Cosine Edge Mode in a Periodically Driven Quantum System
CCosine Edge Mode in a Periodically Driven Quantum System
Indubala I. Satija and Erhai Zhao
Department of Physics and Astronomy, George Mason University, Fairfax, VA 22030, USA
Time-periodic (Floquet) topological phases of matter exhibit bulk-edge relationships that are more complexthan static topological insulators and superconductors. Finding the edge modes unique to driven systems usuallyrequires numerics. Here we present a minimal two-band model of Floquet topological insulators and semimetalsin two dimensions where all the bulk and edge properties can be obtained analytically. It is based on the extendedHarper model of quantum Hall effect at flux one half. We show that periodical driving gives rise to a series ofphases characterized by a pair of integers. The model has a most striking feature: the spectrum of the edgemodes is always given by a single cosine function, ω ( k y ) ∝ cos k y where k y is the wave number along theedge, as if it is freely dispersing and completely decoupled from the bulk. The cosine mode is robust againstthe change in driving parameters and persists even to semi-metallic phases with Dirac points. The localizationlength of the cosine mode is found to contain an integer and in this sense quantized. Robust boundary or edge modes are hallmarks of topo-logical insulators and superconductors [1, 2]. They can beviewed as the “holographic duals” of the bulk through thebulk-boundary correspondence [2]. While the existence of theedge modes are intuitively understood by the standard topo-logical arguments and firmly established mathematically, e.g.,by the index theorem, finding their exact dispersion from thebulk Hamiltonian usually requires involved procedures. Takethe Harper model of integer quantum Hall effect for example[3, 4]. It describes non-interacting fermions on two dimen-sional (2D) lattices in the presence of a magnetic field. Astandard way to obtain its edge mode is to introduce the trans-fer matrix and then solve a higher order equation [5, 6]. Alter-natively, the Hamiltonian of a finite system, such as a slab, isdiagonalized numerically. The dispersions of the chiral edgemodes are rarely given by simple, analytic functions.Recently, time-periodic quantum systems, such as a pieceof graphene irradiated by a driving light field [7], are foundto develop interesting topological phases and edge modes thatmay or may not have a static analog. New concepts are in-troduced to describe the unique properties of these so-calledFloquet topological insulators [7, 8]. A number of topologi-cal invariants have been constructed from the time-evolutionoperator U ( t ) [9–11]. For 2D lattice systems, the point-likesingularities in the phase bands during the time evolution arerelated to the winding number which is equal to the net chi-rality of the edge modes in given quasienergy band gap [12].Non-interacting Floquet topological insulators can be classi-fied according to the Altland-Zirnbauer symmetry classes andspatial dimensions by decomposing the unitary evolution intotwo parts [13]. For example, a 2D Floquet insulator in class Ais characterized by Z × Z , i.e. a pair of integers, rather thanthe familiar Z or Z number. Despite the progress, the bulk-boundary correspondence in Floquet systems remains onlypartially understood [13]. Compared to their static counter-parts, the Floquet edge modes are intrinsically more complex.Finding their dispersion relies even more on numerical analy-sis. It is therefore desirable to construct models for which theFloquet edge modes are described by elementary functions.In this letter, we present an analytically solvable period-ically driven lattice model in two dimensions. It only hastwo bands and generalizes the Harper model at flux one-half( π flux) by allowing the hopping amplitudes to vary periodi- cally in