The anomalous Floquet-Anderson insulator as a non-adiabatic quantized charge pump
Periodically driven quantum systems provide a novel and versatile platform for realizing topological phenomena. Among these are analogs of topological insulators and superconductors, attainable in static systems; however, some of these phenomena are unique to the periodically driven case. Here, we show that disordered, periodically driven systems admit an “anomalous” two dimensional phase, whose quasi-energy spectrum consists of chiral edge modes that coexist with a fully localized bulk - an impossibility for static Hamiltonians. This unique situation serves as the basis for a new topologically-protected non-equilibrium transport phenomenon: quantized non-adiabatic charge pumping. We identify the bulk topological invariant that characterizes the new phase (which we call the “anomalous Floquet Anderson Insulator”, or AFAI). We provide explicit models which constitute a proof of principle for the existence of the new phase. Finally, we present evidence that the disorder-driven transition from the AFAI to a trivial, fully localized phase is in the same universality class as the quantum Hall plateau transition.
Time-dependent driving opens many new routes for realizing and studying topological phenomena in many-body quantum systems. Recently, an intense wave of activity has developed around exploring the possibilities of using periodic driving to realize “Floquet topological insulators”, i.e., driven system analogues of topological insulators Oka and Aoki (2009); Inoue and Tanaka (2010); Kitagawa et al. (2010); Lindner et al. (2011, 2013); Gu et al. (2011); Kitagawa et al. (2011); Delplace et al. (2013); Katan and Podolsky (2013); Titum et al. (2015); Usaj et al. (2014); Foa Torres et al. (2014); D’Alessio and Rigol (2014); Dehghani et al. (2014a, b); Bilitewski and Cooper (2014); Sentef et al. (2015); Seetharam et al. (2015); Iadecola et al. (2015) in a variety of solid state Wang et al. (2013), atomic, and optical contexts Jotzu et al. (2014); Rechtsman et al. (2013). Beyond these analogies, driven systems may also host their own unique types of robust topological phenomena, which have no analogues in non-driven systems Kitagawa et al. (2010); Jiang et al. (2011); Rudner et al. (2013); Kundu and Seradjeh (2013); Carpentier et al. (2015); Asboth et al. (2014). The latter will be at the heart of the present work.
In static (time-independent) two dimensional systems, the appearance of chiral edge states is intimately tied to the topological structure of the bulk Bloch bands, captured by the so-called Chern number Thouless et al. (1982). This well-known bulk-edge correspondence breaks down in periodically driven crystalline systems, where chiral edge states can exist even if the Chern numbers of all the bulk bands are zero Kitagawa et al. (2010); Rudner et al. (2013). Such anomalous edge states are captured instead by a topological invariant that characterizes the time evolution operator of the bulk wave functions Rudner et al. (2013). A system exhibiting this anomalous behavior has been recently realized using microwave photonic networks Hu et al. (2015).
This unique topological phenomenon opens new possibilities, which are inaccessible in static systems. For example, it is well-known that Bloch bands with non-zero Chern numbers cannot be spanned by a complete basis of localized Wannier functions Thouless (1984); Thonhauser and Vanderbilt (2006). Correspondingly, when quenched disorder is introduced, not all the states in a Chern band can be localized; delocalized states must exist at least at one value of energy in the band Halperin (1982). Intuitively, this can be understood from the fact that the chiral edge states in the bulk band cannot “terminate” without hybridizing with a delocalized bulk state. In contrast, a periodically driven system can exhibit chiral edge states even when all the Chern numbers are zero. Moreover, due to the periodicity of quasienergy, it is in principle possible for a chiral edge state to wrap around the entire quasi-energy zone without terminating at a delocalized bulk state. This leads us to hypothesize that, in a disordered periodically-driven system, robust chiral edge states may coexist with an entirely localized bulk. In such a system, which we term an anomalous Floquet-Anderson insulator (AFAI), the chiral edge states form a uni-directional one dimensional system, whose dynamics is decoupled from the bulk at all quasienergies. This situation defies the standard intution from strictly one-dimensional systems that must have an equal number of right and left moving modes. But can an AFAI state really exist? And if so, what are its physical consequences?
In this work, we explore the AFAI phase in periodically driven, two dimensional disordered systems. We construct explicit models that demonstrate its existence, and discuss its topological characterization and its physical properties. Strikingly, the AFAI hosts a unique non-equilbrium topological transport phenomenon: quantized charge pumping in a non-adiabatic setting. Essentially, if all the states in the vicinity of the edge are occupied by fermions (to a distance of several times the bulk localization length), the uni-directional edge states carry a current whose long-time average is quantized in units of one particle per driving period. Importantly, disorder is essential for the quantization of pumping in the AFAI; absent the disorder, generically there is no quantization due to the presence of delocalized states in the bulk.
Quantized pumping is well-known from the work of Thouless on adiabatically driven one-dimensional systems. however, unlike in the Thouless pump, in the AFAI, the driving frequency is not required to be small in order to observe the quantization of the current. The reason the current at the edge of an AFAI can remain quantized even when the adiabatic condition is violated is that the two counter-propagating edge modes that carry the current are spatially separated Thouless (1983); hence, they cannot backscatter into each other even if the driving frequency is not small.
