The stability of localization in the presence of interactions remains an open problem, with finite-size effects posing significant challenges to numerical studies. In this work, we investigate the perturbative stability of noninteracting localization under weak interactions, which allows us to analyze much larger system sizes. Focusing on disordered Anderson and quasiperiodic Aubry-Andr\'e models in one dimension, and using the adiabatic gauge potential at first order in perturbation theory, we compute first-order corrections to noninteracting local integrals of motion (LIOMs). We find that for at least an $O(1)$ fraction of the LIOMs, the corrections are well-controlled and converge at large system sizes, while others suffer from resonances. Additionally, we introduce and study the charge-transport capacity of this weakly interacting model. To first order, we find that the charge transport capacity remains bounded in the presence of interactions. Taken together, these results demonstrate that localization is perturbatively stable to weak interactions at first order, implying that, at the very least, localization persists for parametrically long times in the inverse interaction strength. We expect this perturbative stability to extend to all orders at sufficiently strong disorder, where the localization length is short, representing the true localized phase. Conversely, our findings suggest that the previously proposed interaction-induced avalanche instability, namely, in the weakly localized regime of the Anderson and Aubry-Andr\'e models, is a more subtle phenomenon arising only at higher orders in perturbation theory or through nonperturbative effects.
The stability of localization in the presence of interactions remains an open problem, with finite-size effects posing significant challenges to numerical studies. In this work, we investigate the perturbative stability of noninteracting localization under weak interactions, which allows us to analyze much larger system sizes. Focusing on disordered Anderson and quasiperiodic Aubry-Andr'e models in one dimension, and using the adiabatic gauge potential at first order in perturbation theory, we compute first-order corrections to noninteracting local integrals of motion (LIOMs). We find that for at least an O(1) fraction of the LIOMs, the corrections are well-controlled and converge at large system sizes, while others suffer from resonances. Additionally, we introduce and study the charge-transport capacity of this weakly interacting model. To first order, we find that the charge transport capacity remains bounded in the presence of interactions. Taken together, these results demonstrate that localization is perturbatively stable to weak interactions at first order, implying that, at the very least, localization persists for parametrically long times in the inverse interaction strength. We expect this perturbative stability to extend to all orders at sufficiently strong disorder, where the localization length is short, representing the true localized phase. Conversely, our findings suggest that the previously proposed interaction-induced avalanche instability, namely, in the weakly localized regime of the Anderson and Aubry-Andr'e models, is a more subtle phenomenon arising only at higher orders in perturbation theory or through nonperturbative effects.
Anderson localization, where quenched disorder leads to the localization of single-particle wave functions, has been studied extensively. However, the effect of interaction and non-random potentials on many-body localization (MBL) has gained attention recently. We investigate the dynamics of the interacting Aubry-André model to assess the stability of the many-body localized (MBL) phase. Our motivation to consider the Aubry–André potential as the disorder potential is to eliminate the avalanche mechanism, which often destabilizes MBL in systems with quenched disorder. We analyze the time evolution and the system-size scaling across a range of disorder strengths for the entanglement entropy, number entropy, and the Hartley number entropy. In the non-interacting case, our results confirm an Anderson localization transition at a critical disorder strength Dc = 2, which is consistent with earlier studies. However, when interactions are introduced, the number entropy appears to grow unboundedly as SN ~ ln ln t in the thermodynamic limit, even for large disorder. Meanwhile, the entanglement entropy grows logarithmically, S ~ ln t, and both entropies saturate in a correlated manner at finite times td ~ e^(L·D). These findings suggest that using a quasiperiodic potential to remove the avalanche mechanism does not stabilize true MBL behavior. Instead, the long-time dynamics are qualitatively similar to those found in systems with random disorder. In this thesis, we first introduce the fundamental concepts of thermalization, quantum quench, and symmetry-resolved entropies, then provide a study of localization, specifically in the Aubry-André model. Finally, we place our numerical results in the broader context of the current debate on the stability of many-body localized phases.
Whether disordered and quasiperiodic many-body quantum systems host a long-lived localized phase in the thermodynamic limit has been the subject of intense recent debate. While in one dimension, substantial evidence for such a many-body localized (MBL) phase exists, the behavior in higher dimensions remains an open puzzle. In disordered systems, for instance, it has been argued that rare regions may lead to thermalization of the whole system through a mechanism dubbed avalanche instability. In quasiperiodic systems, however, rare regions are altogether absent and the fate of a putative many-body localized phase has hitherto remained largely unexplored. In this work, we investigate the localization properties of two many-body quasiperiodic models, which are two-dimensional generalizations of the Aubry-Andr\'e model. By studying numerically the out-of-equilibrium dynamics of large systems, we find very long-lived localization on experimentally relevant time scales. Surprisingly, we also observe large-scale transport along deterministic lines of weak potential, which appear in the investigated quasiperiodic models. Our results demonstrate that quasiperiodic many-body systems have the remarkable and counter-intuitive capability of exhibiting coexisting localization and transport properties---a phenomenon reminiscent of the behavior of supersolids. Our findings are of direct experimental relevance and can be tested, for instance, using state-of-the-art cold atomic systems.