time. The time evolution of this Floquet system takesthe form of (momentum-dependent) successive rotations ofpseudo-spin / . This makes it possible to analyze its topo-logical properties analytically. Although the phase diagram ofthe model contains a rich collection of topologically distinctphases characterized by the Chern and winding numbers, theFloquet edge states are invariably described by a single co-sine function across all quasienergy gaps and for all insulatingand semi-metallic phases. Relatedly, the localization (decay)length of the cosine edge mode is dictated by an analyticalformula that resembles a quantization condition. Extended Harper model . Consider spinless fermions hop-ping on a square lattice within the xy plane subject to a mag-netic field in the z -direction [4]. The magnetic flux threadingeach square plaquette is set to φ in unit of the flux quantum.Following Thouless [14], we further include hopping J d be-tween the next-nearest-neighbor sites (along the diagonals ofthe square lattice). The resulting system, referred to as theextended Harper model, is described by the Hamiltonian H s = − (cid:88) r [ J x c † r +ˆ x c r + J y e i nπφ c † r +ˆ y c r + J d e i ( n + )2 πφ ( c † r +ˆ x +ˆ y + c † r − ˆ x +ˆ y ) c r ] + h.c. (1)where r = n ˆ x + m ˆ y labels the lattice sites with n, m beingintegers, and c † r creates a fermion at site r . We set the lat-tice spacing to be one and work in the Landau gauge, so thevector potential A x = 0 and A y = nφh/e . The model forarbitrary flux φ was discussed in detail in Refs. 15 and 16.A particularly fascinating aspect of the system is a new typeof critical phase termed “bicritical” when J d /J x exceeds / [16]. Recently, its highly nontrivial mathematical propertieswere analyzed in Ref. 17.We will focus on the case φ = 1 / as recently realizedin cold atoms experiment [18]. Then H s reduces to a two-band model. It is well known that if only nearest neigh-bor hoppings J x and J y are allowed, its spectrum is gap-less and Dirac-like around zero energy. For finite diago-nal hopping, α ≡ J d /J x (cid:54) = 0 , the energy spectrum E k = ± J x (cos k x + λ cos k y + 4 α sin k x sin k y ) / becomesgapped with Chern numbers of the two bands being ± [15].Here λ = J y /J x is the x − y hopping anisotropy. A two-band model of Floquet topological matter . We now a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p generalize H s into a periodically driven model by allowingthe hopping amplitudes to vary periodically with time t , fol-lowing Ref. 19 and 20. Assume that for < t < T , onlythe x -hopping J x and the diagonal hopping J d are present. Incrystal momentum space, the Hamiltonian is a × matrix, H = − J x cos k y σ x + 4 J d sin k x sin k y σ y . (2)Here the σ ’s are Pauli matrices in the pseudo-spin space de-scribing the sublattice degrees of freedom (each unit cell con-tains two sites). For T < t < T , i.e. for a duration of T ≡ T − T , only J y is turned on, H = − J y cos k y σ z . (3)The Hamiltonian is periodic in time, H ( t + T ) = H ( t ) , andpiecewise constant. We will refer to this as square wave driv-ing [19]. The time evolution operator for a full period T con-sists of two successive rotations [21] in spin space, U ( T ) = e − iH T e − iH T = e iχ σ z e iχ (cos ζσ x +sin ζσ y ) , (4)with the two rotation angles given by χ = 2 J x T (cid:113) cos k x + 4 α sin k x sin k y ,χ = 2 J y T cos k y , (5)and tan ζ = 2 α tan k x sin k y .Alternatively, we can generalize H s to a periodically kickedmodel [19, 22]. Assume J x and J d are held constant, while J y is only turned on when t is multiples of the period T , H ( t ) = H + H (cid:88) m δ ( t/T − m ) , (6)where m is an integer. The one-kick evolution operator U ( T ) is still given by Eq. (4)-(5) with the replacement T , T → T .In fact, periodical kicking can be viewed as the following limitof square