-mode manifold protected by a chiral-reflection index theorem, \(\{H(0),C\}=0\) and \([H(0),P]=0\), with \(\mathrm{Dim\,Ker}[H(0)]\ge |\mathrm{Tr}_P[C]|\sim e^{cN_p}\). On an \(L\times 4\) torus, one can write explicit type-I “lego scars” as tensor products of localized three-row blocks,
\[
|\Psi_s\rangle=\bigotimes_{j=1}^n |\ell_{\alpha_j}\rangle,\qquad
H(\lambda)|\Psi_s\rangle=\lambda Z|\Psi_s\rangle,\qquad Z=n,
\]
with entanglement cut scaling \(S=k\ln 2\le O(1)\). 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 [2202.03451].

A further large-\(N\) scar sector appears in lattice Majorana systems. There the Hilbert space decomposes under \(O(N)\times O(M)\), and the scars are \(SO(N)\) singlets, organized into \(\eta\)- and \(\zeta\)-families. For \(M=6\), explicit formulas yield
\[
S_A(k_{12},k_{13},k_{23})\simeq \tfrac12
\log[(2\pi L)^3 \nu_{12}\nu_{13}\nu_{23}(1-\nu_T)]
\sim \tfrac32 \log L,
\]
and more generally
\[
S_A\sim \alpha \log L+O(1),\qquad
\alpha=\frac{M(M-2)}{16}.
\]
The \(\eta\)-family dimension grows as \(\dim_\eta\sim N^{M(M-2)/8}\), while the scar fraction tends to zero because \(\dim \mathcal H\sim 2^{MN/2}\). This suggests a group-theoretic route to thermodynamic scarring with parametrically subthermal entanglement and a vanishing density of scar states [2212.11914].

## 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
\[
\hat{\mathbb U}^{(g)}=\hat U_o^{(g)}\hat U_e^{(g)},
\]
with
\[
\hat U_e^{(g)}=\exp[i\,\hat P^{(g)}\hat h_e\hat P^{(g)}],\qquad
\hat U_o^{(g)}=\exp[i\,\hat P^{(g)}\hat h_o\hat P^{(g)}],
\]
compiled into native single-qubit rotations and entangling
\[
\widehat{ZZ}_\eta=\exp\!\Bigl[i\frac{\eta}{2}Z\otimes Z\Bigr]
\]
gates, exploiting all-to-all connectivity and up to five \(ZZ\) gates in parallel. Simulations reached \(N=20\) qubits and up to \(418\) entangling \(ZZ\) gates over \(t=10\) Floquet cycles. The monitored observable was the total magnetization
\[
M_Z(t)=\sum_{n=0}^{N-1}\langle Z_n(t)\rangle.
\]
The product scar \(|0\rangle^{\otimes N}\) remained stationary, the non-scar \(|1\rangle^{\otimes N}\) rapidly thermalized to \(M_Z\approx 0\), a boundary-QMBS probe \(|L\rangle=|1,0,\dots,0\rangle\) showed slower decay as \(|g|\gg 1\), and the asymptotic-scar state \(|\mathcal A_1\rangle\) displayed decay that slowed as \(N\) increased. The observed size-dependent suppression of thermalization is the central experimental signature distinguishing AQMBS from conventional scars [2604.12296].

## 5. Open-system asymptotic scars

In open quantum systems, the scar concept can be formulated within the Lindblad equation
\[
\dot\rho=\mathcal L(\rho)
=-\,i[H,\rho]+\sum_j\gamma_j\Bigl[l_j\,\rho\,l_j-\tfrac12\{l_j^2,\rho\}\Bigr].
\]
The relevant algebraic structure is the commutant \(\mathcal C\) of the bond algebra generated by the local Hamiltonian terms and Hermitian jump operators. The stationary-state manifold is the linear span of \(\mathcal C\); when this commutant isolates a one-dimensional singlet projector \(\Pi_s=|\psi\rangle\langle \psi|\), 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
\[
\lim_{L\to\infty}|\dot F(0)|=0,
\qquad
F(t)=\mathrm{Tr}\!\left[\;|\psi_n\rangle\langle \psi_n|\,\rho(t)\right].
\]
Two short-time scaling Ansätze were identified:
\[
F(t)=f\!\Bigl(\frac{t^2}{L^2}\Bigr),\qquad
F(t)=g\!\Bigl(\frac{t}{L^2}\Bigr),
\]
corresponding respectively to \(\tau_{\rm rel}\propto L\) and \(\tau_{\rm rel}\propto L^2\). In the second case, the slow mode can be understood from a one-magnon energy \(\epsilon_{k-\pi}\sim (2\pi/L)^2\), so the smallest decay rate scales as \(\Gamma_{\min}\sim \gamma (2\pi/L)^2\), giving
\[
\tau_{\rm rel}\sim \frac{L^2}{\gamma\,4\pi^2}.
\]
Numerical data collapse for \(L=4,5,6,7,8\) 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 [2509.18023].

## 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 \(L=1\), \(8\pi G=1\), and scalar mass \(m^2=-2\), 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
\[
I_1\equiv \int_0^\infty \frac{dr}{A(r)N(r)},
\]
the large-\(\Omega\) branches obey
\[
\Omega_n^{(\pm)}\simeq (n+\tfrac12)\pi/I_1 \mp \omega,\qquad
\Omega_n^{(\mathrm{grav})}\simeq (n+\tfrac12)\pi/I_1.
\]
For intermediate and large \(\Phi_0\), the average gap ratio \(\langle r\rangle\) approaches the GOE/GSE values \(\approx 0.536\) or \(0.676\), signaling strong level repulsion at low \(n\), while the high-\(n\) sector develops sharp spacing peaks near \(\Delta\Omega\approx \pi/I_1\). Entanglement diagnostics based on vacuum-subtracted Ryu–Takayanagi entropy show \(\delta S_{\rm half}(\mathrm{star})\ll \delta S_{\rm half}(\mathrm{BH})\) 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 [2605.02446].

A distinct usage of “scars” appears in Berry’s random-wave model on \(\mathbb R^d\), 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
\[
H=\frac{1-d}{2}.
\]
After rescaling by \(R^{-(d+1)/2}\), these observables converge in law to \(\mathrm{FGF}_{1/2}\), equivalently
\[
h=(-\Delta)^{-(d+1)/4}W,
\]
with covariance kernel proportional to \(|x-y|^{-(d-1)}\). 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 [2606.07842].

Source: https://www.emergentmind.com/topics/asymptotic-scars