PXP Model: Quantum Many-Body Scar Dynamics

• The PXP model is a paradigmatic Hamiltonian that captures Rydberg blockade physics and many-body scar dynamics.
• It employs projector constraints to enforce Rydberg blockade, producing coherent oscillations and nearly uniform scar energy spacing.
• Its revival dynamics and embedded algebraic structures offer insights into weak ergodicity breaking and potential quantum information applications.

Quantum many-body scars (QMBS) constitute a novel class of highly excited, nonthermal eigenstates embedded within the spectra of otherwise thermalizing many-body quantum systems. The PXP model represents the paradigmatic Hamiltonian for the study of quantum many-body scars, having originated from and accurately described coherent oscillations observed in Rydberg-atom quantum simulators. This model underpins the theory of weak ergodicity breaking in isolated quantum systems and continues to inform both experimental realizations and theoretical generalizations in the broader study of non-thermal many-body dynamics (Serbyn et al., 2020).

1. Definition and Physical Motivation

In the context of many-body quantum dynamics, a quantum many-body scar is defined as an atypical eigenstate |ϕ_s⟩ of a nonintegrable Hamiltonian H that exhibits local observable expectation values ⟨ϕ_s|O|ϕ_s⟩ and bipartite entanglement entropy S(ϕ_s) which diverge starkly from the smooth, volume-law scaling predicted by the Eigenstate Thermalization Hypothesis (ETH). The cardinal distinguishing feature of scars is their subextensive proliferation with the system size L (typically polynomial or linear), occupying a set of vanishing measure in the Hilbert space as L→∞. These states are neither the product of integrability nor many-body localization, but rather represent a form of weak violation of ETH—small, emergent subspaces embedded in a thermalizing spectrum but immune to thermalization (Serbyn et al., 2020).

2. The PXP Model: Hamiltonian and Scarring Mechanism

The PXP model precisely captures the essential physics of Rydberg atom chains in the perfect blockade limit. The Hamiltonian is: $$H_{\text{PXP}} = \sum_{i=1}^L P_{i-1}\, \sigma_i^x\, P_{i+1}~,$$ where σi^x is the Pauli-X operator acting on site i, and P_i = |◦⟩_i⟨◦| projects onto the atomic ground state. The P{i-1} and P_{i+1} projectors impose the Rydberg blockade constraint, preventing simultaneous excitation of adjacent sites. Despite being nonintegrable—a fact established via Wigner-Dyson level statistics—the model supports persistent coherent oscillations after initialization in certain product states such as the period-2 density wave |Z₂⟩ = |•◦•◦…⟩ (Serbyn et al., 2020).

Scarring arises as follows:

• A “scar tower” of highly excited eigenstates |ϕ_n⟩ with n = 0,1,…,O(L) is identified.
• These eigentates have anomalously large overlap with |Z₂⟩, in contrast to ETH expectations.
• Their energy spacing is nearly uniform, E_n ≈ E₀ + nω, with ω ≈ 1.33.
• Their bipartite entanglement entropy S_ent(ϕ_n) scales much more weakly than volume law, usually as a logarithmic function or even remaining constant.

3. Dynamical Consequences: Revivals and Weak Ergodicity Breaking

When the PXP chain is initialized in the period-2 density wave |Z₂⟩, time evolution leads to persistent oscillations in the fidelity F(t) = |⟨Z₂|e^{-iHt}|Z₂⟩|² and in local observables, with a characteristic period T = 2π/ω ≈ 4.7 (\hbar = 1). This phenomenon arises because the initial state overlaps predominantly with the scar tower rather than the thermal bulk: $$F(t) = \left| \sum_n |c_n|^2 e^{-i E_n t} \right|^2 \simeq \left| \sum_n |c_n|^2 e^{-i n \omega t} \right|^2~,$$ where c_n = ⟨ϕ_n|Z₂⟩. If the scar state coefficients |c_n|² are approximately symmetric and ω is sufficiently uniform, near-perfect revivals are observed at integer multiples of T. This dynamical signature is the hallmark of weak breaking of ergodicity—the full spectrum is thermalizing except for a sparse tower of nonthermal states (Serbyn et al., 2020).

