Hierarchy of many-body invariants and quantized magnetization in anomalous Floquet insulators

Author response upon resubmission

We thank both referees for taking the time to review our manuscript, and for their constructive comments and suggestions. We also thank the referees for finding our paper interesting and worthy of publication in SciPost.

We apologize for the delay in our resubmission, which was due to significant modifications of our manuscript in response to both referees’ comments.

In particular, as suggested by both referees, the revised manuscript contains data from new simulations that demonstrate the existence of a regime characterized by a nonzero value of the higher-order invariant $\mu_2$. We have moreover included a more thorough discussion of such regimes (or “phases”) where the higher-order invariants are nonzero. In the revised manuscript, we term these phases “Correlation-Induced Anomalous Floquet Insulators” (CIAFIs). All simulations in the revised manuscript are moreover taken with a different model from the first version, where a single parameter (the interaction strength) controls whether the model is in the many-body localized AFI phase ($\mu_1 = 2$, and $\mu_\ell = 0$ for $\ell > 1$), or a CIAFI phase with $\mu_2 = -2$.

In response to both referee’s comments, we have moreover clarified how our classification (and in particular, the CIAFI phases) requires only $k$-particle localization rather than full many-body localization. Here a system is $k$-particle localized if all states holding up to a finite number of particles, $k$, are localized due to disorder. Full many-body localization is a special case of $k$-particle localization in the $k\to \infty$ limit (where particle density is allowed to remain finite in the thermodynamic limit). The existence of $k$-particle localization is well-established, and generically occurs in two dimensions.

Due to the increased length of the text, we finally found it appropriate to include a brief section summarizing our main results in the beginning of the paper (Sec I of the revised manuscript).

We believe these changes, along with the adjustments made in response to the referees’ specific comments have significantly improved the quality of the manuscript and clarified the nature of our results. We therefore again thank the referees for their constructive suggestions. We are confident our modifications have addressed the concerns raised by the referees and thus hope they will accept our manuscript for publication in SciPost.

Sincerely,

Frederik Nathan, Dmitry Abanin, Netanel Lindner, Erez Berg, and Mark S. Rudner

Response to report 2:

Comment 1: “The entire line of arguments in this paper rests on the assumption that MBL exists in 2d and that an effective description in terms of LIOMs exists. While this is a reasonable starting point, it seems at odds with recent arguments (avalanche theory) [Phys. Rev. B 95, 155129] which seem to preclude the stability of MBL in 2d. The authors comment very briefly that full MBL is not necessary, and rather partial MBL is sufficient, meaning that only some LIOMs exist (and I assume the other complementary operators become extended). I am not sure if such a description would be valid in the case of the avalanche picture, but maybe the authors can add a comment on this. Unfortunately, I am not sure the numerical evidence helps here to make the point, since the observation of MBL in small system with only very few particles does not suffice to argue for the existence of MBL in 2d. This being said, the entire discussion in the manuscript is consistent in itself (including the numerics) and I don't think this comment is an obstacle for publication.”

Our response: We thank the referee for raising this point, where we agree clarification is helpful. We would first like to emphasize that “partial localization” refers to the case where all Floquet eigenstates holding up to some finite number of particles, $k$, are localized. We acknowledge the meaning of “partial localization” is not a priori clear. In the revised manuscript, we therefore use the less ambiguous term “$k$-particle localization” to refer to the above situation. We have moreover added a subsection (Sec. II.b) where we define $k$-particle localization and discuss the nature of this phenomenon, including its relationship with many-body localization (MBL).

In the revised manuscript, we have further clarified that our classification (i.e., the quantization and topological protection of $\mu_1,\ldots \mu_k$) applies to any $k$-particle localized-system. The existence of $k$-particle localization is well-established, and it is expected to generically arise in the two-dimensional systems we consider (see e.g. Ref. [42] of the revised manuscript). Hence the conditions for our classification have been proven to hold for a wide range of situations. We subsequently discuss the implications of our classification for systems exhibiting full MBL, which is a special case of $k$-particle localization, if such exist (namely, that for such systems, magnetization density is quantized in fully occupied regions).

Comment 2: “In the same vein as my above comment, I am wondering about the role of the thermal edge states which the authors mention. If the edges really are thermal, would they not be a seed for a thermalization avalance? It seems to me that the edges could have a much stronger effect than the usually discussed random thermal inclusions in the system (which should be present here too). I understand that the authors leave the investigation of the edges to future work, but maybe it is possible to comment on this point. On the torus, investigated in this manuscript, this problem does not exist.”

