Quantum-Scar Models and Exceptional Eigenstates

Quantum many-body scars are atypical, nonthermal eigenstates embedded in a spectrum whose overwhelming majority can still satisfy eigenstate thermalization. They matter dynamically when a specially structured initial state has anomalously large overlap with a near-equally spaced scar tower, producing revivals that generic states do not share.

Required background. Quenches and relaxation fixes finite-size and recurrence tests, and the eigenstate thermalization hypothesis defines the background from which scarred eigenstates deviate. Helpful background. Scars, fragmentation, and exceptions to ETH develops the general classification.

The constrained PXP model

Rydberg blockade forbids adjacent excitations in its ideal nearest-neighbor limit. In the constrained Hilbert space the PXP Hamiltonian is

$$H_{\mathrm{PXP}}=\Omega\sum_{j=1}^{L}P_{j-1}X_jP_{j+1}, \qquad P_j=\lvert0_j\rangle\langle0_j\rvert.$$

The projectors allow a spin flip only when both neighbors are unexcited. Starting from the staggered product state

$$\lvert Z_2\rangle=\lvert1010\cdots\rangle,$$

experiments and ideal-model dynamics show pronounced revivals in the fidelity

$$F(t)=\left\lvert\langle Z_2\rvert e^{-iH_{\mathrm{PXP}}t}\lvert Z_2\rangle\right\rvert^2$$

and in the staggered density. Generic allowed product states relax much faster.

The state-selective revival was observed in the 51-atom Rydberg simulator of Bernien et al. 2017.

Expand $\lvert Z_2\rangle=\sum_n c_n\lvert E_n\rangle$. Revivals require a subset with both large $\lvert c_n\rvert^2$ and approximately equal spacings $E_{n+1}-E_n\simeq\Delta$. Their phases re-align near $t=2\pi/\Delta$. The relevant eigenstates have anomalous local expectation values and unusually low entanglement relative to nearby thermal eigenstates, yet their number is a vanishing fraction of the full constrained Hilbert space.

Their connection to exceptional PXP eigenstates and an approximate tower was established by Turner et al. 2018.

Forward scattering and an approximate algebra

Relative to $\lvert Z_2\rangle$, split $H=H_++H_-$ so that $H_+$ increases the Hamming distance from the initial state and $H_-$ decreases it. The forward-scattering basis is generated by

$$\lvert n\rangle_{\mathrm F} \propto H_+^n\lvert Z_2\rangle.$$

Projecting $H$ into this $O(L)$-dimensional subspace gives an approximately tridiagonal problem. If $H_+$, $H_-$, and their commutator formed an exact $\mathfrak{su}(2)$ algebra, the projected spectrum would be equally spaced and revivals perfect. In the bare PXP model the algebra closes only approximately, so the wave packet leaks into the thermal complement and revivals decay. Carefully chosen longer-range deformations can improve the algebra and strongly enhance revivals; that diagnostic connects the spectral tower to the dynamics rather than merely labeling low-entanglement states after the fact.

Other exact scar constructions embed nonthermal eigenstates using projector constraints or spectrum-generating algebras. The mechanism must be stated: “scar” is a phenomenological family name, not a guarantee that every model shares the PXP forward-scattering structure.

Scar, integrability, or fragmentation?

The distinctions are sharp enough to test:

Mechanism	Portion of spectrum affected	Initial-state dependence	Exact invariant sectors?
Many-body scars	Atypical set, usually measure zero	Strong overlap needed for revivals	Not required
Integrability	Extensive spectral and dynamical structure	Generalized ensemble for broad states	Extensive conserved charges
Hilbert-space fragmentation	Potentially all states, sector by sector	Dynamics confined to the starting Krylov sector	Yes in the exact model
Finite-size recurrence	No special spectral family required	Depends on commensurabilities	No thermodynamic protection

A constrained Hilbert space is not automatically fragmented. PXP blockade removes forbidden basis states, but the allowed-state connectivity graph is largely connected within conventional symmetry sectors. Hilbert-space fragmentation requires further disconnection into many invariant Krylov components.