This paper is organized as follows. In Sec. II we introduce the defining properties of the AFAI; we discuss the topological invariant characterizing the AFAI, and show that the AFAI exhibits edge modes at every quasi-energy. In Sec. III we show how the edge mode structure leads to quantized charge pumping. We then demonstrate, in Sec. IV, the appearance and robustness of an AFAI in a simple, tractable model. In Sec. V we conduct a numerical study of a wider class of models exhibiting the AFAI phase. We numerically demonstrate the properties discussed in sections II and III. At strong disorder, we find a topological transition between the AFAI and a “trivial” Floquet insulator where all states are localized (including at the edges); we speculate that the transition is in the same universality class as the quantum Hall plateau transition, and corroborate this using our numerical results.
Ii The AFAI: Topological Invariants and Edge States
We begin by defining the AFAI, and introducing the topological invariant which characterizes it. The defining characteristic of the AFAI phase is the peculiar relationship between its bulk and edge mode spectra: in the AFAI phase all bulk Floquet states are localized, yet the system still hosts topologically-protected chiral modes along its edges.
We consider a two-dimensional system of non-interacting particles with a time-periodic Hamiltonian, , where is the driving period. No spatial translational symmetry is assumed. The interesting aspects of the AFAI phase are revealed by comparing the Floquet operators for toroidal and cylindrical geometries. In an AFAI phase, all the eigenstates of on a torus are localized. However, as we will show, in a cylindrical geometry there are eigenstates which are localized at the boundaries of the cylinder, but delocalized along the direction of the boundary, at every quasi energy .
The topological invariant which describes the AFAI is a generalization of the “winding number” introduced in Ref. Rudner et al. (2013). As a first step in constructing the topological invariant, we define an associated, time-periodic evolution operator for the system on a torus:
with . Note that, by construction, . The explicit dependence on in the above definitions comes from the necessary choice of a branch cut for ; we use a definition such that if and if . As an additional ingredient, we also consider a family of time-dependent Hamiltonians and the associated evolution operators , in which constant (time independent) fluxes are threaded through the torus foo ().
With these definitions at hand, we can define the “winding number”
The winding number is an integer, which can in principle depend on the quasi-energy . Note that in order for to be well defined, the quasi-energy has to remain in a spectral gap of for every value of the threaded fluxes (otherwise, the operator is discontinuous as a function of ). We argue that for a large enough system, almost all values of satisfy this requirement. This is because, upon changing fluxes and , the quasi-energies of the localized bulk states only change by an amount proportional to , where is the localization length and is the linear system size. In contrast, the average level spacing is proportional to .
Next, we show that if all the eigenstates of are localized, then the invariant is in fact independent of . This follows from the relation between the winding number and the Chern numbers characterizing the eigenstates of Rudner et al. (2013),
In the above, is the total Chern number of the eigenstates with quasi-energies between and :
where is a projector onto the eigenstates of with quasi-energies between and . If all the bulk eigenstates are localized, not (). Therefore in this case by Eq. (3), for every pair of quasi-energies. Then, we can drop the subscript , and refer to the winding number simply as .
We thus define the AFAI as a time-periodic, disordered system in which (1) all the bulk Floquet eigenstates are localized, (2) the quasi-energy independent winding number is non-zero. Below, we argue that the boundaries of the AFAI necessarily support chiral edge states at every value of the quasi-energy.
Having defined the AFAI phase, we briefly mention how it can be reached. A good starting point for accessing the AFAI is a clean (translationally invariant) Floquet-Bloch system, for which all the Chern numbers of vanish, but with for any within one of the gaps in the quasi-energy spectrum. A schematic quasi-energy spectrum of such a system in a cylindrical geometry is shown in Fig. 2(b). We then add a static, spatially disordered potential to . We will argue below that all the bulk states are generically localized even for arbitrarily weak disorder. The crucial point is that the winding number need not be zero, even if all the bulk states are localized. A specific solvable model, which serves as a proof of principle for the existence of the AFAI phase, is given in Sec. IV.
We finish this section with a discussion of the edge structure of the AFAI. In the clean limit, there are chiral edge states in any bulk quasi-energy gap with a non-zero winding number Rudner et al. (2013). Clearly, these edge states cannot localize when disorder is added. Moreover, intuitively, if all the bulk states are localized, the chiral edge states must persist even within the bulk bands. To see this, consider a system in a cylindrical geometry. Upon inserting a flux quantum through the hole of the cylinder, the chiral edge states exhibit a non-trivial “spectral flow”: i.e., even though the spectrum as a whole is periodic as a function of flux, every state evolves into the next state in the spectrum [Fig. 2(d)]. The spectral flow cannot terminate in the bulk bands. Since all the bulk states are localized, they are insensitive to the flux, and hence there must exist a delocalized, chiral edge state at every quasi-energy within the bulk bands that “carry” the spectral flow.
To make this argument more precise, we define a topological invariant that directly characterizes the spectral flow of the edge states. This topological invariant turns out to be equal to the bulk invariant ; we will show this in detail in Sec. III and Appendix C, where we demonstrate that both invariants are related to quantized charge pumping along the edge.