Anderson localization is known to be inevitable in one-dimension for generic disordered models. Since localization leads to Poissonian energy level statistics, we ask if localized systems possess 'additional' integrals of motion as well, so as to enhance the analogy with quantum integrable systems. We answer this in the affirmative in the present work. We construct a set of nontrivial integrals of motion for Anderson localized models, in terms of the original creation and annihilation operators. These are found as a power series in the hopping parameter. The recently found Type-1 Hamiltonians, which are known to be quantum integrable in a precise sense, motivate our construction. We note that these models can be viewed as disordered electron models with infinite-range hopping, where a similar series truncates at the linear order. We show that despite the infinite range hopping, all states but one are localized. We also study the conservation laws for the disorder free Aubry–Andre model, where the states are either localized or extended, depending on the strength of a coupling constant. We formulate a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using this procedure in the Aubry–Andre model, we show that integrals of motion given by our construction are well-defined in localized phase, but not so in the extended phase. Finally, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction.
The fate of Many-Body Localization (MBL) in the thermodynamic limit remains elusive, partly because numerical studies suffer from unexplained finite-size effects. We introduce and numerically study the charge transport capacity (CTC) -- a quantity that upper bounds the number of particles that can ever be transported across a central cut of a 1D lattice. For ergodic systems, the CTC is linear with the system size L, while we expect it to be O(1) for localized models. Surprisingly, in the interacting Anderson model for numerically accessible L, the disorder-averaged CTC is small, but grows with L at an increasing rate. Moreover, this growth rate appears to be independent of the disorder strength W at very large W. We find that, for these system sizes, this growth occurs because, as L increases, many-body resonances that transport more charge across the cut become more likely. Using a perturbative model for the weakly interacting regime, we provide an understanding of the microscopic origins of the growth of these charge transport resonances (CTRs). We find that the CTRs are sensitive to charge configurations over a spatial region whose size is set by the range of the resonance, not by W, and that numerics cannot access system sizes where their behavior will converge. However, this effective model is consistent with a regime of strong disorder where, for large L, resonances are exponentially suppressed in their size. Finally, we study measures of average charge transport and suggest that for strong enough disorder, average product states can only transfer O(1) charge. Our work suggests that the unsettled growth of short-ranged many-body resonances with L contributes to the numerical drift towards thermalization at numerically accessible system sizes, and provides an understanding of how they can remain controlled or eventually destabilize the MBL phase.
In a recent experiment, Anderson localization was observed directly for non-interacting
Bose-Einstein Condensate in one dimension. The experiment was performed using
quasiperiodic optical lattice. Their system can be described by the one-dimensional noninteracting
Aubry-Andr�e model. This model which simulates disorder with its quasiperiodic
potential shows transition from extended to localised states at a critical disorder
strength. The ground state gets localised around a single point for disorder larger than
the critical value. We find that while inverse participation ratio increases as we increase
disorder and rises to one in insulator phase, super
uid fraction decreases as we increase
disorder and goes to zero sharply in the insulator phase . When we introduce weak
repulsive interaction for Bose gas in the model at the mean field level, we see that the
ground-state shows multiple-site localization for certain disorder range, which lies above
the critical value. For given interaction strength, we find that the inverse participation
ratio increases slowly and super
uid fraction decreases slowly as we increase disorder.
Moreover, by representing our model as a classical two-dimensional nonlinear Hamiltonian
map, we distinguish between localised and extended states by studying the trajectories
in phase-space. Using this, we find out the localization length of eigenstates as a
function of disorder for non-interacting model. Further, we study the time-evolution of
a wave-packet in different regimes of disorder and see the effect of interaction on it.
Many-body localization in disordered systems in one spatial dimension is typically understood in terms of the existence of an extensive number of (quasi) local integrals of motion (LIOMs) which are thought to decay exponentially with distance and interact only weakly with one another. By contrast, little is known about the form of the integrals of motion in disorder-free systems which exhibit localization. Here, we explicitly compute the LIOMs for disorder-free localized systems, focusing on the case of a linearly increasing potential. We show that while in the absence of interactions, the LIOMs decay faster than exponentially, the addition of interactions leads to the formation of a slow-decaying plateau at short distances. We study how the localization properties of the LIOMs depend on the linear slope, finding that there is a significant finite-size dependence, and present evidence that adding a weak harmonic potential does not result in typical many-body localization phenomenology. By contrast, the addition of disorder has a qualitatively different effect, dramatically modifying the properties of the LIOMS.