# Counter-propagating edge modes and topological phases of a kicked quantum Hall system

###### Abstract

Periodically driven quantum Hall system in fixed magnetic field is found to exhibit a series of phases featuring anomalous edge modes with the “wrong” chirality. This leads to pairs of counter-propagating chiral edge modes at each edge, in sharp contrast to stationary quantum Hall systems. We show the pair of Floquet edge modes are protected by the chiral (sublattice) symmetry, and that they are robust against static disorder. The existence of distinctive phases with the same Chern and winding numbers but very different edge state spectra points to the important role played by symmetry in classifying topological properties of driven systems. We further explore the evolution of the edge states with driving using a simplified model, and discuss their experimental signatures.

###### pacs:

Cyclic time-evolutions of quantum systems are known to have interesting topological properties Berry (1984); Aharonov and Anandan (1987). Several groups recently showed that periodic driving can turn an ordinary band insulator (superconductor) into a Floquet topological insulator (superconductor) Oka and Aoki (2009); Lindner et al. (2011); Kitagawa et al. (2010); Lindner et al. (2013); Kitagawa et al. (2011); Jiang et al. (2011); Gu et al. (2011); Rudner et al. (2013); Thakurathi et al. (2013). This provides a powerful way to engineer effective Hamiltonians that stroboscopically mimic stationary topological insulators Lindner et al. (2011); Kitagawa et al. (2010); Rechtsman et al. (2013). Moreover, a large class of topological phenomena in periodically driven many-body systems are unique and have no stationary counterparts. An early example is Thouless’s one-dimensional charge pump, where he showed that the charge transport is quantized and related to a topological invariant Thouless (1983). Other topological invariants for the time evolution operator in two and three dimensions have been constructed recently Oka and Aoki (2009); Kitagawa et al. (2010); Rudner et al. (2013). Yet a systematic classification of these invariants analogous to the periodic table of symmetry protected topological phases Kitaev (2009); Schnyder et al. (2008) is still to be achieved.

In this paper, we identify new topological phenomena in a lattice integer quantum Hall (QH) system under cyclic driving with period . For fixed magnetic field, variations of the driving parameter induce topological phase transitions where the Chern numbers of the quasienergy bands change. We find multiple phases of the driven QH system featuring counter-propagating chiral edge modes at the each edge, and show they are robust against disorder. In particular, there appear “-modes”, pairs of edge modes with opposite chirality at quasienergy . These anomalous edge modes differ from those found previously in other driven two-dimensional (2D) lattice models, where the edge modes at quasienergy all propagate in the same direction and subsequently their number can be inferred either from the Chern number or the winding number Kitagawa et al. (2010); Rudner et al. (2013). Here, these known topological invariants can not predict the number of edge modes of each chirality, but only their difference. For example, we find two phases (phase A and D below) having the same set of Chern and winding numbers but very different edge state spectra. Our analysis suggests that symmetry of the time evolution operator has to be included to fully characterize and understand the topological properties of driven systems.

Our work is motivated by recent experimental achievements of artificial magnetic field for ultracold atoms Lin et al. (2009); Aidelsburger et al. (2011) and temporal modulation of optical lattices Struck et al. (2012); Hauke et al. (2012). We consider a model consisting of (spinless) fermionic atoms loaded onto a square optical lattice. Each site is labeled by vector , where , are integers, () is the unit vector in the () direction, and the lattice spacing is set to be the length unit. The tight binding Hamiltonian has the form

(1) |

Here, is the Wannier state localized at site . () is the nearest neighbor hopping along the () direction. We assume a uniform synthetic magnetic field is applied in the direction, and work in the Landau gauge, , . The flux per plaquette, in units of the flux quantum , is . Field gives rise to the Peierls phase factor in the hopping. For static , , is the well known Hofstadter model Hofstadter (1976).

We investigate a class of periodically driven quantum Hall systems described by above, but with and being periodic functions of time . We will focusing on the following driving protocol

(2) |

Namely, within one period , the hopping along is turned on during the interval , while the hopping along is turned on during the interval .We then have two independent driving parameters, . While it is hard to achieve in solid state systems, temporal modulation of or is straightforward to implement for cold atoms in optical lattices, e.g., by simply tuning the intensity of the laser. In the limit and const, the driving protocol becomes

(3) |

i.e., the hopping is only turned on when , with any integer. In this limit, , . We will simplify refer to systems described by (2) or (3) as kicked quantum Hall systems, because (3) resembles the well studied kicked rotors Thakurathi et al. (2013).

