Quantum Hall network models as Floquet topological insulators
QQuantum Hall network models as Floquet topological insulators
Andrew C. Potter, J. T. Chalker, and Victor Gurarie Department of Physics, University of Texas at Austin, Austin, TX 78712, USA Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom Department of Physics and Center for Theory of Quantum Matter,University of Colorado, Boulder, Colorado 80309, USA
Network models for equilibrium integer quantum Hall (IQH) transitions are described by unitaryscattering matrices, that can also be viewed as representing non-equilibrium Floquet systems. Theresulting Floquet bands have zero Chern number, and are instead characterized by a chiral Floquet(CF) winding number. This begs the question: How can a model without Chern number describeIQH systems? We resolve this apparent paradox by showing that non-zero Chern number is recoveredfrom the network model via the energy dependence of network model scattering parameters. Thisrelationship shows that, despite their topologically distinct origins, IQH and CF topology-changingtransitions share identical universal scaling properties.
Disorder and localization often play a vital role in sta-bilizing topological matter. In particular, these featuresare essential for the experimental observation of quan-tized Hall conductance in a variety of experimental sys-tems such as GaAs quantum wells [1, 2], graphene [3], andmagnetically doped topological insulator thin films [4].These systems all allow tuning between Hall plateauswith different quantized conductance via changing gatevoltages or external magnetic fields, resulting in a topo-logical phase transition. This transition is marked bya jump in the Chern number of the occupied electronstates and a change in the number of chiral edge statescirculating around the perimeter of the sample. It is ac-companied by a divergence in the localization length ata critical value of the chemical potential.Disorder and localization can also stabilize driven sys-tems against drive-induced heating, giving access tonew regimes of quantum coherent dynamics. Recenttheoretical advances have shown that time-periodicallydriven (Floquet) systems can exhibit new types of non-equilibrium topological phases with inherently dynami-cal properties that could not arise in the ground state ofstatic (time-independent) Hamiltonians [5–7]. Strikingexamples include chiral Floquet (CF) phases [6], whichexhibit chiral edge states, despite having only topologi-cally trivial bulk bands.Although CF edge states are reminiscent of those inIQH systems, there are important differences. Crucially,the number of CF edge states for a given Floquet op-erator is the same at all values of the (compact) quasi-energy. By contrast, the number of edge states for a(time-independent) IQH Hamiltonian is a function of(non-compact) energy, and is given by the net Chernnumber of all bulk states at lower energies. As a corollary,CF phases differ from IQH phases in that they do notexhibit charge pumping through bulk states upon adia-batic insertion of magnetic flux, and hence have vanishingHall conductance and Chern number. Instead, for non-interacting systems, the unitary time-evolution opera-tors for CF phases are characterized by an integer-valuedwinding number, χ , the chiral unitary invariant [6].In this paper, we explore a relation between these two distinct types of topological phenomena via the networkmodel introduced by Chalker and Coddington [8] to de-scribe the scattering dynamics of electrons near the quan-tum Hall plateau transition. The Chalker-Coddingtonnetwork model (CCN) is defined by a unitary matrixthat acts on a wavefunction sampled at discrete spatiallattice points in a continuum IQH system. This uni-tary can also be interpreted as the Floquet operator ofa time-periodically driven system [9], as previously dis-cussed in the context of photonic networks where it wasfound that these network models could realize both CFphases and Chern bands for appropriate network geome-tries and parameters [10–12]. We construct a periodicallytime-dependent Hamiltonian, acting on the same lattice,whose Floquet operator coincides with the CCN unitary,and demonstrate that this Floquet system hosts a CFphase, and find vanishing Chern number for all bands atall fixed network-model parameters.This correspondence uncovers a puzzle: IQH systemsare characterized by a Chern number but the CF systemhas Chern number zero. We resolve this puzzle