Quantum Many-Body Scar States

• Quantum Many-Body Scar States are atypical eigenstates within a thermal spectrum that exhibit low entanglement and coherent oscillatory revivals, defying the eigenstate thermalization hypothesis.
• They are constructed in models like the PXP chain using matrix product states and variational quasiparticle approximations, revealing robust nonergodic dynamics.
• Experimental realizations in Rydberg-blockaded chains, lattice gauge theories, and kinetically constrained models highlight their potential for quantum simulation and novel quantum control.

Quantum many-body scar (QMBS) states are highly atypical eigenstates embedded within the spectrum of otherwise nonintegrable and thermalizing quantum systems. Such states weakly violate the eigenstate thermalization hypothesis (ETH), persisting at finite energy density and displaying nonthermal features such as low entanglement, coherent dynamical revivals, and pronounced structure in their correlation functions and spectral responses. QMBS have been identified in Rydberg-blockaded atom chains, kinetically constrained spin models, lattice gauge theories, and various systems with emergent algebraic or fragmented Hilbert space structures. They provide a unifying platform for exploring nonergodic quantum dynamics, with implications for quantum simulation and information processing.

1. Foundational Models and Experimental Signatures

The quantum many-body scar phenomenology emerged from experiments on one-dimensional Rydberg atom chains, where persistent oscillatory dynamics from an antiferromagnetic (Z₂) initial state were observed, in direct contrast with rapid thermalization from generic initial conditions. The effective Hamiltonian for this setting is the PXP model,

$H = \sum_i P_{i-1}\, \sigma^x_i\, P_{i+1}\,,$

where $\sigma^x_i$ is the transverse Pauli operator at site $i$ and the projector $P_{i} = (1 - \sigma^z_i)/2$ enforces the Rydberg blockade constraint disallowing adjacent excitations ("no 11" configuration).

Scar states in the PXP model have been rigorously constructed as matrix product states (MPS) at zero energy (E = 0) and as symmetry-breaking superpositions with simple analytical forms. For periodic boundary conditions with even system size $L$, two exact E = 0 scar states are given by

$$|\Phi_1\rangle = \sum_{\{\sigma\}} \mathrm{Tr}[B^{\sigma_1}C^{\sigma_2}\cdots B^{\sigma_{L-1}}C^{\sigma_L}]|\sigma_1\ldots\sigma_L\rangle$$

with explicit definitions for the B and C matrices, and a translated twin $|\Phi_2\rangle = T_x|\Phi_1\rangle$. These states are characterized by constant (area-law) entanglement entropy—a stark deviation from the volume-law scaling expected for infinite-temperature states—and period-2 bond-centered correlation patterns breaking translation symmetry even at maximal entropy density (Lin et al., 2018).

Experiments revealed strong, persistent oscillations following quenches from charge density wave product states, a phenomenon naturally explained by the high overlap of such states with scar manifolds. Numerical and analytical studies verified that these scar eigenstates are responsible for revivals and atypical coherence at high energy density.

2. Quasiparticle Framework and Excitation Structure

Beyond exact MPS scars, the spectrum of the PXP and related models hosts an approximate tower of nonthermal states—quasiparticle excitations atop the exact scar vacuum. These can be obtained using the single-mode (SMA) and multimode (MMA) variational approximations, by inserting additional variational "defect" matrices at select locations in the MPS chain, effectively generating a dispersing quasiparticle band (Lin et al., 2018). The SMA ansatz yields excited scar states with fixed translation and inversion quantum numbers and energy separation matching the experimentally observed oscillation frequencies (e.g., ω ≈ 1.31 for the primary excitation).

This generative approach underlies the observed dynamical revivals: initial states such as the Z₂ density wave have large spectral weight in the scar subspace, and subsequent time evolution exhibits oscillations at frequencies set by the tower's energy spacing. Extensions to "quasiparticle-excited" initial states lead to sequences of periodic partial revivals, further confirming the robustness of the quasiparticle picture.

3. Analytical Construction and Symmetry Properties

The algebraic structure underlying QMBS often manifests through special raising (Q†) and lowering operators forming a restricted spectrum generating algebra (RSGA) that generates an exact or approximate scar tower: $$|S_n\rangle = \frac{1}{n! \sqrt{\mathcal{N}(L,n)}} (Q^\dagger)^n | \Omega \rangle\,,$$ where Q† inserts localized excitations subject to strict exclusion constraints (such as the Rydberg blockade or "Fibonacci" constraint: no two excitations adjacent) (Iadecola et al., 2019, Kunimi et al., 7 May 2025).

In higher dimensions, similar tower states can be built from product states of local dimers or valence bond solids (VBS), covering one sublattice of the system, leading to simple eigenstates at exactly integer-valued energies and sub-volume (area-law) entanglement (Lin et al., 2020).

For bipartite lattices, a projector-embedding construction recasts the Hamiltonian as a sum of terms vanishing on the scar subspace, and all exact scar states are projections (via the constrained subspace projector) of maximal spin (S = N/2) Dicke or spin-coherent states. Precession of the macroscopic pseudospin in an effective Zeeman field induces the observed periodic quantum revivals when projected back to the constrained subspace (Omiya et al., 2022).

4. Entanglement and Violation of the ETH

A defining diagnostics of QMBS is their violation of the strong ETH: while the majority of eigenstates at any given energy density are thermal and display volume-law entanglement entropy, scar states at the same energies exhibit logarithmic (S ∼ ln L) or constant entanglement, making them highly nonthermal. This entanglement anomaly persists even at infinite temperature and is directly measurable via reduced density matrices or mutual information scaling (Lin et al., 2018, Iadecola et al., 2019).

