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arXiv:2103.11699 · Physical review. B./Physical review. B

Many-body localization in tilted and harmonic potentials

Ruixiao Yao, Titas Chanda, Jakub Zakrzewski

Published Jul 1, 2021• 40 citations
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Abstract

We discuss nonergodic dynamics of interacting spinless fermions in a tilted optical lattice as modeled by a XXZ spin chain in a magnetic (or electric) field changing linearly across the chain. The time dynamics is studied using exact propagation for small chains and matrix product state techniques for larger system sizes. We consider both the initial N\'eel separable state and the quantum quench scenario, in which the initial state may be significantly entangled. We show that the entanglement dynamics is significantly different in both cases. In the former a rapid initial growth is followed by a saturation for sufficiently large tilt $F$. In the latter case the dynamics seems to be dominated by pair tunneling, and the effective tunneling rate scales as $1/{F}^{2}$. In the presence of an additional harmonic potential the imbalance is found to be entirely determined by a local effective tilt ${F}_{\text{loc}}$; the entanglement entropy growth is modulated with frequency that follows $1/{F}_{\mathrm{loc}}^{2}$ scaling first, but at long time the dynamics is determined rather by the curvature of the harmonic potential. Only for large tilts or sufficiently large curvatures, corresponding to the deeply localized regime, do we find logarithmic entanglement growth for the N\'eel initial state. The same curvature determines the long-time imbalance for large $F$, which reveals strong revival phenomena associated with the manifold of equally spaced states, degenerate in the absence of the harmonic potential.