by show-ing that the Chern number of an IQH system is recoveredby properly accounting for energy-dependence of scatter-ing phases and tunneling amplitudes in the model. Thisrelationship provides a fresh perspective on IQH systems,building on the fact that the CF Floquet operator is char-acterized by the chiral unitary invariant. Since the CCNis represented by the same unitary matrix as a CF sys-tem, values of the chiral unitary invariant can be used tolabel IQH phases within the CCN description.These observations show that the network modelequally describes the fixed-energy scattering behavior ofa wave-packet in an IQH system, and the full dynam-ics of a CF system. In particular, this implies thatthe topology changing phase transitions for disordered,non-interacting IQH and CF phases share the same uni-versal scaling properties, in agreement with recent field-theoretic analysis [13] relating sigma-models for CF andIQH phases. Network model definition –
Consider the groundstate of a non-interacting electron gas in a uniform mag- a r X i v : . [ c ond - m a t . d i s - nn ] M a y netic field and a disordered potential, with the lowestLandau level partially occupied on average. If the po-tential is smooth on the scale of the magnetic lengthand has fluctuations smaller in amplitude than the cy-clotron energy, the system is divided into spatial regionsaround potential minima where the Landau level is lo-cally fully occupied, and regions around maxima whereit is empty. Chiral edge states at the chemical potentialcirculate along the boundaries between these occupiedand empty regions. Tunneling between distinct edge seg-ments occurs near saddle-points in the potential wheretheir spatial separation is small.The network model [8] describes a simplified version ofthis picture, in which the potential is chosen so that oc-cupied and empty regions form alternate plaquettes of aregular square lattice. Edge states propagate on directedlinks of this lattice, meeting at nodes that correspondto potential saddle-points. Consider a stationary statein this continuum problem. Its wavefunction is sampledat a single point r i on each link, and represented by acurrent amplitude ψ i . Amplitudes on incoming and out-going links at a node are related by a scattering matrix,so that (referring to Fig. 1): (cid:18) ψ ψ (cid:19) = (cid:18) e iϕ e iϕ (cid:19) (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) ψ ψ (cid:19) . (1)Here, θ parameterizes tunneling while ϕ and ϕ areAharonov-Bohm phases. A similar S -matrix with θ → ¯ θ ≡ π/ − θ describes scattering at the ¯ θ nodes. Disorderis modeled by taking ϕ i to be an independent randomvariable on each link i , uniform in [0 , π ). The modelcan be specified for a closed system of N links by an N × N unitary matrix U , which is composed of 2 × r i , is represented by an eigenvector of U with eigenphase zero (mod 2 π ), such that Eq. (1) is sat-isfied at every node. In order to find the energies at whichscattering eigenphases vanish, and hence relate eigenvec-tors of U to Hamiltonian eigenstates, it is necessary toconsider the energy dependence of θ and ϕ i . From theshape of equipotentials near a saddle-point, one sees that θ increases from 0 to π/ π times) the numberof flux quanta passing through this plaquette. Random-ness in φ i arises from small random variations in the areaof plaquettes, and energy-dependence of φ i arises due toto the change in area of a plaquette with a change in thechemical potential.The behavior of the model is simplest at the extremelimits θ = 0 and θ = π/
2. In the first case, the sys-tem consists only of isolated plaquettes enclosing occu-pied regions. In the second case, it is made up of isolatedplaquettes enclosing empty regions, together with a chi-ral edge state at a boundary where the system meets anexternal empty region. Detailed numerical studies showthat the edge state is present for all θ > π/ ¯ ✓
Network model and Floquet band structure– a) schematic of Chalker-Coddington network model, b)Floquet bands ε ( k ) of the clean network model evolution op-erator at criticality, θ = π/ Floquet perspective –
