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# Interaction instability of localization in quasiperiodic systems

Edited by David A. Huse, Princeton University, Princeton, NJ, and approved March 13, 2018 (received for review January 11, 2018)

## Significance

Understanding how small imperfections affect a system’s dynamics is one of the central questions of theoretical physics—namely, do properties change in a smooth way, such that small perturbation leads to small changes, or do they change discontinuously? Localization in disordered many-particle quantum systems has been shown to be stable to interactions. On a single-particle level, one can also achieve localization with a quasiperiodic potential—a system without disorder but with rich properties. It is believed that localization in disordered as well as in quasiperiodic potentials behaves in essentially the same way, in particular, that both are stable against interactions. We show that this is not so. A quasiperiodic localized system discontinuously changes from localization to diffusion upon introducing interactions.

## Abstract

Integrable models form pillars of theoretical physics because they allow for full analytical understanding. Despite being rare, many realistic systems can be described by models that are close to integrable. Therefore, an important question is how small perturbations influence the behavior of solvable models. This is particularly true for many-body interacting quantum systems where no general theorems about their stability are known. Here, we show that no such theorem can exist by providing an explicit example of a one-dimensional many-body system in a quasiperiodic potential whose transport properties discontinuously change from localization to diffusion upon switching on interaction. This demonstrates an inherent instability of a possible many-body localization in a quasiperiodic potential at small interactions. We also show how the transport properties can be strongly modified by engineering potential at only a few lattice sites.

Science tries to describe nature in the simplest possible terms. Models that can be solved exactly—for instance, integrable systems—play a special role. Although generic systems are not integrable, physicists have developed a plethora of methods that successfully describe phenomena in terms of slightly perturbed integrable models. An important question that we address is how stable integrable systems are to perturbations.

One of the finest results in classical mechanics is the Kolmogorov–Arnold–Moser (KAM) theorem (1). It shows that in a finite-dimensional classical system, integrability breaks smoothly—that is, upon small perturbation, most orbits retain their integrable quasiperiodic character and only few become irregular. Even though irregular orbits can form a connected ergodic component, the so-called Arnold web (2), their measure goes to zero as the perturbation strength decreases. Another important question is that of quantum localization. For noninteracting electrons in a periodic potential, one can use the Bloch (Floquet) theorem to show that transport is ballistic (zero resistance), corresponding to extended eigenstates. It was believed that such systems would become diffusive in the presence of disorder, so it came as quite a surprise when Anderson showed (3) that such ballistic transport is completely unstable against disorder; disorder can cause an immediate change from an ideal conductor to an ideal insulator—the so-called Anderson localization. The KAM theorem of classical mechanics and quantum localization both use the same “KAM techniques” in their proofs (4, 5)—namely, dealing with a small-denominator problem of perturbation series.

For systems with many particles—i.e., in the thermodynamic limit (TDL) —things are less clear. A prevailing opinion is that general integrable systems are not stable, except perhaps localized ones, for which one can in some cases show that small nonlinearity—e.g., preserves some localized orbits (6) or causes very slow transport (7). For quantum systems, even finite dimensional ones, no “quantum KAM theorem” is known, and due to an inherent “discreteness” of quantum mechanics, even the very definition of integrability is not so clear-cut (8). While traditional criteria from quantum chaos (1)—for instance, the distribution of nearest eigenenergy spacings might suggest that the change is not smooth (9)—one has to note that these smallest energy scales in the TDL describe exponentially large (unphysical) times. A more relevant question is that of dynamics on a physically accessible time scale.

