Thermalization of exact quantum many-body scars in spin-1 XY chain under perturbation

TL;DR

The paper addresses the stability of quantum many-body scars in a spin-1 XY chain under a U(1)-preserving, off-diagonal perturbation that breaks bipartiteness. It employs perturbation theory up to third order and finite-size scaling of perturbation-matrix elements to track how the scar states hybridize with ETH states and how their dynamical signatures degrade. The key finding is that scars thermalize in larger chains, as indicated by the $D_H^{-1/2}$ scaling of perturbation matrix elements and the decay of scar-induced revivals and ODLRO, with a weak-perturbation relaxation time $\tau \sim c\lambda^{-2}$. This work highlights the fragility of QMBS under generic perturbations and connects their robustness to an emergent spectrum-generating algebra governing the scar subspace.

Abstract

Quantum many-body scars are special eigenstates that violate the eigenstate thermalization hypothesis while residing at finite energy density along with thermalizing eigenstates. The spin-1 XY model is known to host a family of such exceptional states originating from long-lived quasiparticle excitations that exhibit anomalously low entanglement entropy and long-time periodic revivals, resulting in weak ergodicity breaking. We study the stability of these scarred states against typical U(1) symmetry preserving perturbation in the XY chain. While perturbation theory can describe the deformed scar states at small system sizes, finite-size scaling of the perturbation matrix elements indicate that the scars ultimately thermalize in larger chains. Nonetheless, we demonstrate that the long-range order associated with the scars decays under the perturbation, and we estimate the relaxation timescale of oscillatory dynamics in certain local observables to be of order $λ^{-2}$, where $λ$ is the perturbation strength.

Figures (13)

• Figure 1: (a) Bipartite entanglement entropy $S_A$ of eigenstates of $H_{0}$ for an $L=10$ spin-1 XY chain with OBC and $(J_1,J_3,D,h)=(1.0,0.1,0.1,1.0)$. Density of states in the zero-magnetization sector is shown by color. Red circles denote $S_A$ for scarred states Eq. (\ref{['first_NN_scar']}). The dashed line at the top is the entropy for a random state, $S^{\mathrm{ran}}_A=(L/2)\ln 3-\tfrac{1}{2}$. The scar states are equally spaced in the spectrum and have low entanglement entropy compared to the other eigenstates. (b) Scaling of off-diagonal long-range order ${\mathcal{C}{}_n}$ Eq. (\ref{['odlro']}) calculated numerically for different scar states, with chain length $L$. The Hamiltonian parameters are the same as in (a).
• Figure 2: Time evolution of an $L=8$ chain with rest of the parameters identical to those in Fig. \ref{['fig:unp_entang']}: (a) Fidelity $\mathcal{F}(t) = |\langle{\psi(0)}|{\psi(t)}\rangle|^2$ for different initial states. The Néel state shows perfect periodic revivals in contrast to other non-special states, which display a very fast decay. (b) Entanglement entropy of both the nematic ferro state and $S_z$ product state quickly converge to the maximal entropy (shown by the black dashed line) under time evolution with $H_0$, but the Néel state shows no change in entanglement dynamics.
• Figure 3: Late time dynamics of expectation value of the local observable $O_1$ Eq. (\ref{['nematic_director_param']}) for both Néel and ferro initial states for $L=8$, with the remaining parameters same as in Fig. \ref{['fig:unp_entang']}. When initialized in the Néel state, perfect oscillations are observed in $\braket{O_1}$ over time $t$, whereas the observable shows no oscillation for the ferro state.
• Figure 4: Long-time evolution of an $L=8$ chain under the Hamiltonian $H$ Eq. (\ref{['H']}) after a quench from the nematic Néel state Eq. (\ref{['neel_state']}), with rest of the parameters identical to those in Fig. \ref{['fig:unp_entang']}: (a) Fidelity Eq. (\ref{['fidelity']}) and (b) bipartite entanglement entropy. Results are shown for increasing strength of perturbation Eqs. (\ref{['H']}) and (\ref{['H_p']}) starting from $\lambda=0.02$.
• Figure 5: Heatmap plot of squared overlaps of the exact eigenstates $|E_n(\lambda)\rangle$ of the perturbed Hamiltonian $H_0+\lambda H_p$ for $L=6$, while the remaining parameters are same as in Fig. \ref{['fig:unp_entang']}, with: the scarred states (a), (b), (c) the $|\mathcal{S}{}_1\rangle$, $|\mathcal{S}{}_2\rangle$, $|\mathcal{S}{}_3\rangle$ respectively, and (d) the thermal state $|\psi_{th}\rangle$. The thermal state is chosen as the eigenstate of the unperturbed Hamiltonian $H_0$ with eigenindex four more than the index of $|\mathcal{S}{}_3\rangle$ without any symmetry resolving. The plots simply show the projection of weights of the considered state on the perturbed manifold.