These scattering dynamics de-fine a stroboscopic Floquet evolution in which we intro-duce a discrete time variable t and take ψ ( t + 1) = U ψ ( t ) , (2)where ψ ( t ) is the vector of all the link amplitudes atthe time step t [15]. Writing the eigenvalues of U as e − iε , the phases ε are referred to in this context as quasi-energies, located within a compact Brillouin zone ε ∈ ( − π, π ]. The fact that quasi-energies lie on a circle ratherthan an open line changes the topological classification ofFloquet dynamics compared to that of gapped ground-states of static Hamiltonians [5–7].Non-interacting Floquet bands in systems with con-served particle number are exhaustively classified by twointeger-valued topological invariants:1. the Chern number, C n , defined separately for eachFloquet band, n , and2. the chiral unitary index, χ ( U ), defined for the fullFloquet unitary, which characterizes the numberof chiral edge states that wrap around the quasi-energy Brillouin zone [6].The Chern numbers for the Floquet-bands of U areidentically zero [10, 11]. This can be verified by inspec-tion at the trivial and topological limiting points, θ = 0and θ = π/
2. At these points, the bulk motion consists ofshort loops around the elementary plaquettes of the net-work model (Fig. 1), indicating that the Floquet bandsadmit a strictly localized Wannier basis of orbitals, eachhaving support only on the four links of a single pla-quette. Such a localized Wannier basis implies vanish-ing Chern number [16], immediately implying that eachindividual eigenstate has zero Chern number. This con-clusion extends to all values of θ (cid:54) = π/
4, since Chernnumber may only change at a delocalization transition,which in this model occurs only at θ = π/
4. The re-gions either side of this critical point inherit the vanish-ing Chern number of their limiting points, at θ = 0 , π/ k , the matrix U is block-diagonal with blocks of the form U ( θ, k ) = θe − ik x cos θe − ik y − cos θe ik y sin θe ik x cos θ sin θ − sin θ cos θ . (3)We denote the eigenvalues of U ( θ, k ) by ε n ( θ, k ) for n ∈ { , , , } . At θ = 0 and θ = π/ ε n are inde-pendent of k , taking the values ε n = π/ πn/
2. As θ deviates from these extreme values, the bands disperse,until they touch at θ = π/ k = (0 ,
0) or k = ( π, π ) (Fig. 1). Tuningaway from θ = π/
4, the Dirac points develop a mass-gapwith an alternating sign mass for Dirac points separatedin quasi-energy by ∆ ε = π/
2. Equivalently, viewed in thethree-dimensional parameter space ( θ, k ), these degen-erate points form monopole sources of Berry flux, withoverall cancelling monopole charge, resulting in vanishingnet Chern number for all θ . Chiral Floquet invariant –
Viewing the networkmodel as a lattice Floquet system, since U has only topo-logically trivial bulk bands, any non-trivial topologicalbehavior must emerge from a non-trivial chiral unitaryinvariant, χ (cid:54) = 0. As a first step, we compare behavior at θ = 0 and θ = π/ χ can be computed simply by counting the numberof chiral edge states wrapping the quasi-energy Brillouinzone [6, 10]. There is no edge state in the first case,and one in the second case. The eigenvectors of U cor-responding to the edge state can be given explicitly. Letinteger j label links in order along the boundary, and forsimplicity set all ϕ j = 0. Then the vector with ψ j = e ikj on edge links and ψ i = 0 on all other links is an (unnor-malized) eigenvector of U with quasi-energy ε edge = k .This mode indeed wraps the quasi-energy Brillouin zoneas k varies, and is manifestly absent in the other phaseof the model ( θ ≈ χ ( U CCN ( θ )) = (cid:40) ≤ θ < π/ π/ < θ ≤ π/ χ viathe bulk-boundary correspondence established in [6], itis also reassuring to compute χ directly from the networkmodel in the bulk by considering a system with periodicboundary conditions. Here, we face an obstacle: to com-pute χ from bulk behavior alone, it is not sufficient toexamine stroboscopic times (for which the bulk motionis always trivial when all C n = 0). Instead, one must ex-amine the micro-motion within a single period. The net-work model as formulated is blind to this micro-motion,and additional choices are needed to define it (though thefinal result will be independent of these choices). V
Chern bands from the scattering network – a)Semiclassical orbits for an electron in a periodic potential anduniform magnetic field, b) Partial-set of energy bands fromthe scattering network with θ ( E ) = π (cid:16) tanh (cid:104) E + π/ π (cid:105) + 1 (cid:17) and ϕ ( E ) = E . The horizontal axis is placed at the criticalenergy E = E c for which θ ( E c ) = π/
4, c) Berry curvaturefor two bands near E = 0 for which θ respectively crosses(left panel) and does not cross (right panel) π/
4, giving totalChern number 1 or 0. All other bands have Chern numberzero.