As a dynamical criterion by which we judge the smoothness, we study transport (10) in a one-dimensional system with a quasiperiodic potential (11). Transport of interacting many-particle systems is of obvious high interest and has been discussed ever since the discovery of single-particle localization (3). Due to the difficulty, results were few (12, 13), often focusing on two-particle (14, 15) or single-particle problems with nonlinearity (that is supposed to describe interaction in an effective mean-field way) (16⇓–18). For the two-particle case, results for small interactions and random (14, 19) as well as for (noncritical) quasiperiodic potential (15) show that localization is preserved [at larger interactions two particles, though, can delocalize (15)], while nonlinearities induce subdiffusive (or slower) transport. More recently, many-body localization (MBL)—that is, localization of many particles in the presence of interactions—has gained increased attention; for review, see refs. 20 and 21. MBL has been proven for small interactions in a random system (22). Furthermore, many experiments probing noninteracting localization in fact use a quasiperiodic disorder (23, 24), so it is important to understand how quasiperiodic disorder modifies localization properties. An important property of deterministic quasiperiodic potential is that there are no (rare) local configurations of small or large potential that could influence dynamical properties (25). Quasiperiodic systems have been probed experimentally (26), including the first demonstration of MBL (27) [of certain degrees of freedom (28)]. Experiments with quasiperiodic systems are by now well controlled (29, 30) and can even be performed in 2D (31). A number of recent theoretical studies discussed MBL (32⇓⇓⇓⇓⇓⇓⇓⇓–41) or localization (42⇓–44) in the cosine quasiperiodic potential. A common conclusion from all these few-particle as well as MBL studies seems to be that systems with quasiperiodic and random potential behave rather similarly [apart from a possibly different universality class of the transition point (37)]. In particular, a noninteracting quasiperiodic localized system will behave smoothly as one adds interactions. This would be in line with mathematical reasoning that a point spectrum (localization) is stable. We show that the situation is, in fact, the opposite.

By studying transport properties of a one-dimensional interacting system in the presence of a quasiperiodic potential at infinite (high) temperature and half-filling, we clearly demonstrate that the noninteracting localization discontinuously breaks down to diffusion for infinitesimal interactions (Fig. 1). This surprising fragility of localization in a quasiperiodic potential seems to be due to long-range correlations (and resonances) in the single-particle spectrum and must be contrasted with a continuous behavior for an uncorrelated potential. The result has several strong implications: (*i*) It shows that there cannot exist a KAM-like quantum smoothness theorem that would hold in general for localized systems—depending on the disorder type, one can have smooth or nonsmooth behavior; (*ii*) a possible MBL in quasiperiodic systems at small interactions is likely always unstable, again in contrast to the Anderson model which is stable (22); and (*iii*) by manipulating potential only at a few sites, we can significantly affect the transport, opening the door to transport engineering.

## The Model

We are going to study magnetization transport in the interacting Aubry–André–Harper (AAH) model (45, 46),**1** is equivalent to spinless fermions with the interaction strength being given by U, while magnetization transport is equal to the particle transport. For irrational β [we use

Note that it is expected that a ballistic (integrable) system immediately breaks down to diffusion upon breaking its integrability. This is suggested by Fermi’s golden rule: Perturbation matrix elements are nonzero for extended states, leading to nonzero scattering and diffusion, as well as by explicit results—for instance, numerical studies of a gapless anisotropic Heisenberg chain in the presence of an extra coupling (48) or staggered (49) or random fields (50) all show diffusion. Similarly, exact results for free particles in the presence of a dissipative spin-conserving dephasing (51) are also diffusive [there are rigorous results showing greater sensitivity of extended states, e.g., the von Neumann theorem about instability of continuous spectra (52)]. In line with these expectations, we also observe that the AAH model becomes diffusive for

Transport is studied by using two different settings (Fig. 1). One is a true nonequilibrium steady-state (NESS) situation, akin to what an experimentalist would do to measure conductivity. Boundary spins at each end of the chain are coupled to an effective bath that tries to induce an imbalance of magnetization between the left and the right end, causing a nonzero magnetization current. In technical terms, we use the Lindblad equation (53, 54), whose solution gives the NESS density matrix *SI Appendix*) that asymptotically induces small magnetization

The second method we use is a unitary evolution of an inhomogeneous initial state, studying how the variance increases with time. For reasons of numerical efficiency, a good choice (55) seems to be a weakly polarized domain wall density matrix

## Diffusion

We first check the transport type for small interaction U and *A* that, provided the system is large enough, one obtains diffusive *A*) up to increasingly larger L, when finally the asymptotic diffusion emerges. Similar results are also obtained for the unitary spreading shown in Fig. 2*B*, where increasingly longer times are needed with decreasing U. We have an interesting situation where transport is diffusive at high energies, while it is insulating at zero temperature (34, 35).