wave driving: T → T , T → with J y T fixed atsome constant. The topological properties of these two typesof models are thus identical. Without loss of generality, wewill focus on the kicked model below. Its parameter spaceincludes the hopping ratio α and two dimensionless drivingparameters ¯ J x ≡ J x T /π and ¯ J y ≡ J y T /π .The eigenvalues of U ( T ) have the form e − iω n T with ω n called the quasienergy. The effective Hamiltonian is definedby U ( T ) = e − iH eff T . Even though H and H do not com-mute, the two rotations in U ( T ) can be combined into a singlerotation around some axis ˆ n by an angle ωT , U ( T ) = e iωT σ · ˆ n . (7)Eq. (7) automatically diagonalizes U . The quasienergies arejust ± ω , reflecting particle-hole symmetry, with ω given by cos( ωT ) = cos χ cos χ . (8)Eq. (8) is one of the key results of our paper. The ef-fective Hamiltonian has the form of quantum spin 1/2 in a k -dependent magnetic field, H eff ( k ) = σ · B eff ( k ) with B eff ( k ) = ωT ˆ n ( k ) . The direction ˆ n ( k ) is given by n x = sin χ cos( χ + ζ ) / sin( ωT ) , n y = sin χ sin( χ + ζ ) / sin( ωT ) , and n z = cos χ sin χ / sin( ωT ) .The periodically kicked model Eq. (6) features a rich col-lection of (Floquet) phases as the parameters α , ¯ J x and ¯ J y arevaried. Each phase has its own characteristic bulk quasienergyspectrum and the associated topological invariants and edgestates. The phase diagram can be determined by analyzing Eq.(8). Two examples along different cuts in the parameter spaceare given in Fig. 1. Fig. 2 illustrates the quasienergy spectraof four representative phases in the slab geometry where boththe bulk band structure and the edge states are visible. Insulating phases characterized by a pair of integers . Thephase diagram Fig. 1(a) for ¯ J x < / and fixed α < / is very simple. A series of phases, labeled by I , , I , etc.,appear consecutively as ¯ J y goes through integer multiples of / . All these phases have two finite gaps at quasienergy and π/T and two well separated bands. The 0-gap and π -gapare characterized by the winding number w and w π respec-tively. And we denote the Chern number for the band at posi-tive (negative) energies by C + ( C − ). The Chern numbers andthe winding numbers are related by, e.g., C + = w π − w and C + = − C + . Thus, each phase can be labelled by a pair ofintegers w and w π (or equivalently C + and w etc.). It isa Floquet Insulator (I), so we refer to it as phase I w ,w π . Inparticular, phase I , corresponding to fast driving (small T )is identical to the static quantum Hall state at flux 1/2 ana-lyzed in Ref. 15. As shown in Fig. 2(a), w π = 0 implies noedge states inside the π -gap. In contrast, all the other Floquetphases in Fig. 1(a) have finite number of chiral edge modesinside the π -gap [e.g. I , in Fig. 2(c)] and therefore they haveno static analog .One notices that in Fig. 1(a), the Chern number C + simplyalternates between 1 and -1, and the phase transition pointsare equally distributed. These can be understood from Eq. (8).For small ¯ J x and χ , cos( ωT ) (cid:39) cos χ . So the quasienergy ω crosses or π/T when χ equals to nπ or (2 n +1) π for in-teger n . In either cases, the two quasienergy bands touch eachother, triggering a change in the band Chern number. Accord-ing to Eq. (5), this occurs at J y T = nπ where the gap closesat zero energy and w changes by 2, or at J y T = ( n + 1 / π where the π -gap closes and w π jumps by 2.The sequence of odd w appearing at the 0-gap and even w π at the π -gap found here in Fig. 1(a) is reminiscent of a similarsequence in the static extended Harper model in the vicinityof flux / as described in Ref. [23]. It seems as if by vary-ing the parameter ¯ J y , the driven system for fixed flux is able to“access” the topological edge states of various gaps of the cor-responding static system with a flux value