To construct the edge topological invariant, we consider a cylinder that extends from to , with a flux inserted through the hole of the cylinder. The evolution operator of the system on the cylinder is denoted by . We now isolate the topological features of the edge states by deforming the evolution operator in the regions away from the edges, such that the evolution in the bulk takes a simple universal form, while the evolution near the edges is unaffected. In particular, we âflattenâ the bulk evolution such that all Floquet eigenstates localized sufficiently far (at least a distance ) from the edges have quasienergy . The resulting evolution operator interpolates smoothly between in the vicinity of the edge and of Eq. (1) in the bulk (an explicit formulation of the deformation procedure appears in Appendix A). The deformed evolution operator takes a block-diagonal form:
where in the above, the sub-blocks and correspond to sites with and , respectively; the unity block acts on sites with . The precise value of is not important, as long as it is much larger than the bulk localization length of the original evolution operator, . The integer-valued “edge winding number” is defined as
where the sum in the second line runs over all the eigenstates of , and are their corresponding eigenvalues. The edge winding number (6) counts how many times the spectrum of “wraps” around the quasi-energy zone, , as varies from to . A schematic example of a spectrum with a non-zero winding number is shown in Fig. 2(d). Note that the total winding number of the system, , must vanish Kitagawa et al. (2010). Hence, the winding numbers of and must sum to zero.
A non-zero necessarily implies that there are delocalized states along the edge; if all states were localized, their quasi-energies would be almost insensitive to , and hence would be zero. Note also that, since in the AFAI all the bulk states are localized, changing would not change ; this amounts to adding a few localized states to the spectrum of , and cannot change its winding number.
Iii Quantized charge pumping
We now discuss the physical implications of the AFAI phase. Consider an AFAI placed in a cylindrical geometry, as in Fig 2(c). Fermions are loaded into the system such that in the initial state all the lattice sites are filled up to a distance of from one edge of the cylinder, and all the other sites are empty. Below we show that in the thermodynamic limit, the current across a vertical cut through the cylinder, averaged over many driving periods, is equal to , Eq. (6), divided by the driving period . The exact form in which we terminate the filled region will not matter, as long as all the sites near one edge are filled, and all the sites near the other edge are empty. The system thus serves as a quantized charge pump, but unlike the quantized pump introduced by Thouless Thouless (1983), there is no requirement for adiabaticity.
In Appendix C.2 we furthermore show by a direct evaluation that the long-time average of the pumped charge per driving period is also equal to , the bulk invariant. In particular, this implies that .
To set up the calculation of the charge pumping in the AFAI, we choose coordinates such that is the direction along the edges of the cylinder, and is the transverse direction. We denote the initial many-body (Slater determinant) state, in which all sites up to a distance of from the edge are filled, by . Then, the charge pumped across the line between and is given by
Here, is the flux through the cylinder and is the corresponding Hamiltonian. For Eq. (7), we use a gauge such that on the lattice, every hopping matrix element that crosses the line has a phase of .
The initial state clearly does not return to itself after a single driving period. Therefore, we cannot expect that the pumped current to be identical between different periods along the evolution, nor can we expect it to be exactly quantized. However, we find that the average pumped charge over periods approaches a quantized value in the limit of a large number of periods, where the correction to the quantized value decays as :
Here, (where , are the bulk and edge topological invariants, respectively, defined in Sec. II). Note that is independent of , i.e., the charge pumped across any line parallel to the axis leads to the same .
In order to compute the charge pumped per period, it is useful to express as a superposition of the Floquet eigenstates. As we show in Appendix C.1, when averaging the pumped charge over periods, the contribution of the off-diagonal terms between different Floquet eigenstates decays at least as fast as . The diagonal terms yield a contribution that depends on the evolution over a single period, giving
In the above, are the single particle Floquet states, which evolve in time as (where is periodic in time), and are the Floquet state occupation numbers in the initial state, , where is the creation operator corresponding to . [Note that if fermions were initialized in the Floquet eigenstates, such that or , we would obtain , without the correction terms in Eq. (8)].
Straightforward manipulations yield . At this point, the average current per period depends on . In the thermodynamic limit, we expect this dependence to disappear. As in the case of the quantization of the Hall conductance Avron and Seiler (1985), we average over Spi (). We therefore get
Equation (10) relates the average current in a period to the spectral flow of the Floquet spectrum as the flux is threaded. It is reminiscent of the expression for the edge topological invariant, , Eq. (6), defined in terms of the “deformed” evolution operator . Below, we give a heuristic argument that indeed , up to corrections that are exponentially small in . A more rigorous (but technically cumbersome) derivation of the relation between the pumped charge and the bulk invariant is presented in Appendix C. Numerical evidence for the quantization of the pumped charge is shown in Sec. V.
Our strategy in analyzing the pumped charge is to deform the evolution operator into the “ideal” form, of Eq. (5), for which the pumped charge is exactly quantized, and to put bounds on the correction to the pumped charge due to the deformation. We define the deformation process according to Appendix A, with , the width of the strip beyond which the quasi-energy spectrum becomes flat, chosen such that . Clearly, for the deformed evolution operator, for every eigenstate of . Therefore, the deformed evolution operator has an exactly quantized pumped charge, equal to .
Now, consider the pumped charge of the original (undeformed) evolution. We can roughly divide the Floquet states that contribute to Eq. (10) into three categories:
States that are localized far from occupied region, . For these states, is exponentially small, and hence their contribution to is negligible.
States that are localized near the edge, . These states have . Their wavefunctions and quasi-energies, and hence their contribution to , are essentially unaffected by the deformation process.
States that are localized near the boundary between occupied and unoccupied sites, . For such states, is neither close to nor to ; however, these states are localized in the direction (as are all the bulk states in the AFAI). Therefore, of these states is exponentially small, and they contribute negligibly to .
