Kicked-Harper model vs On-Resonance Double Kicked Rotor Model: From Spectral Difference to Topological Equivalence
Recent studies have established that, in addition to the well-known kicked Harper model (KHM), an on-resonance double kicked rotor model (ORDKR) also has Hofstadter’s butterfly Floquet spectrum, with strong resemblance to the standard Hofstadter’s spectrum that is a paradigm in studies of the integer quantum Hall effect. Earlier it was shown that the quasi-energy spectra of these two dynamical models (i) can exactly overlap with each other if an effective Planck constant takes irrational multiples of and (ii) will be different if the same parameter takes rational multiples of . This work makes some detailed comparisons between these two models, with an effective Planck constant given by , where and are coprime and odd integers. It is found that the ORDKR spectrum (with two periodic kicking sequences having the same kick strength) has one flat band and non-flat bands whose largest width decays in power law as , where is a kicking strength parameter. The existence of a flat band is strictly proven and the power law scaling, numerically checked for a number of cases, is also analytically proven for a three-band case. By contrast, the KHM does not have any flat band and its band width scales linearly with . This is shown to result in dramatic differences in dynamical behavior, such as transient (but extremely long) dynamical localization in ORDKR, which is absent in KHM. Finally, we show that despite these differences, there exist simple extensions of KHM and ORDKR (upon introducing an additional periodic phase parameter) such that the resulting extended KHM and ORDKR are actually topologically equivalent, i.e., they yield exactly the same Floquet-band Chern numbers and display topological phase transitions at the same kick strengths. A theoretical derivation of this topological equivalence is provided. These results are also of interest to our current understanding of quantum-classical correspondence considering that KHM and ORDKR have exactly the same classical limit after a simple canonical transformation.
pacs:05.45.Df, 05.45.Mt, 71.30.+h, 74.40.Kb, 05.45.-a
As one important paradigm in the studies of quantum chaos and quantum-classical correspondence, the kicked rotor (KR) model KR () has received tremendous theoretical and experimental interest in the last three decades KR (); Izrailev (). For some experimental activities on KR within the last three years, we would like to mention those listed in Ref. recentKR (). A one-dimensional KR is described by the Hamiltonian
in terms of dimensionless variables, where and are conjugate (angular) momentum and angle variables, and are the kick strength and the period of the -kicks. The dynamical evolution of the system for a period from time to can be expressed as a quantum map, which is given by the following unitary Floquet operator
For our considerations below, we confine ourselves to a rotor Hilbert space defined by the periodic boundary condition in , with . The Hilbert space can then be represented by the eigenfunctions of , with , , being an integer, and being a dimensionless effective Planck constant. Through extensive numerical simulations and mathematical analysis, it is now well known that in general the KR dynamics can be classified into two categories Izrailev (). For an irrational (hence generic) value of the system can diffuse in (angular) momentum space only for a short time due to “dynamical localization”, regardless of the kick strength. This hints at a discrete spectrum of and is closely related to Anderson localization ATr (). On the other hand, for being a rational multiple of (except for odd multiples of ), has continuous bands: A time-evolving state would keep spreading out in (angular) momentum space ballistically. This category of dynamics was termed as “quantum resonance” QR ().
Another important quantum chaos model is the kicked Harper model (KHM) leboeuf (); KH1 (); KH2 (), originally introduced in Ref. zas () as an approximation of the problem of kicked charges in a magnetic field. Remarkably, the KHM and even a whole class of its generalized versions were shown to be equivalent to the problem of a charge kicked periodically in the presence of a magnetic field dana-pla (). The associated KHM quantum map for each period is given by
with being an additional system parameter. Throughout we assume the KHM is also treated in the same Hilbert space as the KR and is quantized on a rotor Hilbert space. The dynamics of KHM differs from that of KR as described above in several aspects. For example, for all irrational values of , the system in general tends to delocalize (localize) in (angular) momentum space for () KH2 (). Of particular interest is the symmetric case of , for which the quasi-energy spectrum of is fractal-like in general. Scanning the spectrum collectively for fixed versus a varying forms a pattern that resembles the Hofstadter’s butterfly spectrum hofstadter (), a paradigm in studies of the integer quantum Hall effect. The associated dynamics is extended in general and may be connected with the fractal dimensions of the Floquet spectrum.