Our response: As we explained above, our classification only requires $k$-particle localization, which we do not expect is destroyed by thermal edge states. For example, single-particle localization in the bulk is not destroyed by the chiral edge states of the AFAI. However, the edge states may play a role for thermalization in the bulk, in the case where the particle density is finite in the thermodynamic limit. We discuss the implications of edge states in the discussion. As in our previous work (Ref. [37]) we argue that such may provide an interesting platform for studying the consequences of thermalizing subregions for MBL.

Comment 3: “The authors argue that interacting systems exhibit nontrivial topological phases different from the noninteracting case which are visible in a nonzero $\mu_i, i > 1$ and provide an example in Sec III. Is there a reason for the choice of a different model in Sec. IV. which yields $\mu_1= 1$,, $\mu_i =0$ for $i>1$ in the numerical simulations? It would be nice to see numerical evidence for a case where higher invariants are nonzero.”

Our response: We thank the referee for this suggestion. In the revised manuscript, we have included data from simulations of a new model which realizes a regime where $\mu_2 = -2 $. The new data are presented in Fig. 1(cd), while the simulations are discussed both in the “Summary of results” (Sec. I), and in the numerics section (Sec. IV).

Comment 4: “The numerical method used for the calculation of the topological invariants $\mu_i$ is indirect and it seems that a very high precision for the total magnetization is required. In fact, even in the presented case, the numerical result does not allow to fix $\mu_4$ and $\mu_5$. Is there a more direct way to calculated the topological invariants?”

Our response: We are grateful to the referee for highlighting this point. In the revised manuscript, we do not seek to infer the values of the higher-order invariants from our data, but simply demonstrate that our results are consistent with the model realizing the AFI phase with $\mu_1 = 1$.

Response to report 1:

Comment 1:
“The existence of an MBL stabilized interacting AFI is assumed as a starting point. However, as the authors acknowledge themselves, the stability of MBL in the presence of a thermalizing edge can be problematic. They attempt to sidestep this issue by working with periodic boundary conditions. However, it wasn't clear to me whether this actually solves the problem or just "hides" it. For example, in the ideal model of the AFAI, the Floquet operator over one period looks trivial with periodic boundary conditions --- but the non-trivial signatures of the phase can be gleaned by examining the micromotion within a period. Likewise, even if the Floquet unitary over one period is localized with PBCs for the AFI (ignoring avalanches etc.), is it clear that the micromotion within the period will not see signatures of delocalization and affect the invariants that the authors are trying to construct?”

Our response: We wish to make clear that localization of the eigenstates of the Floquet operator, $U(T)$ implies that dynamics are localized at all times, regardless of the properties of the micromotion. Specifically, the stroboscopic dynamics of the system (i.e. the evolution at integer multiples of the driving period, $T$) are equivalent to those generated by the effective Hamiltonian $H_{\rm eff} = \frac{i}{T} \log[U(T)]$, whose eigenstates are identical to those of $U(T)$, and hence localized. As a result, a particle initially located at a given site in the lattice must remain within a distance $\sim\xi$ from this site after each driving period, where $\xi$ denotes the localization length in the system. At intermediate times, the finite Lieb-Robinson velocity of the system, $v$, means that the particle effectively cannot travel more than a distance $\sim vT/2$ away from this site. Hence the localization of the stroboscopic dynamics mean that the dynamics are also localized within the driving period. In the revised manuscript, we have added a clarifying footnote with this point (Ref. [46]).

Comment 2: “One of the central steps is equation (5) which says that the current through a cut is only sensitive to plaquettes near the boundary of the cut. However, this does not address resonances in any way, which will inevitably be present in a large enough system and presumably invalidate equation (10). These will certainly be rare, but to what extent do they interfere with the `topological' characterization?”

Our response We thank the referee for highlighting this issue. While resonances can in principle occur in disordered systems, in the presence of many-body- or $k$-particle localization, such resonances by definition do not play a role for the spread of time-averaged operators (in the Heisenberg picture) in the thermodynamic limit. If such resonances did exist, the spread of, e.g., particle density would mean that the system was not localized due to these resonances. While disorder realizations exist where resonances between far-away sites allow for particles and operators to travel over distances comparable to the system size (hence destroying localization), we expect the stability of localization implies that such realizations have measure zero in the thermodynamic limit. In the revised manuscript we have modified footnote [48] to make this point clearer.