Specifically we seek a continuous path in the space ofunitary matrices, from the identity to U . This can begenerated by a local Hamiltonian H ( t ) that acts on theHilbert space of the network model lattice for all times0 ≤ t ≤ T within the Floquet period T = 1. That is,we require U ( t ) = T e − i (cid:82) t H ( t ) dt (where T denotes timeordering) such that U ( T ) = U , the CCN unitary. This re-lation does not uniquely fix H ( t ). However, if we demandthat H ( t ) is spatially local, any such choice of H ( t ) willproduce the same value of χ . In Appendix. A, we con-struct a specific example via a coupled-wire perspective,and explicitly verify Eq. A2 via evaluation of the bulkchiral Floquet winding-number [6]. The completeness ofthe Floquet classification for lattice models [6, 18, 19]also implies that any such local generating H ( t ) mustbe explicitly time-dependent (in contrast to the staticHamiltonian for continuum Landau levels). Recovery of Chern bands –
These observations natu-rally raise the question: how can a model with zero Chernnumber describe the quantum Hall transition? We nowshow that a non-zero Chern number for the Landau bandof the continuum Hamiltonian is correctly recovered fromthe eigenvectors of U when the network model parame-ters are allowed to vary with energy in a realistic manner.For clarity, we use the clean model in this discussion al-though the results are general. Without disorder, the linkphases obey ϕ i = ϕ ( E ) for all i , where ϕ ( E ) is a mono-tonic function of energy E with an increment Φ acrossthe energy width of the disorder-broadened Landau level.If there are many magnetic flux quanta per unit cell (thenatural regime for the network model) then Φ (cid:29)
1. In-cluding ϕ ( E ), the quasi-energies of U are ε n ( k , θ ) − ϕ ( E ).Eigenenergies of the continuum problem form bands de-fined by: ϕ ( E ) = ε n ( k , θ ) + 2 πm (5)for integer m . For large Φ there are many such bands.Recalling that θ ≡ θ ( E ) is a (slowly varying) func-tion of E , the solution to Eq. (5) for each m, n defines asurface θ m,n ( k ) in the space ( θ, k ) that was introducedfollowing Eq. 3. For all but one of these surfaces, bothof the oppositely charged ( θ, k )-space monopole sourcesof Berry flux lie on the same side of the surface. Thenet Berry flux through these surfaces is therefore zero,implying zero Chern number for the associated band ofeigenstates in the continuum problem. However, thereis one exceptional pair ( m, n ) for which θ m,n ( k ) < π/ k = (0 ,
0) but θ m,n ( k ) > π/ k = ( π, π ) (or vice-versa, depending on the parity of n ). For this exceptionalpair, oppositely charged monopoles lie on opposite sidesof the surface, which therefore has a full unit of Berryflux passing though it. The associated band of eigen-states hence has unit Chern number, and summing overall bands we recover unit Chern number for the Landaulevel. A numerical demonstration for a particular energydependence is shown in Fig. 2. Discussion –