Given the above-mentioned differences between KR and KHM, the work of Ref. JJ08 () by two of the authors emerged somewhat unexpectedly. There it was shown that a variant of KR also has Hofstadter’s butterfly spectrum. In particular, motivated by the double-kicked rotor model studied both experimentally and theoretically in Ref. DKRM (), which is a special case of “multiple KR’s” first introduced in Ref. dana-flat-band2 (), Ref. JJ08 () studied a double-kicked model under a quantum-resonance condition. For a total period of (), a double kicked rotor model is associated with two periodic -kicks of strengths and , separated by a time interval set to be unity, yielding the following Floquet operator
In Ref. JJ08 (), is chosen to satisfy the quantum resonance condition . Then due to the discreteness of the momentum eigenvalues. This leads us to an on-resonance double kicked rotor model (ORDKR), whose Floquet operator is given by JJ07 ():
Note that we have deliberately used symbols and in both and because in this paper, parameter or parameter from both models will always be assigned the same value. Experimental realization of such an ORDKR propagator in atom optics is possible by loading a Bose-Einstein-Condensate (BEC) in a kicking optical lattice, with the initial quasi-momentum spread of the BEC negligibly small as compared with the recoil momentum of the optical lattice exp-note (). Interestingly, for being an irrational multiple of , the ORDKR and the KHM share the same quasi-energy spectrum JMP (); GuarDKR ().
Our main plan for this paper is to make some detailed comparisons between KHM and ORDKR as two closely related dynamical models, both possessing Hofstadter’s butterfly spectrum. Our motivations are as follows. First of all, in Refs. JJ08 (); JMP (), it was shown that and have different spectra if is a rational multiple of . On the other hand, as approaches an arbitrary irrational number, the spectral difference between and , which is characterized by a Hausdorff metric in Ref. JMP (), was shown to approach zero. It is therefore highly worthwhile looking into the actual spectral differences for rational values of , because, up to a classical canonical transformation, ORDKR and KHM have exactly the same classical limit JMO () (obtained by letting approach zero while fixing and ). Indeed, given their equivalence in the classical limit, the spectral differences we analyze constitute beautiful examples to illustrate how quantization of classically equivalent systems may lead to remarkable system-specific consequences. Second, by working on the details we hope to find some clues as to why the dynamics of ORDKR can be so different from that of KHM. We indeed succeed in doing this, finding that even on a qualitative level, the Floquet bands of ORDKR behave much differently from that of KHM, for , with and being coprime and both odd. In particular, we shall prove the existence of a flat Floquet band dana-flat-band1 (); dana-flat-band2 () for ORDKR with , which may be of interest to current studies of strongly correlated condensed-matter systems with an almost flat energy band flat-band (). The existence of a flat Floquet band has been shown elsewhere to be important in explaining the intriguing exponential quantum spreading dynamics in ORDKR JiaoPRL11 (); Hailongwork (). Third, motivated by recent interests in topological characterization of periodically driven systems derek12 (); floquetTI () and given the interesting relationship of the two models described previously, we ask whether, after all, ORDKR and KHM have any interesting topological connections. Based on our numerical and analytical studies, the answer is yes and we shall claim that ORDKR and KHM are topologically equivalent in the sense that their extended Floquet bands (obtained upon introducing a phase shift parameter defined in Sec. III) always have the same band Chern numbers.