As varies, there are avoided crossings in the spectrum, in which the character of the eigenstates changes. E.g., an eigenstate localized around may undergo an avoided crossing with an eigenstate localized around . When is tuned to such degeneracy points, the two eigenstates hybridize strongly, and do not fall into either of the categories discussed above. Such resonances affect both and the occupations of the resonant states. However, since the eigenstates that cross are localized in distant spatial areas, the matrix element that couples them is exponentially small. Therefore significant hybridization requires their energies to be tuned into resonance with exponential accuracy, limiting the regions of deviation to exponentially small ranges of , of order . The number of such resonances increases only polynomially with the size of the system, and therefore for and , their effect on is exponentially small.
We conclude that, in the thermodynamic limit, all the contributions to in Eq. (10) that are not exponentially suppressed are also exponentially insensitive to the deformation process. Therefore, .
Iv Model for an anomalous Floquet-Anderson phase
In this section, we study a simple model which allows us to explicitly demonstrate the existence and robustness of the AFAI phase. We start from a solvable model introduced in Ref. Rudner et al. (2013), which exhibits perfectly flat bulk Floquet bands, and hosts chiral edge modes at its boundaries. Adding a specific kind of disorder to this model results in localization of all the bulk states, while preserving the edge states; the system is thus in the AFAI phase. We then argue that this phase is robust to generic small perturbations (i.e., the bulk states remain localized, and the chiral edge states persist).
We consider a system on a square lattice with a periodic, piecewise-constant Hamiltonian of the form: , for , . The square lattice is divided into two sublattices, and (shown as filled and empty circles in Fig. 3, respectively). During each of the first four segments of the driving, , hopping matrix elements of strength between the and sublattices are turned on and off in a cyclic, clockwise fashion, as shown in Fig. 3: during segment , , , or , each site in the A sublattice is connected by hopping to the site above, to the right, below, or to the left of it, respectively. In the fifth segment of the period, all the hoppings are set to zero, and an on-site potential is applied on the and sublattice sites, respectively.
We choose the hopping strength such that . For this value of , during each hopping segment of the driving period a particle that starts on one of the sites hops to the neighboring site with unit probability. The on-site potential, applied only while all hopping matrix elements are turned off, is chosen to be . With this time-dependent Hamiltonian, it is easy to find the Floquet eigenstates and quasi-energies. The bulk spectrum consists of two flat Floquet bands with quasi-energies , with the corresponding eigenstates localized on either the or sublattice. The winding number invariant can be computed for this model at , yielding Rudner et al. (2013). In a cylindrical geometry the two edges host linearly dispersing chiral modes in the quasi-energy gaps between the two bulk bands.
We now introduce a specific form of a time-dependent disorder potential, , which still allows for an exact solution. The full time-dependent Hamiltonian is given by . During the fifth segment of the driving period, we let , where is a uniformly distributed in the range , and is the annihilation operator on site . During segments 1–4, . We choose .
By following the evolution of a state that is localized on a single bulk site at time , one can easily verify that this state is a Floquet eigenstate, whose quasi-energy is [here () refers to a site in the () sublattice]. The Floquet spectrum consists of two bands, with quasi-energies in the range . One can similarly follow the evolution of a state that is initially localized on a site at the edge; in the geometry of Fig. 3 (viewed as a “strip” geometry with edges parallel to the horizontal axis), a state initialized on the () sublattice at the top (bottom) edge moves by two lattice constants to the right (left) every driving period. Therefore, chiral edge states persist even in the presence of the disorder potential.
As long as the gaps in the quasi-energy spectrum at remain open, the winding numbers at these gaps cannot change. Therefore, at least over a finite range of the disordered potential strength, we have as in the clean limit Rudner et al. (2013). Moreover, in the disordered system, all the bulk Floquet states are localized. Therefore, as argued in the previous section, the winding number is actually independent of the quasi-energy: for all . We conclude that by the definition presented in Sec. II, the Hamiltonian realizes the AFAI phase.
Clearly, the above model utilizes a very specific form of the periodic driving and of the added disorder. Nevertheless, we argue that the AFAI is a robust phase that does not require fine-tuning. To demonstrate the robustness of the phase, we now consider a generic local perturbation of that preserves the periodicity in time, , and show that the AFAI phase survives up to a finite value of .
The perturbation is assumed to be periodic in time and short-ranged in real space, such that the matrix elements of vanish beyond the th neighbor on the square lattice. For (no disorder) and , the bulk eigenstates of are generically dispersive and delocalized. However, we argue that for and for a sufficiently small , all the bulk Floquet states remain localized. To see this, we derive a time-independent effective Hamiltonian for the Floquet problem (on the torus) with :
where denotes time ordering. We further write the effective Hamiltonian as , where corresponds to the unperturbed () effective Hamiltonian, defined such that its eigenstates lie in the range . Here, we are considering a system with periodic boundary conditions; we will comment on the edge states later.
The key point, which we show below, is that for sufficiently small and , the hopping matrix elements of the effective static Hamiltonian decay exponentially with distance. If, in addition, , then all of the Floquet eigenstates remain localized Anderson (1958).
To find the effective Hamiltonian for we need to solve for . The unperturbed effective Hamiltonian is of the form
where for on the sublattice. We express as a power series in ,
To find we expand both sides of Eq. (11) in powers of , and compare them order by order. The details of the calculation are given in Appendix B. The results can be summarized by considering the explicit representation of the operators as a tight-binding “Hamiltonian,”
Using the explicit form for given in Eq. (12), we find for the lowest-order term
with , where is the evolution operator, and is the zeroth-order quasi-energy difference between the states localized at sites and .