DOI
10.1103/physrevb.104.014201

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We discuss nonergodic dynamics of interacting spinless fermions in a tilted optical lattice as modeled by a XXZ spin chain in a magnetic (or electric) field changing linearly across the chain. The time dynamics is studied using exact propagation for small chains and matrix product state techniques for larger system sizes. We consider both the initial N'eel separable state and the quantum quench scenario, in which the initial state may be significantly entangled. We show that the entanglement dynamics is significantly different in both cases. In the former a rapid initial growth is followed by a saturation for sufficiently large tilt FFF. In the latter case the dynamics seems to be dominated by pair tunneling, and the effective tunneling rate scales as 1/F21/{F}^{2}1/F2. In the presence of an additional harmonic potential the imbalance is found to be entirely determined by a local effective tilt Floc{F}_{\text{loc}}Floc​; the entanglement entropy growth is modulated with frequency that follows 1/Floc21/{F}_{\mathrm{loc}}^{2}1/Floc2​ scaling first, but at long time the dynamics is determined rather by the curvature of the harmonic potential. Only for large tilts or sufficiently large curvatures, corresponding to the deeply localized regime, do we find logarithmic entanglement growth for the N'eel initial state. The same curvature determines the long-time imbalance for large FFF, which reveals strong revival phenomena associated with the manifold of equally spaced states, degenerate in the absence of the harmonic potential.
We discuss properties of interacting spinless fermions in a tilted optical lattice as modeled by Heisenberg spin chain in magnetic (or electric) field changing linearly across the chain. The system shows genuine features of strong localization for a sufficiently large tilt with distinct features different from disorder induced localization. In particular, Stark many-body localization is characterized in time dynamics by a linear increase of the entanglement entropy followed by a saturation at a constant, time independent value -- both the number (classical) and the configuration (quantum) entropies saturate to a constant value. Similarly, the correlation with the initial state (the so called imbalance) stabilizes at a constant value (modulo large fluctuations coming from short time Bloch-type oscillations). The presence of an additional harmonic potential profoundly affects the dynamics of the model -- its behavior resembles standard MBL characteristics -- the logarithmic growth of the entanglement entropy in particular. In that case, we find strong revival phenomena associated with the manifold of equally spaced states, degenerate in the absence of the harmonic potential.
The quench dynamics of the Hubbard model in tilted and harmonic potentials is discussed within the semiclassical picture. Applying the fermionic truncated Wigner approximation (fTWA), the dynamics of imbalances for charge and spin degrees of freedom is analyzed and its time evolution is compared with the exact simulations in a one-dimensional lattice. Quench from charge or spin density waves is considered. We show that introduction of harmonic and spin-dependent linear potentials sufficiently validates fTWA for longer times. Such an improvement of fTWA is also obtained for the higher-order correlations in terms of quantum Fisher information for charge and spin channels. This allows us to discuss the dynamics of larger system sizes and connect our discussion to the recently introduced Stark many-body localization. In particular, we focus on a finite two-dimensional system and show that at intermediate linear potential strength, the addition of a harmonic potential and spin dependence of the tilt results in subdiffusive dynamics, similar to that of disordered systems. Moreover, for specific values of harmonic potential, we observed phase separation of ergodic and nonergodic regions in real space. The latter fact is especially important for ultracold-atom experiments in which harmonic confinement can be easily imposed, causing a significant change in relaxation times for different lattice locations.
We investigate the dynamics following a global parameter quench for two one-dimensional models with variable-range power-law interactions: a long-range transverse Ising model, which has recently been realized in chains of trapped ions, and a long-range lattice model for spinless fermions with long-range tunneling. For the transverse Ising model, the spreading of correlations and growth of entanglement are computed using numerical matrix product state techniques, and are compared with exact solutions for the fermionic tunneling model. We identify transitions between regimes with and without an apparent linear light cone for correlations, which correspond closely between the two models. For long-range interactions (in terms of separation distance rrr, decaying slower than 1/r1/r1/r), we find that despite the lack of a light cone, correlations grow slowly as a power law at short times, and that---depending on the structure of the initial state---the growth of entanglement can also be sublinear. These results are understood through analytical calculations, and should be measurable in experiments with trapped ions.
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 Nkinks{N}_{\mathrm{kinks}}Nkinks​ 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 Nkinks{N}_{\mathrm{kinks}}Nkinks​ such that the decay exponent $\ensuremath{\gamma}$ in the density imbalance increases with increase in Nkinks{N}_{\mathrm{kinks}}Nkinks​. 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, hNkinks{h}_{{N}_{\mathrm{kinks}}}hNkinks​​, which is a monotonically increasing function of Nkinks{N}_{\mathrm{kinks}}Nkinks​. A one-kink state always underestimates the value of the disorder at which the MBL transition takes place but h1kink{h}_{\text{1}\phantom{\rule{0.16em}{0ex}}\text{kink}}h1kink​ 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)aln(Vt)aln(Vt) not only in the MBL phase but also in the delocalized phase and in both the phases the coefficient aaa increases with Nkinks{N}_{\mathrm{kinks}}Nkinks​ as well as with the interaction strength VVV. 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.
We numerically investigate the dynamics of entanglement in a chain of spinless fermions with nonrandom but long-range hopping and interactions, and with random on-site energies. For moderate disorder in the absence of interactions, the chain hosts delocalized states at the top of the band which undergo a delocalization-localization transition with increasing disorder. We find an interesting regime in this noninteracting disordered chain where the long-time entanglement entropy scales as $S(t)\ensuremath{\sim}lnt$ and the saturated entanglement entropy scales with system size LLL as $S(L,t\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\sim}lnL$. We further study the interplay of long-range hopping and interactions on the growth of entanglement and the many-body localization (MBL) transition in this system. We develop an analogy to higher-dimensional short-range systems to compare and contrast such behavior with the physics of MBL in a higher dimension.
We investigate the dynamics of two bosons in a tilted one-dimensional optical lattice with both standard and density-assisted tunnelling. The relative sign of the two tunnelling processes strongly affects the Bloch dynamics, leading to distinct reinforcing and competing regimes. We examine how the tunnelling processes generate correlations and find that, in the absence of onsite interactions, the early-time entanglement growth follows the same scaling with the density-assisted tunnelling amplitude. Finite onsite interaction changes this picture by breaking the symmetry between the reinforcing and competing regimes and enhancing the correlation differently in two regimes. This shows how the interplay between the two tunnelling mechanisms can change dyanmics and entanglement in a controlled few-body setting. Particularly, at equal tunnelling amplitudes, the doublon and singlon sectors decouple completely for arbitrary onsite interaction, lattice tilting, and initial state.