This paper is organized in the following order. In Sec. II we present detailed results regarding a spectral comparison between KHM and ORDKR, for , and with and being coprime and odd integers. Numerical findings will be described first, followed by analytical considerations when possible (e.g., band width scaling for a three-band case and the general proof of a flat band for ORDKR). The implications of peculiar spectral properties of ORDKR for its dynamics are also discussed via some numerical studies. In Sec. III we study the KHM and ORDKR by extending them to accommodate a new periodic parameter and demonstrating the topological equivalence of the resulting extended models. Section IV concludes this paper.
Ii Spectral differences and their dynamical implications
ii.1 Summary of main numerical findings
As far as numerics are concerned, the spectrum of the unitary operators can be obtained in a straightforward manner. For completeness we describe some details here. The key step is to take advantage of the periodic property of or in the (angular) momentum space, which arises naturally for being a rational multiple of . Denote to refer to either or . Letting , one easily finds for . This indicates a unit cell in (angular) momentum space, with a size of . The spectrum is then equivalent to that of a reduced matrix , whose elements are given by , with being the Bloch phase in momentum space and running over all integers. As off-diagonal elements of decay exponentially, the summation in can be truncated safely at certain large enough value of (in our analytical studies below, we do not do such truncations). Numerical results are then checked by further increasing the truncation radius. Once is numerically obtained, the standard diagonalization algorithm for a unitary matrix can be exploited to obtain values of quasi-energy . By varying in we have Floquet bands.
In Fig. 1 we show our obtained quasi-energy values of and as a function of the kick strength . Though for each fixed value of , we only show the quasi-energy values for a limited number of Bloch phase choices, the locations of the bands, the band width, and a few avoided band crossings can already be seen clearly for not too large values of . In particular, at , nice Floquet bands can be identified clearly for both ORDKR and KHM, though for very large values of the merging of the bands does occur.
Spectral differences between and in the shown example are also obvious. Based on the results shown in Fig. 1, we have carried out extensive numerical investigations for other cases with , with and coprime and both odd. Some of the main features are presented and commented on below.
First, the band structure of is symmetric with respect to the zero quasi-energy axis, which is however not the case for . This interesting symmetry is absent in both and . We shall prove this property below.
Second, consistent with the above-mentioned symmetry, is seen to have a flat band with . By flat band, we mean that this quasi-energy value is independent of the Bloch phase . So the overall picture is that the bands can be classified into pairs, with each pair having opposite quasi-energy values, plus a flat band in the middle. Again, this is not the case for . The existence of a flat Floquet band was previously observed in studies of the quantum antiresonance phenomenon in kicked systems dana-flat-band1 (); dana-flat-band2 (). However, unless in the case of ( odd) that also corresponds to a quantum antiresonance condition, here the flat band of coexists with other nonflat bands. This coexistence of a flat band with nonflat bands constitutes an interesting feature. As a side note, Ref. GuarORKR () suggested that for a KR defined in this paper under the quantum resonance condition of any order (i.e., , with and arbitrary coprime integers), none of the Floquet bands of is flat. So the existence of one single flat band of is also remarkable as compared with .
Third, the largest bandwidth of the other non-flat Floquet bands of ORDKR scales with as , in the limit of . In sharp contrast, the bandwidths of KHM scale with linearly. Representative numerical results are shown in Fig. 2, where the bandwidth of the widest band is plotted against small values of , for , with , . The power law decay of the ORDKR bandwidth in the form of can be clearly identified, whereas the bandwidth of KHM remains a linear function of , irrespective of the value of . This being the case, in the small regime (), the maximum bandwidths of ORDKR is times narrower than that of KHM.
ii.2 Flat band and Band symmetry in ORDKR
Flat bands in solid-state systems are of vast interest in condensed matter physics because they offer new opportunities for understanding strongly correlated systems without Landau levels. For this reason the existence of a flat band in a periodically driven system can be useful, too. To further understand the flat band of ORDKR, we present a theoretical proof in this subsection. In doing so we shall also prove the band symmetry noted above. We shall also discuss how an eigenstate on a flat band, which is infinitely degenerate, may be numerically found.
