Long-Range Quantum Many-Body Scars

• Long-range quantum many-body scars are atypical nonthermal eigenstates that exhibit persistent memory and logarithmic-in-time entanglement growth due to collective interactions.
• The (PXP)² model combines short-range Rydberg blockade with long-range cavity-mediated couplings, forming a minimal framework for kinetically constrained quantum dynamics.
• These scars defy conventional thermalization, providing insights into ergodicity breaking and suggesting experimental pathways for robust quantum information storage.

Long-range quantum many-body scars are a class of atypical nonthermal eigenstates that display persistent, non-ergodic dynamics in quantum systems whose interactions extend beyond nearest neighbor—often featuring all-to-all or power-law decaying couplings. In systems that combine local kinetic constraints with collective or cavity-mediated interactions, these scars give rise to characteristic slow entanglement growth and enduring memory of initial conditions. The (PXP)$^2$ model, which captures the physics of Rydberg atoms coupled to an optical cavity, constitutes a paradigmatic setting for long-range quantum many-body scars, where photonic and Rydberg constraints coexist. In contrast to their short-range analogs, these long-range scarred states exhibit logarithmic-in-time entanglement entropy dynamics and display robust nonthermal persistence even in the presence of system-wide chaotic or thermalizing behavior (Hosseinabadi et al., 2 Oct 2025).

1. Model Framework and Hamiltonian Structure

The theoretical foundation for long-range many-body scars in this context is provided by the (PXP)$^2$ Hamiltonian constructed for a chain of two-level atoms in a single-mode optical cavity. Here, atoms experience both strong nearest-neighbor Rydberg blockade interactions and global coupling to cavity photons:

• Kinetic constraints: The Rydberg blockade prohibits simultaneous excitations of adjacent sites, restricting dynamics to the subspace where $n_{i} n_{i+1} = 0$.
• Cavity-mediated interactions: The photon mode induces collective, all-to-all couplings, with effective strength scaling as $1/\sqrt{L}$ for $L$ atoms.

Upon eliminating the fast cavity degrees of freedom in the strong blockade limit, the effective Hamiltonian reads: $$\hat{\mathcal{H}} = -\frac{1}{L} \sum_{i,j} \tilde{\sigma}_i^x \tilde{\sigma}_j^x + \Delta \sum_{i} \sigma_i^z$$ where projection operators $\tilde{\sigma}_i^x = P_{i-1} \sigma_i^x P_{i+1}$ enforce the blockade constraint ($P_j = 1 - n_j$), and $\Delta$ is the atomic detuning. Alternatively, the model can be written as: $$\hat{\mathcal{H}} = -\frac{1}{L} \left[ H_{PXP} \right]^2 + \Delta \sum_{i} \sigma_i^z$$ with $H_{PXP} = \sum_{i} \tilde{\sigma}_i^x$ representing the well-known PXP model Hamiltonian, whose own short-range scars have been studied extensively (Hosseinabadi et al., 2 Oct 2025).

2. Interaction Channels and Physical Mechanisms

Two interaction channels define the essential physics:

Mechanism	Range	Effect
Rydberg blockade	Short-range	Nearest-neighbor kinetic constraint; inhibits simultaneous excitations, enforces strong local correlations
Cavity photon coupling	Long-range	All-to-all, collective coupling of atomic excitations; mediates effective global interactions, enhancing quantum coherence

The interplay produces a system where quantum dynamics is shaped both by local mobility restrictions and by delocalized, collective coherence. The (PXP)$^2$ structure realizes long-range coupled, kinetically constrained spin dynamics in a minimal form.

3. Nature and Signatures of Long-Range Quantum Many-Body Scars

Within the spectrum, the Hamiltonian supports a band of nonthermal ("scarred") eigenstates with the following features:

• High overlap with density-wave product states, for example the Néel state $|Z_2\rangle = |1010\dots\rangle$, which lies deep in Hilbert space constrained by Rydberg blockade.
• Violation of the eigenstate thermalization hypothesis (ETH): The scarred eigenstates show lower bipartite entanglement entropy and deviate from the smooth ETH-predicted statistical distribution characteristic of chaotic eigenstates.
• Spectral statistics: The levels associated with scars display semi-Poisson statistics, indicating their nonergodic origin, as opposed to the Wigner-Dyson statistics seen in the thermal spectral bulk.
• Persistence of memory: When initialized in a density wave, such as $|Z_2\rangle$, real-time dynamics show suppression of thermalization, with the system retaining signatures of the initial state over long times.