4. Semiclassical and Algebraic Structures

The origin of scarring in the PXP model can be interpreted via both semiclassical dynamics and embedded algebraic structures:

• Semiclassical Quantization: Projecting the quantum dynamics onto a variational manifold of matrix-product states (lowest bond dimension ansatz) using the time-dependent variational principle (TDVP) yields effective classical equations of motion for two variational angles per site. The phase portrait constructed from these equations displays an unstable periodic orbit passing through variational representations of |Z₂⟩ and its symmetry partner. Along this orbit, quantum leakage—i.e., the deviation from the variational manifold—remains small, providing a robust semiclassical explanation for the observed quantum revivals.
• Embedded (Spectrum-Generating) Algebras: The scar tower may arise from the presence of an approximate or exact algebra (often su(2)), with generators Q^± and Q^z satisfying

$$[H, Q^\pm] = \pm\omega Q^\pm~, \quad [Q^+, Q^-]=2Q^z~, \quad [Q^z, Q^\pm]=\pm Q^\pm~.$$

Starting from a simple reference state |ψ₀⟩ annihilated by Q^-, one builds the scar states (Q^+)^n|ψ₀⟩, with energies distributed in an equally spaced ladder E₀ + nω (Serbyn et al., 2020).

• Krylov Subspace Construction: In certain instances, the scar tower spans a Krylov subspace {H^k|ψ₀⟩} of finite dimension which is only weakly coupled to the rest of the thermalizing spectrum.
• Projector Embedding (Shiraishi–Mori Construction): It is also possible to “embed” arbitrarily chosen low-entanglement states as exact eigenstates within an otherwise thermal spectrum by engineering the Hamiltonian to be of the form H = ∑_i P_i h_i P_i + H', with local projectors P_i annihilating the desired scar subspace.

5. Extensions: Related Models and Sectors

Generalizations and analogs of the PXP model have been constructed in diverse settings:

• The one-dimensional spin-1 Kitaev chain, with appropriate symmetry fragmentation, hosts scar sectors that can be exactly mapped onto the PXP Hamiltonian, while other sectors display even more robust scarring and longer-lived revivals than the ground-state mapping (Mohapatra et al., 2023).
• Models hosting an SU(2) spectrum-generating algebra, such as the AKLT chain and spin-1 XY magnets, admit analytic towers of exact scarred eigenstates.
• Variants in higher-dimensional and gauge-theoretic contexts (e.g., ℤ₂ and U(1) lattice gauge theories) allow for the embedding of scarred subspaces, with robust protection against symmetry-breaking errors possible via quantum Zeno mechanisms (Halimeh et al., 2022, Budde et al., 13 Mar 2024).

6. Experimental Realizations and Quantum Technology Applications

The PXP model's dynamics and scarring phenomena have been directly observed in Rydberg atom array experiments, where initialization in period-2 density wave product states leads to striking coherent oscillations and fidelity revivals, thus validating the theoretical framework. Applications of scars in quantum technology include:

• Preparation of entangled “Schrödinger cat” states by monitoring the timing of the many-body revivals and applying additional pulses at revival maxima.
• Potential for protected quantum information storage within the nonthermal subspaces generated by scars, due to their slow thermalization rates.
• Exploitation of scars for coherent state transfer protocols in quantum simulators and possibly for enhanced quantum sensing, due to the extended retention of phase coherence (Serbyn et al., 2020).

7. Outlook and Theoretical Implications

The PXP model serves as both a prototype and a test bed for the detailed study of quantum many-body scars and weak ergodicity breaking. The interplay between embedded algebraic structures, semiclassical dynamics, and experimental accessibility continues to illuminate the mechanisms underpinning the failure of thermalization in certain subspaces of nonintegrable systems. These advances have implications for understanding thermalization, eigenstate properties, and the engineering of long-lived coherent dynamics beyond traditional paradigms such as many-body localization and integrability (Serbyn et al., 2020).