Quantum state complexity meets many-body scars

TL;DR

This work investigates how spread (Krylov) complexity can diagnose and quantify many-body scar dynamics in the non-integrable PXP model. It contrasts naive Lanczos-based complexity with Forward Scattering Approximation (FSA) to construct closed Krylov subspaces hosting scar states, and analyzes complexity for Z2, Z3, and vacuum initial states. The study shows that FSA captures revival structures in complexity more faithfully than standard Lanczos, and demonstrates how optimal perturbations can restore near SU(2)-like symmetry to strengthen revivals; for the vacuum, carefully engineered few-body terms are required to stabilize oscillations. Overall, the results link dynamical symmetry, Krylov subspace structure, and complexity, offering a diagnostic pathway for weak ergodicity breaking in many-body systems and guiding perturbation design to sustain scar dynamics.

Abstract

Scar eigenstates in a many-body system refers to a small subset of non-thermal finite energy density eigenstates embedded into an otherwise thermal spectrum. This novel non-thermal behaviour has been seen in recent experiments simulating a one-dimensional PXP model with a kinetically-constrained local Hilbert space realized by a chain of Rydberg atoms. We probe these small sets of special eigenstates starting from particular initial states by computing the spread complexity associated to time evolution of the PXP hamiltonian. Since the scar subspace in this model is embedded only loosely, the scar states form a weakly broken representation of the Lie Algebra. We demonstrate why a careful usage of the Forward Scattering Approximation (or similar strategies thereof) is required to extract an appropriate set of Lanczos coefficients in this case as the consequence of this approximate symmetry. This leads to a well defined notion of a closed Krylov subspace and consequently, that of spread complexity. We show how the spread complexity shows approximate revivals starting from both $|\mathbb{Z}_2\rangle$ and $|\mathbb{Z}_3\rangle$ states and how these revivals can be made more accurate by adding optimal perturbations to the bare Hamiltonian. We also investigate the case of the vacuum as the initial state, where revivals can be stabilized using an iterative process of adding few-body terms.

Figures (11)

• Figure 1: Panel (b) and (c) shows $\{\beta_{n}\}$ coefficients (Note that the set $\{\alpha_{n}\}$ is identically 0) as a function of iteration number $n$, respectively for the Lanczos and FSA method. The analysis is performed for $\mathcal{H} = \mathcal{H}_{PXP} + \lambda \mathcal{H}_{P, Z_2}$, with a system size of $L=18$. As clearly seen, the $\{\beta_{n}\}$ coefficients are identically 0 after $L+1$ steps whereas those for the straightforward Lanczos method are finite for any $n$ (even at the 19th step the value of $\beta\sim 0.13$ for optimal perturbation); which is an important difference between FSA and Lanczos. However, irrespective of the value of $\lambda$, as long as one is interested only in the non-zero $\{\beta_{n}\}$ set for FSA, identical results are produced by Lanczos, as shown in panel (a).
• Figure 2: Panel (a) shows the return probability $\mathcal{R}(t)$ defined in Eqn. \ref{['RP']} as a function of $t$ for different values of perturbation strengths $\lambda$ for the system $\mathcal{H} = \mathcal{H}_{PXP} + \lambda \mathcal{H}_{P, \mathbb{Z}_2}$, with $|\mathbb{Z}_2\rangle$ initial state. We see that revival is strengthened at/near a very special value of $\lambda=0.108$, which signifies the best possible restoration of the $su(2)$ symmetry at the aforementioned point. Larger $\lambda$ ($\gtrsim 0.15$) can significantly destroy the revivals. Panel (b) shows the quantum complexity $\mathcal{C}_{L}(|\mathbb{Z}_2\rangle, \lambda, t)$ computed via Lanczos method. At $\lambda=0$ i.e., for the unperturbed PXP model the (imperfect) revivals in $\mathcal{R}(t)$ are also captured via $\mathcal{C}_{L}(t)$. The destruction of revival for larger $\lambda$'s are also reflected via $\mathcal{C}_{L}(t)$. In panel (c), we show the complexity $\mathcal{C}_{FSA}(|\mathbb{Z}_2\rangle, \lambda, t)$ computed from FSA. Not only does it capture perfect and imperfect revivals and the destruction of the same, but also the the nature of oscillation has rather striking qualitative similarity with that for $\mathcal{R}(t)$. Finally in panel (d), we demonstrate $\mathcal{C}_{L}(t)$ and $\mathcal{C}_{FSA}(t)$ for the special perturbation strength $\lambda=0.108$, which restores the $su(2)$ symmetry to a very good extent within scar subspace leading to perfect revivals.
• Figure 3: Comparison of FSA-like and Lanczos methods for $|\mathbb{Z}_3\rangle$ case are shown in the left panel. $\vec{\lambda} = \vec{\lambda}^{*}_{\mathbb{Z}_{3}}$ denotes optimal symmetry restoring values of the perturbations given in \ref{['z3pertstrength']}. Coefficients calculated from FSA and vanilla Lanczos at this value (blue and green dotted lines) almost coincide, however only the FSA curve closes at $2L/3$. The associated revivals for fidelity and complexity are plotted on the right hand panel. Note that the bare model shows a beating pattern on top of the revivals in $\mathcal{R}(t)$, indicating a modulation in oscillation amplitudes, this is however absent in the perturbed model. Certainly, FSA complexity shares more structural similarity with $\mathcal{R}(t)$ compared to Lanczos complexity.
• Figure 4: Lanczos coefficients (upper panel) and complexity calculated for the system described in \ref{['exactz3']}. Note that FSA closes after $L/3 +1$ steps (with system size $L=18$). The exact$su(2)$ symmetry is very apparent in both these figures.
• Figure 5: Lanczos and FSA from the vacuum state. As seen before for $|\mathbb{Z}_2\rangle$ case, naive Lanczos from PXP does not close, while FSA closes after $\frac{L}{2}$ iterations. Adding a set of perturbative higher spin terms approximately restore the $su(2)$ symmetry to the system.