For with and being coprime integers, the spectrum becomes that of a reduced Floquet matrix with elements . After performing some necessary integrals and using the fact that both and are odd, one can express as a summation of finite terms (see Appendix for details). In the following discussions regarding the existence of a flat band and the band inversion symmetry, we shall restrict ourselves to the cases of (note however, in the next section, the notation introduced here will be extended to the cases with ). We first introduce diagonal unitary matrices , , and unitary matrix , with matrix elements , , and , where indices and take values . Note that in obtaining our expression for , we made use of the fact that . We then have the following compact form for the reduced Floquet matrix
To prove that there is a flat band for ORDKR, we show that has an eigenvalue equal to one, regardless of the value of . Consider then a matrix transformed from by a unitary operation , which takes the form
The eigenvalue equation of may be rewritten as
where , denotes an eigenvector, and is an eigenvalue of . Detailed calculations show that is a symmetric matrix (see Appendix for details) and since is a diagonal matrix, must be an antisymmetric matrix of odd dimension. It immediately follows . Thus, regardless of the Bloch phase , is a permissible solution to Eq. (8). We have thus shown that always has a unity eigenvalue or zero quasi-energy for . This is nothing but the existence of a flat Floquet band.
Our considerations above also lead us to a proof of the band inversion symmetry of ORDKR for odd and . Specifically, because , we see that if , then as well. That is, both and are solutions to the eigenvalue equation of Eq. (8). As such, if we have a quasi-energy , we must have in the spectrum. This completes our proof of the inversion symmetry of the ORDKR.
A flat band is infinitely degenerate as states on the band can still have a continuous Bloch phase . Due to such an independence upon the Bloch phase, the band dispersion relation directly yields a zero group velocity in the (angular) momentum space, thus indicating a zero mobility in the (angular) momentum space. Further, the infinite degeneracy allows us to construct a flat-band eigenstate that is localized in the (angular) momentum space (though the Floquet operator itself is periodic in momentum with a period ). It is interesting to outline a simple approach to the construction of flat-band states. It is found that, highly localized flat-band states can be obtained by directly truncating the full Floquet matrix to a small size, such that there is only one eigenstate whose eigenvalue is real and still equals to unity (thus not affected by the truncation). Other localized states on the flat band can be obtained by shifting it by a multiple of N sites, or by superimposing these states localized at different locations. Figure 3 depicts one computational example of a flat-band eigenstate strongly localized in the (angular) momentum space. We have checked that if we use a flat-band state we constructed as the initial state for time evolution, then indeed this state does not evolve with iterations of our ORDKR quantum map. This situation is more subtle than the quantum antiresonance phenomenon dana-flat-band1 (); dana-flat-band2 (): for ORDKR with multiple bands, only special states prepared on the single flat band can remain localized, whereas in the case of quantum antiresonance an arbitrary state should remain localized.
ii.3 A theoretical bandwidth result and its dynamical consequence
For with and being coprime integers, the reduced Floquet matrices and (see our general expressions in the Appendix) can be obtained analytically. To further understand and confirm the bandwidth scaling of ORDKR and KHM, we have also carried out analytical studies for a three-band case, with and .
For ORDKR, the three eigenvalues are found to be and , where . One finally finds where . For it can be shown that the edges of the band correspond to and . The bandwidth can thus be determined to be . Taylor expanding this expression for the bandwidth, we find the first nonzero term to be , a clear power-law scaling of .
For KHM, the eigenvalues can be deduced from the equation . The resulting explicit expression of the eigenvalue equation is
where . Note that all eigenvalues are in the form of , since the reduced Floquet matrix is always unitary. The three eigenvalues are found to be , and where . For , the edges of the band correspond to and . The band width can thus be determined to be , and . Taylor expanding the expressions of eigevalues for and keeping the lowest order in , we have , and , a clear linear scaling of K.