Comment 3: “I found the discussion below equation (10) very confusing. The authors discuss slowly perturbing some region $R$, and assert that this does not change the expectation values for $\bar m_p$ in the infinite size limit. They rely here on the assumption that $\bar m_p$ only depends on the region near $p$, and hence is affected by an exponentially small amount if it is located a distance $\sim \mathcal{O}(L)$ from $R$. However, slowly changing $R$ should lead to all sorts of level crossings, resonances and global rearrangements in the system --- why should these leave $\bar m_p$ unchanged? “The authors do mention such resonances later in Section IID when examining the response to a changing magnetic field. Here they seem to imply that the resonances can simply be "gauged away", which I found quite surprising -- how does a gauge choice get around the resonances that will certainly be created? Further, this mechanism (even if true), only seems to apply to a changing magnetic field while the discussion below (10) is seemingly supposed to hold for all weak deformations that preserve MBL. Surely not all of these lead to resonances that can be "gauged away"? Since this is one of the central steps of the proof, the authors should clarify these concerns.”

Our response We thank the referee for raising this question, which indeed touches a subtle point and led us to improve our line of arguments. After consideration of the referee’s comment, we found a simpler line of arguments to establish the topological invariance of $\Tr_{\ell}\bar m_p$.

To recap, the goal of the paragraph below Eq. (10) [Eq. (6) in the new version] is to show that $\Tr_{\ell}\bar m_p$ remains invariant under a finite, local change of parameters in some subregion $R$ far away from $p$ as long as $\ell$-particle localization is preserved. (Note that this implicitly assumes that no accidental resonances between far-separated sites are induced by the specific perturbation; see footnote 43 of the new manuscript.) To see why this implies topological invariance, recall that $\Tr_{\ell}\bar m_p$ must take the same value for all plaquettes in the system; hence even for a plaquette $q$ within $R$, the value of $\Tr_{\ell}\bar m_q$ remains unaffected by the perturbation. As a result, the universal value of $\Tr_{\ell}\bar m_p$ is invariant under any finite, local perturbation that preserves $\ell$-particle localization, and thus (in this sense) is a topological invariant.

In the previous version of the manuscript we sought to prove that $\Tr_{\ell}\bar m_p$ is invariant under a finite perturbation in $R$ by considering the behavior of $\Tr_{\ell}\bar m_p$ under a smooth interpolation between the unperturbed and perturbed situations. However, after considering the referee’s comment, we realized it was enough to consider the direct response of $\Tr_{\ell}\bar m_p$ to a finite perturbation in the region $R$, without having to introduce the smooth interpolation (which indeed results in resonances of the many-body spectrum). The fact that $\bar m_p$ only has support within a finite region of the lattice (up to an exponentially suppressed correction), along with the characteristic properties of $\ell$-particle localization, means that $\bar m_p$ is unaffected by any finite perturbation that is applied sufficiently far away from $p$ (up to an exponentially suppressed correction). As a result, following the arguments in the above paragraph, $\Tr_{\ell}\bar m_p$ is a topological invariant of the system. In the revised manuscript, we have included this new, more direct line of arguments below Eq. (6).

We thank again the referee for highlighting this point, which led to a significant improvement of our line of arguments.

We also wish to clarify that we do not claim that resonances induced by the perturbing magnetic field $B_0$ can be gauged away. Rather, in Appendix B, we demonstrate that the probability of $B_0$ inducing a resonance anywhere in the quasienergy spectrum approaches zero in the thermodynamic limit. Hence in the thermodynamic limit, $B_0$ does not induce any resonances, except for a measure zero set of disorder realizations. The gauge transformation we mention is only required to relate the Floquet eigenstates of the one- and zero-flux systems to each other. Specifically, we show that each Floquet eigenstate of the zero-flux system is approximately identical to a Floquet eigenstate of the one-flux system within an appropriately chosen (state-dependent) gauge for the one-flux system. In the revised manuscript, we have changed the part of the main text referring to Appendix B, in order to further clarify the derivation. Comment 4: “The authors make the point that the (new) higher order invariants only exist in models that cannot be continuously deformed to an AFAI by tuning down interactions. If this is the case, what is the basis for the assumption of MBL in such models? Are there any known examples of MBL systems --- especially those with a LIOM description, which the authors rely on --- that cannot be deformed to a non-interacting localized system? I think this is a very strong assumption that is largely left unjustified - and the central result in this paper (the existence of higher order invariants) turns on this.”