The time evolution operator of the system, defined by , has the formal solution , where denotes time-ordering and we set throughout. The discrete translation symmetry leads to a convenient basis , defined as the eigenmodes of Floquet operator ,

Here the quasienergy , by definition, is equivalent to for any integer and lives within the quasienergy Brillouin zone (QBZ), . For rational flux , is a matrix in momentum space and there are quasienergy bands. For convenience, we label the lowest band within the QBZ with , and the subsequent bands at increasingly higher quasienergies with . Correspondingly, we call the gap below the -th band the -th gap. For example, the gap around is the first gap. The Chern number for the -th quasienergy band can be defined analogous to the stationary case Thouless et al. (1982)

where the integration is over the magnetic Brillouin zone, and is the -th eigenwavefunction of .

Figure 1 displays four representative quasienergy spectra of a finite slab of length in the direction under periodic driving (2). As in static QH systems, we observe edge states forming within the quasienergy gaps. Consider the left edge () and let us denote the number of chiral edge modes propagating in the () direction by (). For driven 2D systems, the Chern numbers are generally insufficient to predict . Instead, as shown by Rudner et al Rudner et al. (2013), the net chirality of the edge modes inside the -th quasienergy gap, , is given by the following winding number

Here corresponds to respectively, and is a smooth extrapolation of Rudner et al. (2013)

where is the effective Hamiltonian with the branch cut of the logarithm chosen at quasienergies within the -th gap. Ref. Rudner et al. (2013) showed the Chern numbers can be inferred from the winding numbers by .

The quasienergy spectra (Fig. 1) manifest a few nice symmetries of the Floquet operator . Related symmetries have been discussed for the stationary Hofstadter Hamiltonian Wen and Zee (1989). Firstly, magnetic translational symmetry of (and ) dictates that an isolated band has -fold degeneracy for flux and its Chern number satisfies the Diophantine equation, where is an integer Dana et al. (1985). For and , etc. This forces all the quasienergy bands to have nonzero Chern numbers differing by multiples of 3. Secondly, is invariant under spatial inversion , (in the slab geometry). Thus, an edge state solution implies another edge state at with the same quasienergy and localized at the opposite edge. Thirdly, has a discrete chiral (sublattice) symmetry Schnyder et al. (2008): , where stands for staggered gauge transformation, , with . In recipocal space, amounts to a shift in , Wen and Zee (1989). It follows that for in the slab geometry, , where operator performs the local gauge transformation . Therefore, if is a quasienergy eigenvalue, e.g. an edge state solution, so is at shifted momentum . Two such edge states at and reside at the same edge. This will have a significant consequence for edge modes at the QBZ boundary, where and become equivalent to each other.

Applying the theoretical analysis outlined above, we obtain Fig. 2, the “phase diagram”
of the kicked quantum Hall system in terms of two independent
driving parameters, and .
It showcases four representative phases ^{1}^{1}1The term phase used here is not to be confused with the many body ground state or the thermodynamic
phase. It refers to parameter regimes of periodically driven systems with characteristic spectral and topological properties., labelled by A to D, for flux .
All of them feature three well defined quasienergy bands and three gaps, while
the spectrum in the rest of the phase diagram is largely gapless.
The corresponding spectrum of each phase in the slab geometry can be found
in Fig. 1. The table in Fig. 2 summarizes what we know about each phase:
the number of edge modes on the left edge propagating in the
direction, , inside the -th gap;
the winding number of the -th gap; and the Chern number
of the -th band. Note that and are calculated independently
from the bulk spectrum.
At the phase transition points where the gap closes,
the Chern numbers always change by a multiple of ,
consistent with the Diophantine equation Dana et al. (1985).
In what follows, we discuss in turn each of these phases.

(A). The main features of phase A can be understood by considering the fast driving limit, . The effective Hamiltonian , takes the same form of in Eq. (1), only with the bare hopping replaced by the effective hopping The driven system in phase A stroboscopically mimics a static QH system with the same flux but renormalized hopping. In particular, there is no edge state crossing the gap centered round .

