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arXiv:1808.05764 · Physical review. B./Physical review. B

Spin-mediated particle transport in the disordered Hubbard model

Ivan V. Protopopov, Dmitry A. Abanin

Published Mar 7, 2019• 20 citations
arXivHTMLPDF

Abstract

Motivated by the recent experiments that reported signatures of many-body localization of ultracold atoms in optical lattices [M. Schreiber et al., Science 349, 842 (2015)], we study dynamics of highly excited states in the strongly disordered Hubbard model in one dimension. Owing to the $SU(2)$ spin symmetry, spin degrees of freedom form a delocalized thermal bath with a narrow bandwidth. The spin bath mediates slow particle transport, eventually leading to delocalization of particles. The particle hopping rate is exponentially small in $t/W\phantom{\rule{4pt}{0ex}}(t,W$ being hopping and disorder scales) owing to the narrow bandwidth of the spin bath. We find the optimal length scale for particle hopping and show that the particle transport rate depends strongly on the density of singly occupied sites in the initial state. The delocalization rate is zero for initial states with only doubly occupied or empty sites, suggesting that such states are truly many-body localized, and therefore the Hubbard model may host both localized and delocalized states. Full many-body localization can be induced by breaking spin rotational symmetry.

DOI
10.1103/physrevb.99.115111

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Motivated by the recent experiments that reported signatures of many-body localization of ultracold atoms in optical lattices [M. Schreiber et al., Science 349, 842 (2015)], we study dynamics of highly excited states in the strongly disordered Hubbard model in one dimension. Owing to the SU(2)SU(2)SU(2) spin symmetry, spin degrees of freedom form a delocalized thermal bath with a narrow bandwidth. The spin bath mediates slow particle transport, eventually leading to delocalization of particles. The particle hopping rate is exponentially small in t/W(t,Wt/W\phantom{\rule{4pt}{0ex}}(t,Wt/W(t,W being hopping and disorder scales) owing to the narrow bandwidth of the spin bath. We find the optimal length scale for particle hopping and show that the particle transport rate depends strongly on the density of singly occupied sites in the initial state. The delocalization rate is zero for initial states with only doubly occupied or empty sites, suggesting that such states are truly many-body localized, and therefore the Hubbard model may host both localized and delocalized states. Full many-body localization can be induced by breaking spin rotational symmetry.
We study spin transport in a Hubbard chain with a strong, random, on-site potential and with spin-dependent hopping integrals ${t}_{\ensuremath{\sigma}}$. For the SU(2) symmetric case ${t}_{\ensuremath{\uparrow}}={t}_{\ensuremath{\downarrow}}$, such a model exhibits only partial many-body localization with localized charge and (delocalized) subdiffusive spin excitations. Here, we demonstrate that breaking the SU(2) symmetry by even weak spin asymmetry, ${t}_{\ensuremath{\uparrow}}\ensuremath{\ne}{t}_{\ensuremath{\downarrow}}$, localizes spins and restores full many-body localization. To this end we derive an effective spin model, where the spin subdiffusion is shown to be destroyed by arbitrarily weak ${t}_{\ensuremath{\uparrow}}\ensuremath{\ne}{t}_{\ensuremath{\downarrow}}$. Instability of the spin subdiffusion originates from an interplay between random effective fields and singularly distributed random exchange interactions.
The many-body localization (MBL) proximity effect is an intriguing phenomenon where a thermal bath localizes due to the interaction with a disordered system. The interplay of thermal and nonergodic behavior in these systems gives rise to a rich phase diagram, whose exploration is an active field of research. In this paper, we study a bosonic Hubbard model featuring two particle species representing the bath and the disordered system. Using state-of-the-art numerical techniques, we investigate the dynamics of the model in different regimes, based on which we obtain a tentative phase diagram as a function of coupling strength and bath size. When the bath is composed of a single particle, we observe clear signatures of a transition from an MBL proximity effect to a delocalized phase. Increasing the bath size, however, its thermalizing effect becomes stronger and eventually the whole system delocalizes in the range of moderate interaction strengths studied. In this regime, we characterize particle transport, revealing diffusive behavior of the originally localized bosons.
We experimentally study localization and dynamics of ultracold fermions in speckle and optical lattice potentials to explore Anderson localization, many-body localization, and relaxation dynamics in strongly correlated systems. Anderson localization is probed by releasing non-interacting, spin-polarized gases into three dimensional, anisotropic disordered potentials produced from optical speckle. A fraction of the atoms are localized by the disorder, and a mobility edge is found separating localized from extended states. The length scale of the speckle is varied, and the localized state is found to scale linearly with the geometric mean of the speckle autocorrelation length. We realize the Fermi Hubbard model by loading atoms in a cubic optical lattice. Non-equilibrium momentum distributions are created via Raman transitions, and the excitation relaxation rate is measured in the lattice. Transport experiments were performed in a disordered optical lattice to explore the disordered Hubbard model. These experiments reveal localization in the presence of strong interactions and an interaction driven metal-to-insulator transition. The localized state is found to be insensitive to a doubling in the temperature of the gas and is consistent with predictions of many-body localization.
Anderson localization1 in a random system is sensitive to a distance dependence of the excitation transfer amplitude V(r). If V(r) decreases with the distance r slower than 1/r^d in a d-dimensional system then all excitations are delocalized at arbitrarily strong disordering, due to the resonant interaction of far separated quantum states (Fig. 1). At finite temperature T>0 the density of excitations is finite and they can influence each other by means of their interaction. Many body excitations involving simultaneous transitions of several single particle excitations create additional channels for energy delocalization and transport. Here we show that if the interaction of excitations decreases with the distance slower than 1/R^(2d) then excitations are delocalized at finite temperature irrespectively to disordering. This delocalization results in the finite decoherence rate in the ensemble of interacting spins 1/2 representing the model of quantum computer, thus restricting the quantum hardware performance. It also leads to the energy and particle delocalization and transport at finite temperature in various physical systems including doped semiconductors, despite of the full localization of electrons at zero temperature.
Many-body localization (MBL) in a one-dimensional Fermi Hubbard model with random on-site interactions is studied. While for this model all single-particle states are trivially delocalized, it is shown that for sufficiently strong disordered interactions the model is many-body localized. It is therefore argued that MBL does not necessarily rely on localization of the single-particle spectrum. This model provides a convenient platform to study pure MBL phenomenology, since Anderson localization in this model does not exist. By examining various forms of the interaction term, a dramatic effect of symmetries on charge transport is demonstrated. A possible realization in a cold atom experiment is proposed.
We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.