These two lines of analysis show thatthe network model equally describes both chiral Flo-quet topological insulators, and quantum Hall phases andtransitions. Specifically, these results establish a preciseequivalence of the dynamics of a wave-packet with near-constant energy in these two settings. Since the criticalproperties at topological phase transitions in both classesarise from delocalization at a fixed critical energy, thisimplies that these topologically distinct phenomena sharethe same critical properties at their (disordered) topolog-ical phase transitions. We emphasize that, despite shar-ing critical scaling properties the CF and IQH systemsare sharply distinct. Beyond their distinct bulk-topologyand edge-state structure, they exhibit qualitatively differ- ent dynamics of spatially localized wave-packets, whichremain indefinitely localized in strongly-disordered CFphases [20], but instead spread sub-diffusively in IQH sys-tems due to overlap with critically extended states [21].In the absence of interactions, the equilibrium classifi-cation and the Floquet classification have related struc-ture [7]: for each equilibrium class with classification G ,there is a corresponding set of purely dynamical Floquetphases also having classification G . Our results raise thequestion: are the critical properties of topological-phasetransitions equivalent for all of these equilibrium/Floquetphase pairs? We conjecture that the answer is affirma-tive. For example, closely related 2 d network models canbe used to establish a similar relation between topologicalphase transitions in equilibrium and Floquet chiral super-conductors with spin-rotation symmetry (spin-quantumHall effect, class C) whose critical properties correspondto percolation [22–24]. Moreover, analogous 1 d scatteringnetwork constructions can be obtained by compactifying2 d examples to extend these results to all 1 d classes [25],leading us to conjecture a general equivalence of univer-sal scaling structure of non-interacting equilibrium andFloquet topological phase transitions.The network model construction is special to non-interacting systems with elastic scattering. In interact-ing settings, the conjectured static/Floquet topologicalquantum criticality correspondence likely continues tohold for interacting MBL systems in 1 d where phase tran-sitions are characterized by RG-flow to infinite random-ness and have equivalent interacting and non-interactingscaling properties [26] By this route the 1 d Floquet modelstudied in Ref. [27, 28] is related to an Ising model, whichis known to have a network model representation [29].However, the correspondence will presumably fail for in-teracting 2 d systems, since in that context CF phasesare governed by rational-fractional-valued topological in-variants with a completely distinct structure from Chernnumber [18, 19], and because the non-interacting CF-to-trivial critical-point will broaden into an interveningthermal phase upon including interactions [30]. Acknowledgements –
We thank Yang-Zhi Chou,Matthew Foster, Itamar Kimchi, Sid Parameswaran, andRomain Vasseur for insightful discussions. This work wassupported by NSF DMR-1653007 (AP), by the EPSRCGrant No EP/S020527/1 (JTC), and by the Simons Col-laboration on Ultra-Quantum Matter, which is a grantfrom the Simons Foundation (651440, VG). [1] W. Li, G. A. Cs´athy, D. C. Tsui, L. N. Pfeiffer, andK. W. West, Phys. Rev. Lett. , 206807 (2005).[2] W. Li, G. A. Cs´athy, D. C. Tsui, L. N. Pfeiffer, andK. W. West, Phys. Rev. Lett. , 216801 (2009).[3] A. Giesbers, U. Zeitler, L. Ponomarenko, R. Yang,K. Novoselov, A. Geim, and J. Maan, Physical ReviewB , 241411 (2009).[4] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, et al., Science , 167 (2013).[5] T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys-ical Review B , 235114 (2010).[6] M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin,Phys. Rev. X , 031005 (2013).[7] R. Roy and F. Harper, Physical Review B , 155118(2017). [8] J. T. Chalker and P. D. Coddington, J. Phys. C-SolidState Physics , 2665 (1988).