Our response: We would like to clarify that the invariants $\mu_1,\ldots \mu_k$ only require localization of all Floquet eigenstates holding up to $k$ particles for their quantization and topological protection. In the revised manuscript we have further clarified this point in the introduction, the newly added summary of results (Sec. I), and the section identifying the invariants (Sec. III). In the previous version of the manuscript, the situation above (where all states holding up to some finite number of particles, $k$, are localized) is termed “partial localization.” In the revised manuscript, we have changed the term to $k$-particle localization, whose meaning we believe is clearer, and have added a subsection discussing this notion of localization (Sec. II.b).

We do not presently know whether the higher-order invariants $\mu_2,\ldots$ are compatible with MBL, though we suspect it is not the case. However, we currently we do not know of a definite proof for their incompatibility. A recently posted preprint on the topological classification of fermionic MBL systems does not contain the higher-order invariants, indicating that they are incompatible with MBL [Zhang and Levin, Phys. Rev. B 103, 064302 (2021)]. However, technically there still is a possibility for them being compatible with MBL: to our knowledge it has not yet been proven that all MBL phases should be deformable into some noninteracting phase without going through a delocalization transition. In Sec. V of the revised manuscript we further discuss the open question of the compatibility of the higher-order invariants with MBL, including considerations such as the above. Crucially, our main results for $k$-particle localized systems are independent of the answer to this question.

Comment 5: “It is also interesting that the numerical example presented is for the AFI that can be connected to the AFAI and not for the new "model" with higher-order invariants. “ Our response: We thank the referee for this suggestion, which was also given in report 2. In the revised manuscript, we have included data from simulations of a model where $\mu_2$ takes value $-2$. We believe these newly added simulations suggested by the referee significantly improves the quality of the manuscript.

Comment 6: “The authors do say at the end of Section III that the "new" model may not support MBL (and hence all the phenomenology discussed) and leave this an interesting open question. But with this left as an open question, I'm not sure what exactly the take home message of the paper is. The title talks about a `hierarchy of invariants', but the conclusion that these actually exist in any model needs the new class of models being discussed to (i) be MBL (ii) with a LIOM description and (iii) without a non-interacting AFAI limit. Without justifying (i)-(iii), the conclusion seems to me to not be warranted (this is even ignoring the issue of resonances raised in the point above).”

Our response. We acknowledge that the structure of our results and the conditions for their validity were unclear in the first version of the manuscript. To be clear, in our paper we obtain two main results:

1. Periodically driven systems of fermions where all $k$-particle states are localized due to disorder (see above for definition), are characterized by $k$ integer-valued invariants $\mu_1, \ldots \mu_k$, with $\mu_\ell$ characterizing quantized contributions to the magnetization density from $\ell$-particle correlations.

2. With full MBL (which is the special case of $k$-particle localization in the limit $k \to \infty$ where particle density remains finite in the thermodynamic limit), magnetization density is quantized in regions where all states are occupied. Hence, we confirm that anomalous Floquet insulators, which are the MBL extensions for the AFAI, are characterized by quantized bulk magnetizations: for these models, only $\mu_1$ is nonzero, and the magnetization density in fully occupied regions is given by $\mu_1/T$.

We acknowledge that this structure of results and the conditions under which they are valid were not sufficiently clear in the previous manuscript. In the revised manuscript, we have therefore clarified these two main results and their conditions for validity. In particular, we discuss these points in the introduction, and in the newly added summary of main results (Sec. I), as well as in the section where we characterize the topology of the system (Sec. III). At these points in the text, we make it clear which of our results rely on $k$-particle localization, and which of our results require many-body localization to be valid, as outlined above.

* Clarified the structure of the main results (namely that we uncover two main results: the quantization of magnetization in the AFI phase, and the existence the CIAFI phases).
* Clarified the conditions for the main results (in particular, we added a subsection defining $k$-particle localization, and emphasized that the CIAFI phases require only $k$-particle localization).
* Added a section summarizing the main results of the paper (Sec I) .
* Replaced the numerical simulations with simulations of a new family of models that realizes both the AFI phase and the CIAFI phases.
* Included numerical simulations demonstrating the existence of a CIAFI phase (characterized by a nonzero value of $\mu_2$) (see Fig. 1).
* Made minor modifications throughout manuscript to improve readability and clarity.