The modern classification of scar mechanisms, their relation to fragmentation, and their stability limits is reviewed by Serbyn, Abanin, and Papić 2021 and Moudgalya, Bernevig, and Regnault 2022.

A decisive numerical test

Exact diagonalization should resolve translation, inversion, and any internal symmetries before comparing level statistics or ETH plots. Within each block:

1. plot a local eigenstate expectation value versus energy and identify outliers;
2. compare their bipartite entanglement with the microcanonical band;
3. compute overlaps of $\lvert Z_2\rangle$ and several control states with every eigenstate;
4. reconstruct $F(t)$ using only the proposed tower and compare it with exact dynamics; and
5. repeat with $L$ and with symmetry-preserving perturbations.

The tower reconstruction is especially important. A visually striking entanglement outlier that carries negligible initial-state weight cannot explain an observed revival. Conversely, revivals at one small $L$ can arise from accidental gap commensurability even without a stable tower.

The 2017 Rydberg-chain experiment exposed state-selective revivals, and the 2018 PXP analyses tied them to exceptional eigenstates. Subsequent work has found exact and approximate scar mechanisms in broader models; a 2025 study extended approximate $\mathfrak{su}(2)$ scar families to longer blockade ranges but also found that some require weakly entangled, rather than product, initial states Kerschbaumer et al. 2025. Breadth of constructions does not remove the need to establish the mechanism in each Hamiltonian.

Evidence status

This page was checked through 10 August 2026. Exceptional-eigenstate and state-selective-revival phenomena are well established in finite quantum simulators and many controlled models. Their perturbative stability and thermodynamic significance are model dependent, and the boundary between scarring and weak fragmentation continues to evolve. Current changes and contrary results belong in the Quantum Matter and Emergence Research dossier.

Exercises

1. Revival from an exact tower. Let $E_n=E_0+n\Delta$ for every eigenstate with nonzero $c_n$ in an initial state. Show that the fidelity returns to one at $2\pi/\Delta$.

Solution

At $t=2\pi/\Delta$, each component acquires $e^{-iE_nt}=e^{-iE_0t}e^{-i2\pi n}=e^{-iE_0t}$. The entire state therefore returns up to a global phase, and $F=1$. Unequal spacings cause relative phase errors and imperfect revivals.

2. Why resolve symmetries? Explain why mixing two independent translation sectors can make an integrable or chaotic spectrum appear to have Poisson-like level spacings.

Solution

Levels from different symmetry blocks do not repel because their eigenstates cannot hybridize. Superposing several individually repelling spectra therefore fills in small spacings and can mimic Poisson statistics. Level statistics must be computed separately in each irreducible symmetry sector before being used as evidence for chaos or integrability.

References

• Bernien, Hannes, Sylvain Schwartz, Alexander Keesling, Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S. Zibrov, Manuel Endres, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. “Probing Many-Body Dynamics on a 51-Atom Quantum Simulator.” Nature 551, 579–584 (2017). DOI.
• Kerschbaumer, Aron, Marko Ljubotina, Maksym Serbyn, and Jean-Yves Desaules. “Quantum Many-Body Scars beyond the PXP Model in Rydberg Simulators.” Physical Review Letters 134, 160401 (2025). DOI.
• Moudgalya, Sanjay, B. Andrei Bernevig, and Nicolas Regnault. “Quantum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results.” Reports on Progress in Physics 85, 086501 (2022). DOI.
• Serbyn, Maksym, Dmitry A. Abanin, and Zlatko Papić. “Quantum Many-Body Scars and Weak Breaking of Ergodicity.” Nature Physics 17, 675–685 (2021). DOI.
• Turner, Christopher J., Alexios A. Michailidis, Dmitry A. Abanin, Maksym Serbyn, and Zlatko Papić. “Weak Ergodicity Breaking from Quantum Many-Body Scars.” Nature Physics 14, 745–749 (2018). DOI.