Abstract
Symmetries play a central role in single-particle localization. Recent research focused on many-body localized (MBL) systems, characterized by new kind of integrability, and by the area-law entanglement of eigenstates. We investigate the effect of a non-Abelian $SU(2)$ symmetry on the dynamical properties of a disordered Heisenberg chain. While $SU(2)$ symmetry is inconsistent with the conventional MBL, a new non-ergodic regime is possible. In this regime, the eigenstates exhibit faster than area-law, but still a strongly sub-thermal scaling of entanglement entropy. Using exact diagonalization, we establish that this non-ergodic regime is indeed realized in the strongly disordered Heisenberg chains. We use real-space renormalization group (RSRG) to construct approximate excited eigenstates, and show their accuracy for systems of size up to $L=26$. As disorder strength is decreased, a crossover to the thermalizing phase occurs. To establish the ultimate fate of the non-ergodic regime in the thermodynamic limit, we develop a novel approach for describing many-body processes that are usually neglected by RSRG, accessing systems of size $L>2000$. We characterize the resonances that arise due to such processes, finding that they involve an ever growing number of spins as the system size is increased. The probability of finding resonances grows with the system size. Even at strong disorder, we can identify a large lengthscale beyond which resonances proliferate. Presumably, this eventually would drive the system to a thermalizing phase. However, the extremely long thermalization time scales indicate that a broad non-ergodic regime will be observable experimentally. Our study demonstrates that symmetries control dynamical properties of disordered, many-body systems. The approach introduced here provides a versatile tool for describing a broad range of disordered many-body systems.
Highlights
The remarkable experimental advances of the past decade open a window into probing highly nonequilibrium dynamics of interacting quantum systems, using platforms such as ultracold atoms [1], trapped ions [2], and NV centers in diamond [3]
For the system sizes analyzed, it is evident that ergodicity has not developed and a single SDRG tree state provides a good approximation to the eigenstates, as we demonstrate in the previous subsection
The goal of this paper is to investigate the effects of continuous non-Abelian symmetries on dynamical properties of disordered systems
Summary
The remarkable experimental advances of the past decade open a window into probing highly nonequilibrium dynamics of interacting quantum systems, using platforms such as ultracold atoms [1], trapped ions [2], and NV centers in diamond [3]. [40,41] is that at sufficiently strong disorder a new kind of dynamical, nonergodic phase may emerge—characterized by an entanglement entropy of excited eigenstates that is subthermal but scales faster than the area law [e.g., as SentðlÞ ∼ c logðlÞ]. Such a phase would display only a partial set of LIOMs, being distinct from the conventional MBL phase. We analyze the number and structure of resonances, finding that they become relevant at large system sizes This result signals that at L → ∞ SUð2Þ-symmetric systems thermalize but via unconventional, multispin processes.
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