(B). Phase B highlights a remarkable consequence of periodic driving: there are now two chiral edge modes inside the second and third gap. This is in sharp contrast to phase A, not only in the number of edge modes, but also in their chirality. Thus, simple periodic modulations of hopping proposed here is sufficient to change both the number and the chirality of edge states, and the Chern numbers of the bands. More importantly, phase B contains a pair of counter-propagating edge modes, dubbed “-modes”, inside the first gap at the QBZ boundary . These two edge modes, shown in blue for the left edge, have to come in pairs due to the chiral (sublattice) symmetry defined above: an edge mode crossing the QBZ boundary at implies another edge mode also crossing the QBZ boundary at . They are guaranteed to have opposite group velocity because they are related by . Such pairs of -modes are reminiscent of, and of course fundamentally different from, the counter-propagating edge modes protected by time-reversal symmetry in quantum spin Hall effect Kane and Mele (2005). The dispersion of the two -modes around quasienergy , labelled by and , can be formally described by a 1D Dirac Hamiltonian with chiral symmetry, . Note that , so in this basis. After a rotation to a basis where is diagonal, , demonstrating that belongs to class AIII of symmetry protected gapless 1D Dirac Hamiltonians as classified systematically by Bernard et al Bernard et al. (2012). Thus, perturbations obeying the chiral symmetry, e.g. small variations in the hopping or the magnetic flux, cannot open a gap Bernard et al. (2012).

We have further examined the robustness of the -modes against static on-site perturbations of the form , which break the chiral symmetry. Kinematically any potential with a finite Fourier component tends to mix the two modes. However, we find static perturbations including single impurity, staggered potential along , and random disorder potential do not open a gap around quasienergy . This is verified by numerically solving for the spectra of finite lattices of dimension . To resolve the number of edge states within the first gap, we define spectral function , where the sum over is restricted to the left half of the slab, and are the -th quasienergy and the corresponding eigenwavefunction, respectively. As shown in Fig. 3, for is peaked at two different values, with a separation by , suggesting two edge modes at and near despite the disorder. These results seem to indicate that the stability of the -modes has a topological origin. A full understanding however is still lacking.

Previous work on driven 2D lattice models Kitagawa et al. (2010); Rudner et al. (2013) also found chiral edge modes at . But those -modes all have the same chirality. As a result, the number of -modes can be predicted from the winding number , demonstrating the bulk-boundary correspondence Rudner et al. (2013). In contrast, here the -modes always come in pairs, so the net chirality is zero, . The knowledge of the winding or Chern numbers therefore is insufficient to predict the number or the chirality of the -modes.

(C) Phase C is very similar to phase B. The only difference is that there are 4 (instead of 2 in phase B) chiral edge modes propagating in the same direction inside the second and third gap. This is yet another example that Chern numbers of the quasi-energy bands can be controlled by periodic driving.

(D) Phase D is qualitatively different from all other phases. Firstly, near the QBZ boundary, there are two pairs of counter-propagating -modes, . Secondly, the edge states within the second and third gap also contain counter-propagating modes: two of the edge modes propagate in the same direction, but the remaining one propagates in the opposite direction. For example, , . Although phase D has exactly the same set of and as phase A, it has counter-propagating edge modes in all three quasienergy gaps that are robust against weak disorder.

The evolution of the edge states and the successive phase transitions as are varied can be captured by a simple model, a two-leg ladder extending in the direction. For flux , the Floquet operator of the ladder is , where the ’s are the Pauli matrices in the orbital space. It follows that the effective Hamiltonian of the ladder

with arccos. Thus, the quasienergy spectrum has two bands (branches),

Figure 3 shows the ladder spectrum for and (phase B), which agrees remarkably with the edge modes shown in Fig. 1. As is increased, both the curvature and the width of the bands increase. Beyond a critical value , the top of the band (and the bottom of the band) grows beyond the QBZ, and re-enters from the opposite side of the QBZ. Consequently, the number of states crossing the QBZ boundary jumps from 0 to 4, marking a transition from phase A to phase B. From this perspective, the pair of -modes results directly from the winding of quasienergy across the QBZ boundary as driving in the -direction () is increased. For , both the top and bottom of exceed the QBZ, giving rise to two pairs of -modes at each edge in phase D. When folded into the QBZ, they intrude into the second and third gap, leading to the anomalous edge mode propagating in the “wrong” direction.

