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arXiv:2202.05217 · Physical review. B./Physical review. B

Initial state dependent dynamics across the many-body localization transition

Yogeshwar Prasad, Arti Garg

Published Jun 6, 2022• 9 citations
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Abstract

We investigate quench dynamics across many-body localization (MBL) transition in an interacting one-dimensional system of spinless fermions with aperiodic potential. We consider a large number of initial states characterized by the number of kinks ${N}_{\mathrm{kinks}}$ in the density profile, such that the equal number of sites are occupied between any two consecutive kinks. We show that on the delocalized side of the MBL transition the dynamics becomes faster with increase in ${N}_{\mathrm{kinks}}$ such that the decay exponent $\ensuremath{\gamma}$ in the density imbalance increases with increase in ${N}_{\mathrm{kinks}}$. The growth exponent of the mean square displacement which shows a power-law behavior $\ensuremath{\langle}{x}^{2}(t)\ensuremath{\rangle}\ensuremath{\sim}{t}^{\ensuremath{\beta}}$ in the long-time limit is much larger than the exponent $\ensuremath{\gamma}$ for one-kink and other low-kink states though $\ensuremath{\beta}\ensuremath{\sim}2\ensuremath{\gamma}$ for a charge density wave state. As the disorder strength increases ${\ensuremath{\gamma}}_{{N}_{\mathrm{kink}}}\ensuremath{\rightarrow}0$ at some critical disorder, ${h}_{{N}_{\mathrm{kinks}}}$, which is a monotonically increasing function of ${N}_{\mathrm{kinks}}$. A one-kink state always underestimates the value of the disorder at which the MBL transition takes place but ${h}_{\text{1}\phantom{\rule{0.16em}{0ex}}\text{kink}}$ coincides with the onset of the subdiffusive phase preceding the MBL phase. This is consistent with the dynamics of interface broadening for the one-kink state. We show that the bipartite entanglement entropy has a logarithmic growth $aln(Vt)$ not only in the MBL phase but also in the delocalized phase and in both the phases the coefficient $a$ increases with ${N}_{\mathrm{kinks}}$ as well as with the interaction strength $V$. We explain this dependence of dynamics on the number of kinks in terms of the normalized participation ratio of initial states in the eigenbasis of the interacting Hamiltonian.

DOI
10.1103/physrevb.105.214202

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Understanding the nature of the transition from the delocalized to the many-body localized (MBL) phase is an important unresolved issue. To probe the nature of the MBL transition, we investigate the universal properties of single-particle excitations produced in highly excited many-body eigenstates of a disordered interacting quantum many-body system. In a class of one-dimensional spinless fermionic models with random disorder, we study the finite-size scaling of the ratio of typical to average values of the single-particle local density of states and the scattering rates across the MBL transition. Our results indicate that the MBL transition in this class of one-dimensional models of spinless fermions is continuous in nature. For various ranges of interactions in the system, the critical exponent $\ensuremath{\nu}$ with which the correlation length $\ensuremath{\xi}$ diverges at the transition point ${W}_{c}, \ensuremath{\xi}\ensuremath{\sim}|W\ensuremath{-}{W}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, satisfies the Chayes-Chayes-Fisher-Spencer (CCFS) bound $\ensuremath{\nu}\ensuremath{\ge}2/d$ where ddd is the physical dimension of the system. We also discuss why the critical exponent obtained from finite-size scaling of the conventional diagnostic of many-body localization, the level spacing ratio, strongly violates the CCFS bound while the single-particle density of states and scattering rates are consistent with the CCFS criterion.
Many-body localization is characterized by a slow logarithmic growth of the entanglement entropy after a global quantum quench while the local memory of an initial density imbalance remains at infinite time. We investigate how much the proximity of a many-body localized phase can influence the dynamics in the delocalized ergodic regime where thermalization is expected. Using an exact Krylov space technique, the out-of-equilibrium dynamics of the random-field Heisenberg chain is studied up to L=28L=28L=28 sites, starting from an initially unentangled high-energy product state. Within most of the delocalized phase, we find a sub-ballistic entanglement growth $S(t)\ensuremath{\propto}{t}^{1/z}$ with a disorder-dependent exponent $z\ensuremath{\ge}1$, in contrast with the pure ballistic growth z=1z=1z=1 of clean systems. At the same time, anomalous relaxation is also observed for the spin imbalance $\mathcal{I}(t)\ensuremath{\propto}{t}^{\ensuremath{-}\ensuremath{\zeta}}$ with a continuously varying disorder-dependent exponent $\ensuremath{\zeta}$, vanishing at the transition. This provides a clear experimental signature for detecting this nonconventional regime.