ScarFinder: a detector of optimal scar trajectories in quantum many-body dynamics

Abstract

Mechanisms that give rise to coherent quantum dynamics, such as quantum many-body scars, have recently attracted much interest as a way of controlling quantum chaos. However, identifying the presence of quantum scars in general many-body Hamiltonians remains an outstanding challenge. Here we introduce ScarFinder, a variational framework that reveals possible scar-like dynamics without prior knowledge of scar states or their algebraic structure. By iteratively evolving and projecting states within a low-entanglement variational manifold, ScarFinder isolates scarred trajectories by suppressing thermal contributions. We validate the method on the analytically tractable spin-1 XY model, recovering the known scar dynamics, as well as the mixed field Ising model, where we capture and generalize the initial conditions previously associated with ``weak thermalization''. We then apply the method to the PXP model of Rydberg atom arrays, efficiently characterizing its mixed phase space and finding a previously unknown trajectory with nearly-perfect revival dynamics in the thermodynamic limit. Our results establish ScarFinder as a powerful, model-agnostic tool for identifying and optimizing coherent dynamics in quantum many-body systems.

Figures (13)

• Figure 1: Schematic illustration of a single step of the ScarFinder algorithm. (a) The initial state $|\psi(0)\rangle$, lying on a low-entanglement variational manifold $\mathcal{M}$, is assumed to be a superposition of a scar component $|\psi_0\rangle$, which evolves periodically (dashed line), and a generic component $|\phi\rangle$. (b) After a unitary time evolution step $\hat{U}(\Delta t)$, the state becomes $|\psi(\Delta t)\rangle$, with the superposition preserved. However, while the scar component $|\psi_0'\rangle$ remains low-entangled, the generic component $|\phi'\rangle$ becomes highly entangled and delocalized. (c) At $t = 0$, both components exhibit low entanglement, characterized by a small number of dominant Schmidt coefficients $\lambda_k$. (d) The delocalization of $|\phi^\prime\rangle$ leads to a broader distribution of Schmidt coefficients. This entanglement difference enables the separation of the two components in $|\psi(\Delta t)\rangle$: the leading Schmidt coefficients are dominated by the scar part, while the subleading terms largely originate from the thermal component. The projection step in ScarFinder leverages this structure to suppress thermal contributions and reinforce the scar trajectory.
• Figure 2: Convergence of scar fidelity $F_\mathcal{S}$ in Eq. (\ref{['eq:scarfid']}) for initial states $|\psi_\alpha\rangle$ in Eq. (\ref{['eq:imperfect_scar']}), simulated using iTEBD. (a) For a fixed projection time step $\Delta t=0.2$, states quickly converge to the scar trajectory for smaller values of $\alpha$, whereas larger $\alpha$ slows convergence. (b) For fixed deviation $\alpha=\pi/2$, larger values of $\Delta t$ generally accelerate convergence.
• Figure 3: Convergence behavior of ScarFinder initialized from random states at target energy $E_\text{target}=0$, with $\Delta t=0.2$. Energy conservation is explicitly enforced during the algorithm's iterations. (a) For the $\hat{V}=\hat{V}_1$ perturbation in Eq. (\ref{['eq:V1']}), among 100 independent trials, only 5 successfully converge. (b) For the $\hat{V} = \hat{V}_1'$ perturbation in Eq. (\ref{['eq:V1prime']}), 9 out of 10 trials converge successfully. (c),(d): Entanglement entropy of eigenstates for (c) $\hat{V} = \hat{V}_1$ and (d) $\hat{V} = \hat{V}_1'$. Improved convergence in (b) correlates with a clearer separation between scar and thermal states in the entanglement entropy spectrum. Data in panels (a),(b) are obtained using iTEBD with a maximal bond dimension $\chi=16$ and evolution step $d t=0.01$. Panels (c)-(d) are exact diagonalization results for a finite system $L=10$, combining momentum sectors $k=0$ and $k=\pi$.
• Figure 4: (a)-(c) Dynamics of the optimal scar initial state for the PXP Hamiltonian with a two-site unit cell and various bond dimensions $\chi$. (a) Entanglement entropy growth. Compared to the $|\mathbb{Z}_2\rangle$ state (black line), $\chi=2$ already shows significantly reduced entanglement growth, while $\chi=8,12$ dynamics are nearly flat, indicating an extremely stable scar. (b) Logarithmic fidelity dynamics for the optimal state at $\chi=12$, shows nearly perfect revivals with no visible decay. (c) Expectation value of the nearest-neighbor observable $\langle Z_i Z_{i+1} \rangle$ for $\chi=12$, showing persistent oscillations throughout the evolution. (d)-(g) Overlap between the optimized scar state $|\psi(\chi)\rangle$ and the eigenstates of the PXP Hamiltonian for different bond dimensions: (d) $\chi=2$, (e) $\chi=4$, (f) $\chi=8$, and (g) $\chi=12$. As $\chi$ increases, a well-defined band of scarred eigenstates becomes more distinct from the thermal bulk, with increasing spectral isolation. Data in panels (a)-(c) are obtained by iTEBD, while (d)-(g) are exact diagonalization results for system size $L=24$ in $k=0$ momentum sector. Since iTEBD simulations in (a)-(c) use a 4-site unit cell, the momentum $k=0$ in exact diagonalization data corresponds to translation by 4 sites.
• Figure 5: (a)-(c) Dynamics of the optimized scar initial states for the PXP Hamiltonian with a 3-site unit cell and different bond dimensions $\chi$. (a) Late-time entanglement entropy growth ($t \in [15,30]$). Increasing $\chi$ systematically suppresses the entanglement growth and stabilizes periodic dynamics. The "*-state" refers to the MPS-optimized initial state from Ref. Michailidis2020, previously shown to have better revivals compared to the $|\mathbb{Z}_3\rangle$ product state. (b) Late-time ($t \in [15,30]$) logarithmic fidelity for the optimized initial state at $\chi=12$, computed from the dominant eigenvalue of the MPS transfer matrix. The fidelity displays persistent revivals, highlighting the stability of the identified scar trajectory. (c) Time evolution of the nearest-neighbor observable $\langle Z_i Z_{i+1} \rangle$ for $\chi=12$. (d)-(g) Overlaps between energy eigenstates and the optimized initial scar state for bond dimensions $\chi=2$ (d), $\chi=4$ (e), $\chi=8$ (f), and $\chi=12$ (g). With increasing $\chi$, distinct eigenstate towers clearly emerge from the thermal continuum, signifying improved scar dynamics. Data in panels (d)-(g) is obtained by exact diagonalization for a system size $L=24$ in the $k=0$ momentum sector w.r.t. 3-site translations.