The anomalous edge modes unique to periodically driven QH system can be detected experimentally by momentum-resolved radio-frequency spectroscopy Stewart et al. (2008), which measures the spectral function . Atoms occupying the -mode at quasienergy absorb radio-frequency photon and undergo a vertical transition to an empty hyperfine state which can be subsequently imaged. For example, in phase B, the measured spectral function will feature peaks at and energy . Alternatively, the edge currents can be probed by quantum quenches that convert them into density patterns Killi and Paramekanti (2012) or following the recent proposal of Ref. Goldman et al. (2013).

###### Acknowledgements.

We thank Brandon Anderson, Michael Levin, Xiaopeng Li, Takuya Kitagawa, and Mark Rudner for helpful discussions. This work is supported by AFOSR (FA9550-12-1-0079), NSF (PHY-1205504), and NIST (60NANB12D244). EZ also acknowledges the support by NSF PHY11-25915 through KITP.## References

- Berry (1984) M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
- Aharonov and Anandan (1987) Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987).
- Oka and Aoki (2009) T. Oka and H. Aoki, Phys. Rev. B 79, 081406 (2009).
- Lindner et al. (2011) N. H. Lindner, G. Refael, and V. Galitski, Nature Physics 7, 490 (2011).
- Kitagawa et al. (2010) T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010).
- Lindner et al. (2013) N. H. Lindner, D. L. Bergman, G. Refael, and V. Galitski, Phys. Rev. B 87, 235131 (2013).
- Kitagawa et al. (2011) T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, Phys. Rev. B 84, 235108 (2011).
- Jiang et al. (2011) L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Phys. Rev. Lett. 106, 220402 (2011).
- Gu et al. (2011) Z. Gu, H. A. Fertig, D. P. Arovas, and A. Auerbach, Phys. Rev. Lett. 107, 216601 (2011).
- Rudner et al. (2013) M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Phys. Rev. X 3, 031005 (2013).
- Thakurathi et al. (2013) M. Thakurathi, A. A. Patel, D. Sen, and A. Dutta, (2013), arXiv:1303.2300 .
- Rechtsman et al. (2013) M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
- Thouless (1983) D. J. Thouless, Phys. Rev. B 27, 6083 (1983).
- Kitaev (2009) A. Kitaev, AIP Conf. Proc. 1134, 22 (2009).
- Schnyder et al. (2008) A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008).
- Lin et al. (2009) Y. J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V. Porto, and I. B. Spielman, Nature 462, 628 (2009).
- Aidelsburger et al. (2011) M. Aidelsburger, M. Atala, S. Nascimbène, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).
- Struck et al. (2012) J. Struck, C. Ölschläger, M. Weinberg, P. Hauke, J. Simonet, A. Eckardt, M. Lewenstein, K. Sengstock, and P. Windpassinger, Phys. Rev. Lett. 108, 225304 (2012).
- Hauke et al. (2012) P. Hauke, O. Tieleman, A. Celi, C. Ölschläger, J. Simonet, J. Struck, M. Weinberg, P. Windpassinger, K. Sengstock, M. Lewenstein, and A. Eckardt, Phys. Rev. Lett. 109, 145301 (2012).
- Hofstadter (1976) D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).
- Thouless et al. (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
- Wen and Zee (1989) X. Wen and A. Zee, Nuclear Physics B 316, 641 (1989).
- Dana et al. (1985) I. Dana, Y. Avron, and J. Zak, J. Phys. C 18, L679 (1985).
- (24) The term phase used here is not to be confused with the many body ground state or the thermodynamic phase. It refers to parameter regimes of periodically driven systems with characteristic spectral and topological properties.
- Kane and Mele (2005) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).
- Bernard et al. (2012) D. Bernard, E.-A. Kim, and A. LeClair, Phys. Rev. B 86, 205116 (2012).
- Stewart et al. (2008) J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature 454, 744 (2008).
- Killi and Paramekanti (2012) M. Killi and A. Paramekanti, Phys. Rev. A 85, 061606 (2012).
- Goldman et al. (2013) N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. 110, 6736 (2013).