As long as is smaller than for every pair of sites (which is the case for ), the factor in Eq. (15) is bounded. Similarly, the matrix elements are all non-singular (see Appendix B). Under these conditions, we expect the expansion in powers of to converge. In Appendix B, we argue that for sufficiently small , the matrix elements of decay exponentially with distance. Therefore, has the form of a tight-binding model with random on-site potentials and weak, short-range hopping. In this context, we expect all states to remain localized up to a critical strength of .
Since all the bulk states remain localized as is turned on, the chiral edge states that exist for cannot disappear; the only way to remove them is by closing the mobility gap in the bulk, allowing the two counter-propagating states at the two opposite edges to backscatter into each other. Hence, we expect the edge chiral states, and the associated quantized pumping, to persist up to a critical value of where the bulk mobility gap closes.
V Numerical results
Numerical simulations substantiate the conclusions of Sections II–IV. We will first briefly summarize our main findings, and then describe the simulations and results in more detail in the subsections below. For the simulations, a variant of the model discussed in Sec. IV, defined on a square lattice, is used:
where is the time-dependent, piecewise-constant Hamiltonian described in Sec. IV (pictured in Fig. 3). Using numerics, we are now able to study the more generic case in which the sublattice potential (denote here by ), as well as the disorder potential are time-independent (in contrast to the model studied in Sec. IV). We define , and take to be uniformly distributed in the interval . The parameters of the model are chosen to be , and .
In the clean case (), the system exhibits an anomalous Floquet-Bloch band-structure: the Chern numbers of all the bulk bands are zero, but the winding number for any value of within each of the band gaps Rudner et al. (2013). Such a band-structure is depicted in Fig. 2(b). When the disorder potential is turned on, however, the system enters the AFAI phase. Below, we show numerically that the bulk states become localized, and coexist with edge states which occur in all quasi-energies. Furthermore, when the system is initialized with fermions filling all of the sites in the vicinity of one edge, while the rest remain empty, as in Sec. III, the disordered system exhibits quantized amount of charge pumped per period, when averaged over long times. Finally, we examine the behavior of the system as the strength of the disorder potential is increased. We find that when the disorder strength reaches a certain critical value, the system undergoes a topological phase transition where the winding number changes from to . For stronger disorder, a “trivial” phase (where all bulk states are localized and there are no chiral edge states) is stabilized.
v.1 Localization, edge modes, and quantized charge pumping in the AFAI
The localization properties of the bulk Floquet eigenstates of (16) can be extracted from the statistics of the spacings between the quasi-energy levels. For localized states, the distribution of the level-spacing is expected to have a Poissonian form. In contrast, extended states exhibit level repulsion and obey Wigner-Dyson statistics Mehta (2004). To distinguish between these distributions, it is convenient to use the ratio between the spacings of adjacent quasi-energies levels Oganesyan and Huse (2007); Atas et al. (2013); DÃAlessio and Rigol (2014). Choosing the quasi-energy zone to be between and (i.e., choosing for ), we label quasi-energies in ascending order. We then define the level-spacing ratio (LSR) as , where . This ratio, , converges to different values for extended and localized states, depending on the symmetries of the system. For localized states, Oganesyan and Huse (2007), while for extended states, DÃAlessio and Rigol (2014). The latter value is obtained when one assumes that the quasi-energies are distributed according to the circular unitary ensemble (CUE) DÃAlessio and Rigol (2014), and in the thermodynamic limit, coincides with the value obtained by the more familiar Gaussian unitary ensemble (GUE).
Since the Floquet problem does not possess any generic symmetries such as time-reversal, particle-hole, or chiral symmetry, we expect its localization properties to be similar to those of the unitary class Evers and Mirlin (2008); Altland and Zirnbauer (1997); Dyson (1962). In analogy with the situation in static Hamiltonians in the unitary class Pruisken (1985), we expect that arbitrarily weak disorder is sufficient to localize the all Floquet states (on the torus). However, for weak disorder, the characteristic localization length can be extremely long, and easily exceeds the system sizes accessible in our numerical simulations. Therefore, the level spacing ratio is expected to show a gradual crossover from having the characteristic of delocalized states, , when , to the value that indicates localized behavior, , when .
This behavior is demonstrated in Figs. 4(a), panels (i)–(iii), where we plot the disorder averaged level spacing ratio and the density of Floquet states, as a function of the quasi-energy for different disorder strengths. For weak disorder, , panel (i) shows that the level spacing ratio is in any spectral region where Floquet states exists. On the other hand, panel (iii) shows that already for , the level-spacing ratio approaches at all quasi energies, as expected from localized states.
Note that, as the disorder strength increases, the level spacing ratio decreases uniformly throughout the spectrum [Fig. 4(a), panels (i–iii); the same behavior is seen at weaker values of the disorder (not shown)]. There is no quasi-energy in which the LSR remains close to , corresponding delocalized Floquet eigenstates. This is consistent with the expectation that the bulk Floquet states become localized even for weak disorder, and the localization length becomes shorter as the disorder strength increases. The behavior of the LSR as a function of system size, Fig. 4(b), also shows behaviour consistent with the above expectation. In contrast, if the bulk bands of the clean systems carried non-zero Chern numbers, delocalized states would persist in the bands up to a critical strength of the disorder, at which point they would merge and annihilate.