slightly away from / . We recall that the winding numbers correspond to the(infinitely many) solutions of the Diophantine equation [24].As a result, the entries in each sequence are related by modulo , the denominator of the flux / . Semi-metallic phases and Dirac points . The phase diagrambecomes more complicated for larger values of ¯ J x and α . Oneexample is shown in Fig. 1(b) along for fixed ¯ J x = 0 . and α = 1 / . Besides the insulating phases with two gaps suchas I , and I , discussed above, new phases emerge whichhave only one well-defined quasienergy gap but Dirac pointsat ω = 0 or π/T in the spectrum. We refer to them as semi-metallic (S) phases. For example, the phase S , is gapped at ω = π/T but gapless at ω = 0 . Its spectrum shown in Fig.2(b) is analogous to the familiar Dirac semimetal: the twobands become degenerate at zero energy for certain k -points.These locations can be found from Eq. (8). A solution for ω = 0 requires that cos χ cos χ = 1 , i.e., χ = n π, χ = n π, n + n = even, (9)which only has solutions at isolated k -points according to Eq.(5). The spectrum of the other semi-metallic phase ˜S , isillustrated in Fig. 2(d). In contrast to S , , it is gapped at ω = 0 but has Dirac points at quasienergy ω = π/T . It istherefore a Floquet semimetal [25]. We distinguish it from Sby tilde to emphasize that it has no counterpart in the staticmodel H s . The locations of its Dirac points are also given byEq. (9) but with n + n = odd .Floquet semimetals have yet to be classified systemati-cally and there is no generally accepted convention to labelthem. Here we find it proper to identify the semi-metallicphases unambiguously using a pair of integers ( w , w π ) , e.g.S w ,w π and ˜S w ,w π , to indicate the number of edge modesat quasienergy 0 and π/T , as done above for the insulatingstates. It is attempting to label the phase S , by only oneinteger w π = 2 . But this is insufficient and misses one im-portant point: as shown in Fig. 2(b), edge modes are presentnot only inside the π gap but also at all other energies coexist-ing with the bulk states , including the Dirac points at ω = 0 .In the latter case, the edge states appear to be “caged”, i.e.bounded in k y by the Dirac points. They are robust and can beviewed as the continuation of the edge states of phase I , de-spite the gap closing at the Dirac points. Similar coexistenceof the edge and bulk states is also observed in phase ˜S , andmany other semi-metallic phases, such as the SS phases withDirac points both at quasienergy zero and π/T shown in Fig.1(c). To summarize, the semi-metallic states found here aredistinct from static Dirac semimetals or gapless Floquet su-perconductors studied in Ref. [21]. They are characterizedby two integers. The persistence of the edge modes into thesemi-metallic phases will become more transparent once wework out its analytical expression below. Cosine edge mode and its localization length . Given theplethora of insulating and semi-metallic phases found in thismodel, it is natural to expect that the energy-momentum dis-persion of the edge states, ω edge ( k y ) , to change from onephase to another or from one gap to another as in the staticHofstadter problem. It thus comes as a surprise that ω edge ( k y ) takes the same simple form for all phases, including the semi-metallic phases, regardless of the driving parameters. Empir-ically, the functional form of ω edge ( k y ) becomes apparent ifthe spectra in Fig. 2 are plotted in the repeated zone schemefor both the quasienergy and the quasimomentum.We solve for ω edge ( k y ) from the bulk dispersion Eq. (8)and show it is simply the cosine function by using a well es-tablished method in band theory [26–28]. In order to enlargethe Hilbert space to accommodate localized states, we allowBloch wavevectors k to take complex values and analytically J y T/ ! ω T/ !
