Slow Quantum Thermalization and Many-Body Revivals from Mixed Phase Space

Abstract

Describing the way strongly interacting quantum systems approach thermal equilibrium remains an important open problem. Recent works discovered systems in which thermalization rates may depend very sensitively on the initial conditions, via a mechanism reminiscent of quantum scars in chaotic billiards. While strongly interacting systems do not always have an obvious quasiclassical limit, time-dependent variational principle (TDVP) allows one to project the unitary dynamics onto the matrix-product state manifold, resulting in a classical nonlinear dynamical system. We show that such dynamical systems exhibit a mixed phase space which includes both regular and chaotic regions. Provided TDVP errors are small, the mixed phase space leaves a footprint on the exact dynamics of the quantum model: when the system is initialized in a state situated on the stable periodic orbit, it exhibits robust many-body revivals. Intriguingly, the initial state giving rise to strongest revivals may be entangled. Surprisingly, even when TDVP errors are large, as in the thermalizing Ising model with transverse and longitudinal fields, initializing the system in the regular region of phase space leads to a slowdown of thermalization. Our work establishes TDVP as a method for identifying interacting quantum systems with anomalous dynamics. Mixedphase space classical variational equations allow one to find slowly-thermalizing initial conditions in interacting models. These results provide an intriguing connection between classical and quantum chaos, pointing towards possible extensions of classical KAM theorem to quantum systems.

Quantum scars as embeddings of weakly broken Lie algebra representations

“…δ + (2),8 = P P σ + 2n P σ z P P + P P σ z P σ + 2n P P + P P σ − 2n+1 P σ z P P + P P σ z P σ − 2n+1 P P. (43) Putting these terms together, we obtain the second order perturbations,H + 2) . Coefficients optimizing fidelity were found to be: λ * i = [0.11135, 0.000217, −0.000287, −0.00717, (44) 0.00827, 0.00336, 0.00429, 0.0103, 0.00118], (45) where the first value is the optimal coefficient for the first order term Eq. (32), while the remaining coefficients correspond to the terms in order of appearance in Eqs.…”

“…30,31 While the collection of models that feature scarred-like eigenstates has recently expanded, [32][33][34][35][36][37][38][39][40][41][42][43] a smaller subset of such models have been demonstrated to display revivals from easily preparable initial states. [44][45][46] Thus, the connection between revivals and the presence of atypical eigenstates remains to be fully understood.…”

Slow thermalization of exact quantum many-body scar states under perturbations

“…References [22,23] developed a systematic construction to embed nonthermal states in the spectrum. Many other systems or models have also been discovered or constructed to have scars or scarlike physics [24][25][26][27][28][29][30][31][32][33][34][35].…”

Statistical localization: From strong fragmentation to strong edge modes

“…Several works investigated the possibility of mimicking similarly localized behavior without explicitly breaking translation invariance [25][26][27][28][29][30][31][32][33], as well as the possibility of intermediate behavior, such as the existence of a small number of ETH-violating eigenstates within an otherwise generic spectrum of states [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. Recently, the authors of the present paper, following earlier work on dipole-conserving random circuits [49], identified a novel mechanism for such non-ergodic behavior, dubbed Hilbert space fragmentation [50,51].…”