In the AFAI phase all the bulk states are localized, but the edge hosts chiral modes at any quasi-energy (cf. Sec. II). To test this, we simulate the time evolution of wavepackets initialized either in the bulk or near the edge of the system. We consider the system in a rectangular geometry. The initial state, , is localized to a single site . To obtain information on quasi-energy resolved propagation, we investigate the disorder-averaged transmission probability, , which is a function both of quasi-energy and the total time of evolution . Here, the bar denotes disorder averaging. The transmission amplitude in each disorder realization, , is obtained by a partial Fourier transform of the real time amplitude, , and is given by
The real time transmission amplitude is computed numerically by a split operator decomposition. Figs. 5(a),(b) show at different quasi-energies, for initial states on the edge and in the bulk, respectively. The simulations are done for a disorder strength . At this disorder strength, the analysis of the level-spacing statistics shown in Fig. 4(a) indicates that all the bulk Floquet bands are localized with a localization length smaller than the system size. Fig. 5(a) shows the value of when the wavepacket is initialized at the edge of the system, . The wave packet propagates chirally along the edge. The figure exemplifies that the edge modes are robust in the presence of disorder, and are present at all quasi-energies. Importantly, edge states are also observed at quasi-energies where the bulk density of states is appreciable, indicating that the chiral edge states coexist with localized bulk states [the density of states in the bulk is shown in Fig. 4(a)].
In contrast, Fig. 5(b) shows for a wavepacket initialized in the middle of the system. The wavepacket remains localized at all quasi-energies, as expected if all bulk Floquet eigenstates are localized. This confirms that the model we study numerically indeed exhibits the basic properties of the AFAI phase: fully localized Floquet bulk states, coexisting with chiral edge states which exist at every quasi-energy.
Next, we numerically demonstrate the quantized charge pumping property of the AFAI. Using the model described above, we numerically compute the value of given by Eq. (9) for a single value of the flux, . When computing , we averaged the charge pumped across all the lines running parallel to the direction of the cylidner (see Fig. 2), as well as over 100 disorder realizations. In Fig. 6(a), we show the cumulative average of the pumped charge per cycle in the limit of long times, [c.f. Eq. (9)] as a function of disorder strength. At weak disorder, when the localization length is smaller than the system size, is clearly not quantized. However, for disorder strength , the value of quickly tends towards unity. This agrees with the results presented in Fig. 4(a.iii), which indicate that at this disorder strength, the localization length is substantially smaller than . Finite size scaling demonstrating that indeed asymptotes to unity in the thermodynamic limit is presented in the inset of Fig. 6(a).
The value of the cumulative average of the pumped charge over periods, [c.f. Eq. (7)] is plotted vs. in Fig 6(b), demonstrating its approach to for large values of (i.e., at long times). As in panel (a), we averaged over all the lines running parallel to the direction, and over 100 disorder realizations. We examine the asymptotic behavior of and find a power law behaviour of the form with , shown in the inset of panel (b). Note that for a single disorder realization and a single vertical cut, is expected to exhibit an oscillatory behaviour with an envelope which decays as , see Appendix C.1. This expectation is indeed confirmed by our numerical simulations, as we show in Appendix D. In contrast, Fig. 6(b) shows a power-law behaviour with a power larger than and no oscillations; this is clearly the result of averaging over the frequencies appearing in for each disorder realization and vertical cut. The above results numerically confirm the discussion in Sec. III, and conclude our numerical analysis of the AFAI phase.
v.2 Strong Disorder Transition
For sufficiently strong disorder, we expect the AFAI to give way to a topologically trivial localized phase in which the winding number vanishes. We now analyze the transition between the AFAI and this “trivial” phase. As explained above, the winding number can only change if a delocalized state crosses through the quasi-energy as disorder is added. In the AFAI phase all of the bulk states are already localized. How does the transition between the two phases occur?
Clearly, at the transition, delocalized states must appear in the quasi-energy spectrum. As disorder is increased, the delocalized states must sweep the full quasi-energy zone, changing the topological invariant as they do so. The transition from the AFAI phase to the trivial phase can therefore occur through a range of disorder strength , where is the disorder strength at which the first delocalized state appears, and is the disorder strength at which all Floquet states are again localized, and for all . Below we will support this scenario using numerical simulations, and furthermore provide evidence suggesting that the transition is of the quantum Hall universality class.
We study the same model used in Sec. V.1 and examine the level-spacing ratio, , as a function of disorder strength and quasi-energy. For this model, our simulations indicate , within our resolution (limited by the system size). In Fig. 7(a), we plot , averaged over disorder realizations and all quasi-energies. We see that at disorder strength the level spacing ratio reaches , indicating delocalization. On either side of this point, approaches as the system size increases, which indicates localization. The peak in the value of as a function of disorder gets sharper for larger system size, which is a signature of a critical point of this transition. In Fig 7(b), we show that at disorder strength , the LSR is independent of the quasi-energy with (for disorder strengths close to , we also find that the LSR is independent of the quasi-energy, but with ). This indicates that all of the Floquet states have a delocalized character at this disorder strength, which leads us to conclude that .