10 0 1 2 3 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 C - w J y T/ ! S ( a )( b ) I C + w π I I I I I I I I I S w π J y T/ ! SS ( c ) I S SS S I ~ FIG. 1. Phase diagrams of the periodically kicked model Eq. (6). (a)Schematic of the quasienergy bands (shaded region) and gaps (emptyregion) as ¯ J y is increased for fixed ¯ J x < / and α < / . Aseries of insulating (I) phases I w ,w π are identified by examining theband Chern numbers C + , C − and the winding numbers w , w π . (b)Topological phases with fixed α = 1 / and ¯ J x = 0 . . S , and ˜S , are semi-metallic (S) phases described in the main text. Theirspectra are shown in Fig. 2(b) and 2(d). (c) Topological phases alongthe line α = ¯ J x = ¯ J y . SS denotes (double) semi-metallic phaseswith Dirac points at both quasienergy 0 and π/T .FIG. 2. Quasienergy spectrum of a slab of finite width L x = 61 in the x -direction but periodic in y -direction. Panel (a), (b), (c), (d)are for ¯ J y = 0 . , . , . and . , corresponding to phase I , , S , ,I , , and ˜S , . Here α = 1 / , ¯ J x = ¯ J y except for (d), ¯ J x = 0 . . continue H ( k ) (and U ) which becomes non-Hermitian in gen-eral. The edge states correspond to real eigenenergies (insidethe bulk gap) of H ( k ) for complex values of k . For the slabgeometry in our case, it is sufficient to let k x = k r + ik i inEq. (8), where k r , k i ∈ R are the real and imaginary part of k x respectively. In order to guarantee a real solution for ω , wemust require χ to be real, which in turn requires that, after alittle algebra, sin(2 k r ) = 0 . It has two types of solutions: I : k r = ± π/ (10) II : k r = 0 . (11)The numerical fact that the edge spectrum does not dependon J x for fixed J y implies that cos χ collapses to a constant.The continuity of the edge spectrum in the limit of J x T → further fixes the constant to be 1. Namely, cos χ = 1 or χ = 2 n e π, n e ∈ Z . (12)Then Eq. (8) simplifies to cos( ωτ ) = cos χ which leads to ω edge ( k y ) = 2 J y cos k y . (13)We have checked that Eq. (13) fits exactly the numericallyobtained edge spectra of finite slabs, e.g., those in Fig. 2. Theinverse localization/decay length k i of the edge mode can befound from Eq. (12) and the expression for χ , sinh k I i = 1 − ( n e / ¯ J x ) − (2 α sin k y ) − , (14) sinh k II i = ( n e / ¯ J x ) − − (2 α sin k y ) , (15)for type I and type II solution respectively.The decay length appears to be “quantized” due to the pres-ence of integer n e in Eq. (15). The natural question is: howdoes the system choose this quantum number ? Our detailednumerical studies of the slab geometry reveals that n e is noth-ing but the integer part of ¯ J x , n e = [ ¯ J x ] . (16)The value of n e seems to encode the condition that the decaylength diverges when the cosine edge mode merges with thebulk bands. Presently, a deeper understanding of this simplebut fascinating result remains elusive.In the fast driving limit, ¯ J x → , the localization lengthsof the edge modes correspond to n e = 0 and therefore thestatic model permits only type I localization. In the drivenmodel, for α < / , the edge states are of type I while for α > / , they switch between type I and type II as k y varies.Interestingly, the switch from type I to II occurs at values of k y = k ∗ y where the edge states cross bulk at special pointswhere the bulk spectrum is “pinched”, i.e. independent of k x .The pinching occurs when J y cos k ∗ y = ± (2 m + 1) π/ , cor-responding to quasienergy ± π/ . We note that the existenceof two distinct types of localized edge modes in our modelis reminiscent of two types of localized regimes in the staticextended Harper model for irrational flux [16]. However, un-derstanding the possible relationships between the localiza-tion characteristics of the static and the driven system is stilllacking. Discussions . Our periodically kicked model thus presentsa marked dichotomy: the dispersion of the cosine edge stateonly depends on ¯ J y , while its decay length only depends on ¯ J x . On the ¯ J x − ¯ J y plane, for increasing ¯ J y but fixed J x , thecosine mode will stretch in amplitude and continuously windcross the quasienergy Brillouin zone boundary π/T , givingrise to ever-increasing number of chiral edge modes in the 0and π gap shown in Fig. 1(a). When ¯ J x is increased for fixed ¯ J y , e.g. ¯ J y = 1 / , each time ¯ J x reaches an integer value, n e jumps by one and the gap closes at ω = 0 . The distinctionbetween the bulk and edge states gets lost but the edge statespectrum stays the same.The simple cosine dispersion of the edge mode in our Flo-quet system is identical to that of a free particle hopping on aone-dimensional chain along the y direction. In other words,the edge states seem “perfectly” localized as if they do notcouple at all to sites away from the edge, even though the hop-ping amplitudes J x and J d are finite. The lack of diffusion isreminiscent of Creutz’s ice-tray model [29], where edge statesform at the ends of a two-leg ladder with J x and J d at flux / due to deconstructive interference. Note however H in ourmodel is periodic in y and not one-dimensional. Furthermore,the wave function of the cosine edge mode varies with time.One single function describing all the edge modes reveals thecontinuity of the edge spectrum and the robustness of the edgestate throughout the phase diagram. For example, the persis-tence of the edge mode into the semi-metallic phases becomevery natural and easy to understand. Similar picture emergesfor other driving protocols, e.g. with J d time-dependent but J x , J y held constant, and also for triangular lattice.In summary, the model introduced and solved here bringsa new perspective to the active field of systematically under-standing the topological properties of time-periodic quantumsystems. Our simple Floquet system exhibits a rich variety oftopological insulating phases I w ,w π characterized by a pair ofintegers as well as semi-metallic phases S w ,w π , ˜S w ,w π withDirac points developed at quasienergy zero or π/T and Flo-quet edge states existing at all quasienergies. Its most remark-able feature is the simple cosine dispersion of the edge stateacross the entire phase diagram. The cosine edge mode alsoshows a nontrivial behavior in its decay length. In additionto these new features and the pedagogical value of the modelitself, the construction may be generalized to analyze drivensystems in other symmetry classes and spatial dimensions.This work is supported by by AFOSR grant numberFA9550-16-1-0006 and NSF PHY-1205504 (EZ). [1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045 (2010).[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057 (2011).[3] P. G. Harper, Proceedings of the Physical Society. Section A ,879 (1955).[4] D. R. Hofstadter, Physical Review B , 2239 (1976).[5] Y. Hatsugai, Phys. Rev. Lett. , 3697 (1993).[6] B. A. Bernevig and T. L. Hughes, Topological insulatorsand topological superconductors (Princeton University Press, 2013).[7] T. Oka and H. Aoki, Physical Review B , 081406 (2009).[8] N. H. Lindner, G. Refael, and V. Galitski, Nature Physics ,490 (2011).[9] T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Physical Re-view B , 235114 (2010).[10] M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, PhysicalReview X , 031005 (2013). [11] D. Carpentier, P. Delplace, M. Fruchart, and K. Gawedzki,Phys. Rev. Lett. , 106806 (2015).[12] F. Nathan and M. S. Rudner, New Journal of Physics , 125014(2015).[13] R. Roy and F. Harper, arXiv preprint arXiv:1603.06944 (2016).[14] D. J. Thouless, Phys. Rev. B , 4272 (1983).[15] Y. Hatsugai and M. Kohmoto, Phys. Rev. B , 8282 (1990).[16] J. H. Han, D. J. Thouless, H. Hiramoto, and M. Kohmoto, Phys.Rev. B , 4272 (1994).[17] A. Avila, S. Jitomirskaya, and C. Marx, arXiv preprintarXiv:1602.05111 (2016).[18] C. J. Kennedy, W. C. Burton, W. C. Chung, and W. Ketterle,Nat Phys , 859 (2015).[19] M. Lababidi, I. I. Satija, and E. Zhao, Phys. Rev. Lett. ,026805 (2014). [20] Z. Zhou, I. I. Satija, and E. Zhao, Phys. Rev. B , 205108(2014).[21] E. Zhao, arXiv preprint arXiv:1603.08822 (2016).[22] D. Y. H. Ho and J. Gong, Phys. Rev. B , 195419 (2014).[23] I. Satija, Butterfly in the Quantum World (IOP Concise, Morganand Claypool, 2016).[24] I. Dana, Y. Avron, and J. Zak, Journal of Physics C: Solid StatePhysics, Volume 18, Issue 22, L679 (1985).[25] R. W. Bomantara, G. N. Raghava, L. Zhou, and J. Gong, Phys.Rev. E , 022209 (2016).[26] W. Kohn, Phys. Rev. , 809 (1959).[27] V. Heine, Proceedings of the Physical Society , 300 (1963).[28] Y.-C. Chang and J. N. Schulman, Phys. Rev. B , 3975 (1982).[29] M. Creutz, Rev. Mod. Phys.73