These features are robust in that they persist despite the quantum chaos present in the remainder of the spectrum.

4. Entanglement Dynamics: Logarithmic Growth

A distinctive dynamical property of long-range quantum scars in models such as (PXP)$^2$ is the rate of entanglement growth. In contrast to linear-in-time growth observed in short-range scarred models (e.g., PXP), the (PXP)$^2$ model displays much slower, logarithmic entanglement scaling:

$S_{L/2}(t) \sim \log t$

where $S_{L/2}$ is the von Neumann entropy of a half-system bipartition, and $t$ is the time after a global quench from a density-wave product state. This behavior stems from the collective cavity-induced long-range coherence, which restricts the rate at which quantum information spreads through the system.

The magnon, or collective spin-wave, picture provides a physical mechanism: magnon excitations proliferate slowly within the blockaded and collective constrained subspace, resulting in highly sublinear entanglement dynamics. This property is evidenced by numerical calculations directly comparing $S_{L/2}(t)$ versus $\log t$ for various initial states and is a direct fingerprint of long-range scarring.

5. Equilibrium Phase Structure and Comparison to Conventional Phases

At equilibrium, the model exhibits a rich phase structure:

Phase	Parameter Regime	Order Parameter	Distinction
Paramagnetic	Large positive $\Delta$	$\langle \sigma^z \rangle \to +1$	Spins polarized, no collective order
Néel-ordered ("Z$_2$")	Large negative $\Delta$	Alternating density-wave order	Translational symmetry broken (period 2 DW)
Blockaded ferromagnetic/superradiant	Intermediate $\Delta$	Nonzero $\langle a \rangle$, enhanced $\langle \tilde{\sigma}^x \rangle$	Breaks spin $Z_2$, not translation (distinct from classical superradiant phase); magnetization may exceed classical bounds due to blockade "dressing"

The blockaded FM/superradiant phase is accessible uniquely due to the combination of kinetic constraints and long-range photon coupling. Unlike the standard Dicke model superradiant transition, the presence of the blockade mechanism "dresses" the mean-field product states, resulting in quantum phases that surpass classical magnetization limits.

6. Experimental Realizations and Theoretical Implications

The (PXP)$^2$ model is experimentally accessible using Rydberg atom arrays inside high-finesse optical cavities, where both the Rydberg blockade and collective photonic coupling have been realized with precise control over detunings, interaction strengths, and particle number.

• Parameter regimes: Realistic Rydberg-cavity experiments can operate in the strong blockade limit $V_{\text{Ryd}} \gg \gamma_{\text{cav}}$, with cavity coupling $g$ and collective detunings that map onto the theoretical model.
• Observables: Both spectral and dynamical features are measurable, including density-wave revivals, entanglement dynamics, and global photon observables (e.g., $\langle a \rangle$).
• Theoretical frameworks: The tractable nature of the model (via Hilbert space restriction) enables soft-spin field-theoretic and LMG-related analyses of the scars and their stability under competing quantum fluctuations and collective drive.

This establishes the model as a minimal yet comprehensive testing ground for further advances in the understanding of long-range scars, quantum nonergodicity, and the stabilization of nonthermal behavior in hybrid quantum systems.

7. Broader Significance: Ergodicity Breaking and Nonthermal Dynamics

Long-range quantum many-body scars in models like (PXP)$^2$ illuminate several core aspects of nonthermalization in many-body quantum systems:

• Mechanisms of weak ergodicity breaking: The coexistence of a small, measure-zero set of nonthermal, low-entanglement eigenstates with a chaotic/thermalizing bulk.
• Dynamics dominated by constraints and long-range coherence: Global cavity coupling combines with local blockade to produce sublinear spreading of entanglement and persistent initial-state memory.
• Distinction from localization or MBL: The slow dynamics and suppressed thermalization do not require disorder or Anderson localization; instead, they emerge from the interplay of constraints and long-range couplings.
• Experimental accessibility: The slow entanglement dynamics and suppression of thermalization make these systems strong candidates for exploring robust quantum information storage and dynamics with current cold-atom and cavity QED technologies.

Overall, the paper of long-range quantum many-body scars provides new insight into the competition between collective coherence and local constraints, and opens new routes to engineer, observe, and exploit nonthermal dynamical regimes beyond the paradigms of both ordinary ergodicity and strong localization (Hosseinabadi et al., 2 Oct 2025).