The very fast decay of the Floquet bandwidth of ORDKR suggests that in a considerable range of the bandwidths will be very narrow. In other words, for a small , all the Floquet bandwidths would be effectively zero for a reasonably long time scale. Therefore, when it comes to the dynamical evolution of the system, effectively the system will not feel its continuous Floquet spectrum and hence displays localization behavior, for a time scale inversely proportional to the bandwidths. We call this the time scale of transient dynamical localization and denote it by . We then have . The overall expectation is the following: within , ORDKR displays localization in the (angular) momentum space, but afterwards it begins to show ballistic behavior in the (angular) momentum space. Because of the power law scaling, the intriguing time scale can be very sensitive to a change in the kick strength . Our numerical calculations indeed confirm this. Figure 4(a) shows an example of the dynamics of the kinetic energy of ORDKR, starting from an initial state with zero momentum. In all three of the shown cases, the kinetic energy is seen to freeze over a time scale before it starts to increase ballistically. The time scale of the freezing stage is shown to increase rapidly as we decrease the value of . As a comparison, Figure 4(b) shows the parallel dynamics of KHM, for the same three values of . There it is seen that the transient stage of localization is only weakly dependent upon , which is again consistent with the linear -dependence of the bandwidth of KHM. Quantitatively, the transient localization time scale is numerically determined from the duration of kinetic energy freezing. The thus obtained numerically and shown in Fig. 4(c) indeed satisfies the scaling for ORDKR, which is in sharp contrast to the scaling for KHM. The results here can also be understood as a quantitative explanation of our earlier finding of transient dynamical localization in Ref. JJ07 (). For future experiments, the observation of the aforementioned scaling of versus may serve as the first piece of evidence of a successful realization of an ORDKR.
Iii Topological Equivalence between ORDKR and KHM
In this section, we devote ourselves to a detailed comparison of the Floquet band topologies of ORDKR and KHM. We first describe our motivation and introduce new notation. Next, we report numerical findings of the Floquet band topological numbers of both models. Finally, an exact analytical proof of the topological equivalence between ORDKR and KHM is presented.
iii.1 Motivation and Notation
One early study leboeuf () suggested that topological properties of the Floquet bands of KHM may be connected with the regular-to-chaos transition in the classical limit. Because ORDKR and KHM share the same classical limit (up to a canonical transformation), we suspect that there should be some similarity in their Floquet band topologies. Our second motivation for a topological study is related to an earlier finding that, when is a rational number, the spectral union of (variant of ORDKR defined below) over all is the same as that of (variant KHM defined below) over all JMP (). This previous mathematical result further suggests a possible topological connection between the two models. Interestingly, as we explore this possible topological connection, we are able to see a connection between KHM propagator and ORDKR propagator for each individual value of along with an individual value of the Bloch phase, thus going beyond Ref. JMP () that considered a unification of all values of and the Bloch phase. Further, as we shall see below, the connection is established by a mapping in the parameter space, which cannot be achieved by a unitary transformation between the two propagators.
Next, we introduce necessary notation for our discussion of band topology. To characterize the band topology for both ORDKR and KHM, we introduce an additional periodic phase parameter to the ORDKR and KHM maps, namely,
For , both operators are periodic in (angular) momentum space with period . Hence, their eigenvalues are -periodic in the Bloch phase and also in , giving rise to extended Floquet bands which disperse as a function of and . These 2-dimensional bands may be topologically characterized by Chern numbers, denoted for the th band. In what follows, we denote as an (generalized) eigenstate of either or , in the th band, with an eigenvalue . Such a generalized eigenstate lives on the entire (angular) momentum space. We then denote as the reduced Floquet matrix constructed from either or using the method described at the beginning of Section II. We next define the state , which is projected onto sites of one unit cell in the (angular) momentum space, i.e., . We further assume that is normalized over one unit cell consisting of sites. Using the above notation, the Berry curvature of the th band is then defined as derek12 ()
where we have suppressed the explicit dependences on and for brevity. From the Berry curvature we obtain the Chern number ,
iii.2 Numerical Findings
We have conducted extensive numerical evaluations of the Floquet band Chern numbers associated with both and . We find that for the same and respectively in both models, the Chern numbers are always equal. For example, for and , Fig. 5 represents the Floquet band Chern numbers for both models versus a varying . The Chern numbers obtained for are identical with those for . Here, we adopt the convention that the band with largest absolute value of Chern number is always represented by the line in the middle. Vertical lines represent collisions between quasi-energy bands, during which Chern number transitions can take place. Note that in some cases band 1 and band 3 can collide directly with each other through the boundary of the quasienergy Brillouin zone. It is also important to stress that the Chern numbers of ORDKR match those of KHM for all values, despite their jumps at various topological phase transition points. We are thus clearly witnessing, albeit numerically, a remarkable topological equivalence between ORDKR and KHM!