To stress the surprising nature of a discontinuous change from localization to diffusion for small U, we compare results to those for random potential at each site obtained by taking *SI Appendix*). To this end we also show the Anderson case at *B*) for the unitary evolution at relatively large *A*) for the Lindblad setting (*A*), while for the AAH model at *A*). Sample-to-sample variation is considerably smaller in the AAH model (essentially no averaging over ϕ is needed) than in the Anderson model (Fig. 3*A*) (37). Although both noninteracting models share at first sight similar single-particle localization, they behave completely differently in the presence of small interactions. We note that the instability of the AAH model has been noted before—namely, one can show that an infinitesimal modification of

Confirming diffusion for small U, we study in detail the dependence of the diffusion constant D on U and λ. We find that D goes to zero as one decreases U for *SI Appendix*). In both cases the dependence follows (Fig. 4*A*)*B*). At fixed interaction, *C*. At small U and large λ, it gets exponentially small; if expressed in terms of a noninteracting single-particle localization length ξ (*SI Appendix*), the dependence would be *SI Appendix*). Our results are thus not compatible with the MBL transition point smoothly connecting to *C* that the curves for *A*), and equal

## Single-Particle Correlations

In the following, we give an explanation why a quasiperiodic potential is so much different from a random one. Taking *SI Appendix*) for the AAH and the Anderson model. Spectral index k is ordered according to the eigenstate’s center-of-mass location *E* and *F*. We can see that in the AAH model these distances are much smaller than in the Anderson case (and have a sharp cutoff with no long tails) and are greatly enhanced at the Fibonacci numbers *A*, where the main resonances are seen at

## Transport Engineering

Finally, we demonstrate that it is indeed the spatial “Fibonacci weave” correlations in *A*) due to the quasiperiodic potential that is crucial for the diffusion. To break these correlations, we take the AAH model with a modified phase at some sites. Namely, at sites j where *SI Appendix*), and the transport immediately changes from diffusive to subdiffusive; Fig. 6. Switching on interactions and modifying the potential at a few sites therefore enables one to alter the global transport type. It would be interesting to see how such changes influence transport in other free models with a correlated potential (60). This suggests a mindset change from that of studying transport of particular models to one where one instead asks how can one engineer a specific transport type? For free models, there are results that allow one to construct a model with any anomalous transport (61⇓–63).

## Conclusions

We have shown that exponential localization in a quasiperiodic potential in the presence of small interactions at half-filling and high energy breaks down discontinuously to diffusion. This inherent instability, which is markedly different from the smooth behavior seen for a random potential, appears to be due to long-range correlations in the single-particle spectrum. They have implications for possible proofs of MBL in quasiperiodic systems, while high sensitivity of global transport to the potential at only a few sites opens the way to transport engineering, where one could significantly modify transport by small targeted changes to the potential. Our findings should be within reach of present-day experiments.

## Acknowledgments

We thank T. Prosen and V. K. Varma for comments on the manuscript. This work is supported by Slovenian Research Agency Grants J1-7279 and P1-0044; and by European Research Council Advanced Grant 694544–OMNES.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: znidaric{at}fmf.uni-lj.si.

Author contributions: M.Ž. designed research; M.Ž. and M.L. performed research; M.Ž. and M.L. analyzed data; and M.Ž. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1800589115/-/DCSupplemental.

Published under the PNAS license.

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