At the critical point, , we expect the wavefunctions to have a fractal character Ludwig et al. (1994). This behavior is manifested in the distribution of the inverse participation ratio (IPR), . We study the distribution of the IPR, , among all the Floquet eigenstates and averaged over disorder realizations. Fig. 7(c) shows the distribution for different system sizes. We note that the shapes of the distributions for different sizes are similar, a signature of criticality. In two dimensions, the average value of the IPR at a critical point is expected to scale like , with Ludwig et al. (1994). Fig. 7(d) shows the scaling collapse of all the distributions. From the collapse we find the fractal dimension, . The inset in this figure also shows a linear scaling . The critical exponent we find in our numerical simulations is close to the value found for the universality class of quantum Hall plateau transitions Ludwig et al. (1994); Huckestein et al. (1992), , indicating that the transition from the AFAI to the trivial phase may belong to this universality class. This is natural to expect, since, like the quantum Hall transition, in the transition out of the AFAI phase a delocalized state with a non-zero Chern number must “sweep” through every quasi-energy, to erase the chiral edge states. We expect that the AFAI transition can be described in terms of âquantum percolationâ in a disordered network model, similar to the Chalker-Coddington model for the plateau transitions Chalker and Coddington (1988). We leave such investigations for future work.
In this paper we have demonstrated the existence of a new non-equilibrium phase of matter: the anomalous Floquet-Anderson insulator. The phase emerges in the presence of time-periodic driving and disorder in a two-dimensional system, and features a unique combination of chiral edge states and a fully localized bulk. Such a situation cannot occur in non-driven systems, where the presence of chiral edge states necessarily implies the existence of delocalized bulk states where the chiral branches of the spectrum can terminate. In a driven system, the periodicity of the quasienergy spectrum alleviates this constraint, allowing chiral states to “wrap around” the quasienergy zone and close on themselves.
One of the key physical manifestations of the AFAI is a new type of non-adiabatic quantized pumping, which occurs when all states near one edge of the system are filled. It is interesting to compare this phenomenon with Thouless’ quantized adiabatic pumping, described in Ref. Thouless (1983).
The complementary relationship between pumping in the AFAI and the Thouless case is best revealed by first viewing the Thouless pump from the point of view of its Floquet spectrum. In Thouless’ one-dimensional pump, a periodic potential is deformed adiabatically such that in each time cycle a quantized amount of charge is pumped through the system. In the adiabatic limit, the quasi-energy spectrum of the pump exhibits one pair of counter propagating one-dimensional chiral Floquet-Bloch bands, which wrap around the quasi-energy Brillouin [see Fig. 8(a)]. The (nonzero) quasi-energy winding number of each band gives the associated quantized pumped charge Kitagawa et al. (2010). Importantly, for any finite cycle time the two counter-propagating states hybridize and destroy the perfect quantization of the charge pumped per cycle [Fig. 8(b)].
In a strip geometry, the AFAI can be viewed as a quasi-one-dimensional system. As discussed in Sec. II, the system hosts chiral edge states that run in opposite directions on opposite edges. Furthermore, as shown by the spectral flow (see Fig. 2), these counterpropagating chiral modes cover the entire quasi-energy zone, analogous to the counter-propagating modes of the Thouless pump [Fig. 8(a)]. Crucially, however, the counterpropagating modes of the AFAI are spatially separated and therefore their coupling is exponentially suppressed: no adiabaticity restriction is needed to protect quantization. Thus quantized pumping at finite frequency can be achieved in the AFAI phase.
How is the AFAI manifested in experiments? First, the localized bulk and chiral propagating edge states could be directly imaged, for example in cold atomic or optical setups. More naturally for a solid state electronic system, the pumping current could be monitored in a two terminal setup. Unlike the case of an adiabatic pump, where a quantized charge is pumped at zero source-drain bias, to observe quantized charge pumping in the AFAI the chiral propagating states of one edge of the system would need to be completely filled at one end of the sample, and emptied at the opposite end. We speculate that this can be achieved using a large source-drain bias. A detailed analysis of such non-equilibrium transport in a two or multi-terminal setup, as well as an investigation of promising candidate systems, are important directions for future study.
The implications of our results go beyond those specific to the class of systems studied in this paper. As a direct generalization of our results, one can consider constructing anomalous Floquet insulators in different dimensions and symmetry classes. Floquet-Bloch band structures which generalize those of Ref. Rudner et al. (2013) can serve as a starting point for constructing such anomalous periodically driven systems. Going beyond the single particle level, an important challenge is to understand how the properties of the AFAI change in the presence of interactions. An exciting possibility is to obtain a topologically non-trivial steady state for an interacting, periodically driven system Dehghani et al. (2014a, b); Bilitewski and Cooper (2014); Seetharam et al. (2015); Iadecola et al. (2015); Khemani and Sondhi (2015). The common wisdom dictates that a periodically system with dispersive modes is doomed to evolve into a highly random state which is essentially an infinite temperature state as far as any finite order correlation functions are concerned DÃAlessio and Rigol (2014); Ponte et al. (2014, 2015); Lazarides et al. (2014). Our results on the single particle level demonstrate that it is possible to obtain a topological Floquet spectrum with no delocalized states away from the edges of the system. It is therefore possible that such periodically driven systems can serve as a good starting point for constructing topologically non-trivial steady states for interacting, disorder (many-body localized) periodically driven systems. What types of topological steady states can be obtained by this method, and what are their observable signatures, will be interesting subjects for future work.
Acknowledgements.We are grateful to Cosma Fulga, Mykola Maksimenko, and Ady Stern for enlightening discussions. P.T. and N.L. acknowledge support from the US - Israel Bi-National Science foundation. N.L acknowledges support from the CIG Marie Curie grant and from and I-Core, the Israeli excellence center âCircle of Lightâ. E. B. was supported by the Minerva foundation, the CIG Marie Curie grant, and the Israel Science foundation. M. R. acknowledges support from the Villum Foundation and from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement PIIF-GA-2013-627838. G. R. and P.T. are grateful for support from NSF through DMR-1410435, as well as the Institute of Quantum Information and matter, an NSF Frontier center funded by the Gordon and Betty Moore Foundation, and the Packard Foundation.