The "entropy gap" between scar and thermal states provides a more robust indicator than energy separation—scar states can thus be "hidden" in otherwise thermal bands, protected not by gaps in the spectrum but by structural differences in entanglement and local correlations. Their existence requires direct violations of canonical typicality, with subsystem density matrices differing markedly from the microcanonical predictions (Larsen et al., 30 May 2024).

5. Experimental Realizations and Spectroscopic Probes

QMBS have been realized and probed in a range of experimental settings:

• Rydberg-blockaded chains and arrays (Lin et al., 2018, Lin et al., 2020).
• Two-dimensional Rydberg atom systems, where exact VBS scars are present (Lin et al., 2020).
• Optical lattice implementations of kinetically constrained spin models, where fragmentation and emergent constraints provide a platform for scar state construction (Yang et al., 12 Jun 2025).
• Stabilizer scar states in lattice gauge theories, where the corresponding eigenstates are exactly characterized, have zero "magic" and are accessible via quantum circuits (Hartse et al., 19 Nov 2024).

Spectroscopic signatures are now a central focus for scar detection. In the spin-1 AKLT chain, for example, a two-magnon ladder operator

$$Q_\pi^\dagger = \frac{1}{\sqrt{L}} \sum_\ell (-1)^\ell (S^+_\ell)^2$$

generates QMBS as

$|S_n\rangle = (Q_\pi^\dagger)^n |G\rangle,$

with the dynamical correlation function displaying a δ-function resonance peak at momentum k = π, superimposed on a continuum background—a distinctive "bow-tie" shape in spectroscopic (RIXS) measurements (Wei et al., 5 Aug 2024).

6. Mechanisms, Robustness, and Relation to Hilbert Space Structure

Scarring is often rooted in the underlying fragmentation of Hilbert space: kinetic constraints partition it into disconnected (or nearly so) Krylov subspaces, with scar towers residing in special measure-zero fragments inaccessible from typical thermal initial conditions (Yang et al., 12 Jun 2025). Quasiparticle excitations propagating within inert backgrounds generate stable scar towers, while interactions between multiple such excitations can lead to inelastic processes and "two-body loss", resulting in approximate (rather than exact) scars whose coherence decays according to the emergent Lindblad dynamics.

Scar state phase transitions are now recognized: by embedding a matrix product state subject to a classical phase transition into a thermal spectrum, one induces a "scar transition" in the excited state (and possibly in a few nearby states), sharp in local observable or entanglement behavior despite the bulk of the spectrum remaining featureless (Larsen et al., 30 May 2024).

Notably, in systems with even infinitesimal but generic interactions (such as weakly interacting fermions), the overwhelming majority of scar states are destroyed: the probability of finding sub-volume-law entangled states drops double-exponentially with system size, confirming the near-absolute stability of the strong ETH in such regimes (2207.13688). In contrast, models with spectrum generating algebras or kinetic constraints retain their scars even under substantial modifications, offering unusually robust families of nonthermal eigenstates.

7. Broader Theoretical Perspectives and Future Directions

The theoretical framework for QMBS draws on algebraic approaches—RSGA, Lie algebras (notably SU(2)), and non-Hermitian projector structures (Omiya et al., 2022, Wang et al., 21 Mar 2024). Critical connections have been established:

• Between scarring and weak ergodicity breaking, with scarred subspaces dynamically decoupled and insensitive to typical perturbations (Pakrouski et al., 20 Nov 2024).
• Between scarring and integrability: models can be constructed such that generalized spin helix scars (i.e., product states with local phase winding) persist after integrability-breaking deformations, stabilized as the null space of local non-Hermitian projectors or via Temperley–Lieb algebra representations (Wang et al., 21 Mar 2024).
• Between scarring and supersymmetry: the parent Hamiltonians governing scar subspaces admit SUSY representations, with the scar ground state mapping onto the SUSY-unbroken vacuum (Kunimi et al., 7 May 2025).

Quantum computational protocols now exist for preparing scar states and their coherent superpositions on NISQ devices, exploiting their finite-bond-dimension MPS structures and low entanglement to design efficient, shallow circuits suitable for quantum simulation experiments (Gustafson et al., 2023).

Open problems include the classification and protection of approximate (asymptotic) scars, the design of models with tunable scarring properties, and the utilization of scarring for enhanced quantum memory or control. The identification of scar states through quantum machine learning architectures, such as enhanced quantum convolutional neural networks (QCNNs), is a frontier area, enabling the discovery of new nonthermal states beyond analytically tractable settings (Feng et al., 11 Sep 2024).

Finally, recent work has established deep ties between scar phenomena and concepts such as phase transitions in excited states, connections to superconductivity via BCS wavefunctions as scar ground states, and the role of quantum resource measures ("magic") in quantifying nonergodic structure and computability (Hartse et al., 19 Nov 2024, Pakrouski et al., 20 Nov 2024, Larsen et al., 30 May 2024).

---

Table: Key Models and Their Scar State Mechanisms

Model/Class	Construction Principle	Scar Entanglement Scaling
PXP Chain	MPS, RSGA, projector embedding	Area/logarithmic law
2D PXP (Rydberg)	VBS (dimer product) construction	Area law
Spin-1 XY/AKLT	Ladder operator (SGA), SMA/MMA	Logarithmic law
Kinetically Constrained Spin Models	Hilbert space fragmentation, quasiparticles	Area law or approximate scars
Bilayer EPR States	Maximally entangled EPR state projections + symmetry projection	Bimodal (maximal between layers or area law in sectors)

---

Quantum many-body scarring is now recognized as a robust, deeply algebraic phenomenon manifesting as exceptional eigenstates with anomalously low entanglement and unique dynamical signatures. It provides a platform for controlled violation of thermalization in nonintegrable quantum systems, with both foundational significance and immediate experimental relevance.