Some insight into this observed topological equivalence may be obtained by comparing the quasienergy dispersions of the two models. In Fig. 6, we present the Floquet band structure for both ORDKR and KHM, in the case of . Interestingly, the ORDKR band profile appears to be the same as that of KHM, up to some translation along the and axes, followed by a rotation of the spectrum about the quasi-energy axis. This observation is consistent with our proof of topological equivalence in the next section.
We have numerically observed that the topological equivalence also occurs for . As one example of this, Fig. 7 depicts a zoo of Chern numbers for ORDKR and KHM, with , fixed but varying. We again see the same equivalence of Chern numbers across a few topological phase transition points. In addition, we found computationally that the Chern numbers are invariant upon an exchange between and . This was found to hold true also in other cases with more bands.
We have also plotted the Floquet band structure for a case in Fig. 8. Here we consider the case of , . It is seen that the band profiles of ORDKR and KHM are once again similar and appear to be related by a rotation and translation.
iii.3 Proof of Topological Equivalence
To strictly confirm our claim of topological equivalence, we present an analytical proof in this subsection. The proof proceeds as follows. We first show that the reduced ORDKR Floquet matrix and the reduced KHM Floquet matrix are equivalent up to a series of unitary transformations and a mapping between their parameter values. We then show that these matrices obtained under the unitary transformations and mapping of parameters still correspond to the same Chern numbers as the original reduced matrices. These steps constitute a proof of topological equivalence.
We consider cases with , with and co-prime and both odd. In these cases, the reduced Floquet matrices of and (see the Appendix for details) can be written compactly as a product of unitary matrices
where , , , are diagonal unitary matrices, with matrix elements , , , , where the index takes values . and are defined as they were in Section II.
We begin the proof by applying a unitary transformation given by to the matrix to obtain . Writing as the exponential of a matrix, we obtain
In the following steps, we will apply a series of unitary transformations to the reduced matrix and show that the result is equivalent to the above unitarily transformed version of provided a condition between and in the two models is obeyed.
Applying a transformation given by to , we obtain , which we simplify as follows.
Denoting , . The explicit expression for is
Next, we introduce the permutation matrix which is made up entirely of zeroes except that in the -th row, the -th column equals 1, with . Here, and take values . Note that is unitary and that the set of values will include all of the values . We apply the unitary transformation to and obtain , where . is a diagonal unitary matrix with diagonal elements . The effect of the permutation matrix on is as follows.
We can see that the structure of the above matrix is very similar to and would be made identical with it if we were to replace all the nonzero elements with 1. This is achieved by a transformation via the diagonal unitary matrix which has diagonal elements . It can be shown that . Denoting and using that and commute due to their both being diagonal, we obtain
From Eq. (14) and (19), we observe that and are identical, provided that and . Summarizing what we have found so far, we have learned that if we unitarily transform from to , where , and unitarily transform from to , where , we find that the two unitarily transformed matrices are identical up to some mapping between and .
Figure 9 represents one example of the quasi-energy band plot for both ORDKR and KHM. Referring to panel (b) and panel (c), we thus directly see that provided that and , the extended Floquet band structure for ORDKR and KHM are the same (though the boundaries on the plane are different).