Appendix A Construction of the deformed evolution operator on the cylinder
In this appendix we construct the deformed evolution operator on the cylinder, of Eq. (5), which is used to demonstrate the existence of delocalized edge states in the AFAI. The deformation is designed such that at it interpolates smoothly between (corresponding to the cylinder) in the vicinity of the edges of the cylinder, and in the bulk of the cylinder. We first define the family of operators , where
Here, we choose , where is the bulk localization length. Analogously to Eq. (1), the family of deformed evolution operators is defined as
Here, is defined as in Eq. (1), i.e., , where is the evolution operator for a full period on the torus. The deformed evolution operator corresponds to .
Appendix B Perturbative derivation of the effective Hamiltonian
Here, we outline the details of the derivation used to demonstrate the perturbative stability of the AFAI phase. We first examine the effective static Hamiltonian defined in Eq. (11), expressed as a power series in . We insert Eq. (13) into (11) and expand both sides in powers of . For the left-hand side, we obtain
where and . Note that the eigenvalues of are only defined modulo (where is an integer). The form of (t) depends on the choice of , while the evolution operator does not. To fix this ambiguity, we choose to lie in the range .
The right–hand side of (11), expanded in powers of , reads
where and .
To find explicitly, we express them in the “tight-binding” form (14); inserting this form into (22,23), and using the fact that contains only on-site potentials and no inter-site hopping, we arrive at
Equating this expression for to the right hand side of Eq. (22) gives Eq. (15). From (24) we see that, as long as for every pair of sites, is non-singular. As , may diverge for all , and the expansion in fails. This reflects the fact that, in general, a Floquet operator whose eigenvalues are spread throughout the quasi-energy zone cannot be generated by a static, local Hamiltonian.
From the form of the right-hand side of Eq. (22), we can analyze the maximum range of the hopping matrix elements in . We denote the maximum range of the hopping matrix elements in by , where corresponds to a nearest neighbor hop, to second neighbors, and so on. Since the matrix elements of the unperturbed evolution operator vanish beyond second neighbor sites on the square lattice, we find that contains matrix elements whose range is at most . Similarly, from Eq. (23), vanishes beyond the th neighbor, and more generally, vanishes beyond range . Hence, the matrix elements of at range contain the exponentially small factor .
Appendix C Quantized charge pumping and the winding number
In this Appendix, we show that for the AFAI, a non-zero value for the winding number implies quantized charge pumping on the edge of the system. As in Sec III, we take an initial state with all sites filled in a strip of width near one edge of the AFAI, [see Fig. 2(c)], and the rest to be empty.
To calculate the time-dependence of the pumped charge, we begin by deriving an expression for the instantaneous current flowing across a longitudinal cut through the cylinder, i.e., across a line parallel to the -axis. The corresponding current operator is found by first allowing a flux to be threaded through the cylinder. Next we pick a gauge where the gauge (vector) potential is nonzero only on the links connecting sites with to sites with . The net current flowing across the cut between and is then described by the operator , where is the Hamiltonian of the system in the presence of the flux . Here the tilde denotes the cylindrical geometry.
Below, we first (Sec. C.1) show that when averaged over many periods, the charge pumped approaches a quantized value equal to the edge topological invariant , expressed in terms of the spectral flow on one edge of the system [Eq. (10)]. We then show (Sec. C.2) that is in fact equal to the bulk topological invariant , given by Eq. (2)
c.1 Quantized charge pumping from spectral flow
We start from Eq.(7), which gives the charge pumped during the time integral . The initial state, which is a single Slater determinant in terms of position eigenstates, is given by a superposition of Slater determinants in term of Floquet states
Inserting this into the expression for the pumped charge in the interval , Eq. (7), we get
The double sum in Eq. (26) contains both diagonal and off diagonal terms for the contributions of single particle Floquet states. Denoting each of the contributions by , and using , the different terms in (26) can be written as
According to Floquet’s theorem, the Floquet states can be written as , where is a periodic function. Substituting this into Eq. (27) we obtain for the diagonal terms,
and for the off diagonal terms,
Clearly, when computing the average charge pumped over periods, , the off-diagonal terms will give a contribution which decays as , while the diagonal terms will give the contribution which does not decay with ,
where is the probability for the Floquet state to be occupied, . Averaging over the flux values, we obtain Eq. (10), which is the result we set out to obtain in this subsection.
c.2 Quantized charge pumping and the winding number
We now show that , the charged pumped over a period averaged over long times, is equal to the winding number , appearing in Eq. (2). We show that in the limit of a large number of cycles, , the average charge pumped per cycle contains a quantized piece equal to the winding number, plus a small correction that decays with the averaging time at least as fast as .
Recall that the many-body initial state that we consider is a single Slater determinant with electrons populating all sites in the strip of width near one edge of the cylinder. At time , the expectation value of the current is given by the sum of contributions from each of these single particle states, propagated forward in time with the evolution operator for the system with the threaded flux . Defining a projector that projects onto all sites within the strip of initially occupied sites, the current is given by
Using Eq. (31), the total charge pumped in the first cycle, , is given by