[9] R. Klesse and M. Metzler, International Journal of Mod-ern Physics C , 577 (1999).[10] M. Pasek and Y. Chong, Physical Review B , 075113(2014).[11] P. Delplace, M. Fruchart, and C. Tauber, Physical Re-view B , 205413 (2017).[12] P. Delplace, arXiv preprint arXiv:1905.11194 (2019).[13] K. K. Woo, D. Bagrets, T. Micklitz, and A. Altland,arXiv preprint arXiv:1910.06892 (2019).[14] R. Klesse and M. Metzler, Europhysics Letters (EPL) ,229 (1995).[15] C.-M. Ho and J. T. Chalker, Phys. Rev. B , 8708(1996).[16] D. Thouless, Journal of Physics C: Solid State Physics , L325 (1984).[17] See supplementary information for an explicit construc-tion of a time-dependent lattice-Hamiltonian that gener-ates the network model unitary.[18] H. C. Po, L. Fidkowski, T. Morimoto, A. C. Potter, andA. Vishwanath, Physical Review X , 041070 (2016).[19] L. Fidkowski, H. C. Po, A. C. Potter, and A. Vish-wanath, Physical Review B , 085115 (2019).[20] P. Titum, E. Berg, M. S. Rudner, G. Refael, and N. H.Lindner, Physical Review X , 021013 (2016).[21] J. Chalker and G. Daniell, Physical review letters , 593(1988).[22] V. Kagalovsky, B. Horovitz, Y. Avishai, and J. T.Chalker, Phys. Rev. Lett. , 3516 (1999).[23] I. A. Gruzberg, A. W. W. Ludwig, and N. Read, Phys.Rev. Lett. , 4524 (1999).[24] T. Senthil, J. B. Marston, and M. P. A. Fisher, Phys.Rev. B , 4245 (1999).[25] A. C. Potter and V. Gurarie, to appear.[26] D. S. Fisher, Physical review letters , 534 (1992).[27] W. Berdanier, M. Kolodrubetz, S. Parameswaran, andR. Vasseur, Proceedings of the National Academy of Sci-ences , 9491 (2018).[28] W. Berdanier, M. Kolodrubetz, S. A. Parameswaran,and R. Vasseur, Phys. Rev. B , 174203 (2018).[29] F. Merz and J. T. Chalker, Phys. Rev. B , 054425(2002).[30] R. Nandkishore and A. C. Potter, Physical Review B ,195115 (2014).[31] D.-H. Lee, Phys. Rev. B , 10788 (1994). Appendix A: Time-dependent generatingHamiltonian for network models
In this supplementary information, we build a local,time-dependent Hamiltonian that generates the networkmodel dynamics. In contrast to a previous construc-tion relating Floquet and network model systems [12] weuse only the Hilbert space of the network model to pro-duce a time-dependent Hamiltonian that directly gener-ates the microscopic dynamics of an arbitrary networkmodel, from which the bulk chiral Floquet invariant canbe directly computed.To construct a generating H ( t ), we adopt a coupled-wire perspective [31] by grouping the chiral segments of the model into pairs of alternating strips (Fig. 3). Webuild H ( t ) from a sequence of piecewise time-independentHamiltonians consisting of particle hoppings between dis-joint pairs α, β of sites. These Hamiltonians are sums ofterms of the form h αβ = Jσ y , where σ y is the Paulimatrix acting on the two amplitudes ( ψ α , ψ β ) T . Whenapplied for time t = λ/J this produces the unitary evolu-tion u αβ ( λ ) = e − iλσ y . Opposite chiral motion at the edgeof each one-dimensional strip is produced in the secondand third portions of the period by applying sequentialSWAP operations, u αβ ( λ = π ), on the horizontal anddiagonal bonds, shown on panels 2 and 3 of Fig. 3. Scat-tering is introduced via an initial partial swap step, byapplying u αβ ( θ ) and u αβ (cid:0) π − θ (cid:1) at the inter- or intra-wire scattering nodes respectively (panel 1 of Fig. 3).With a local generating Hamiltonian in hand, one cancompute χ directly via the formula [6] χ [ ˜ U ] = 18 π (cid:90) dtdk x dk y Tr (cid:16) ˜ U † ∂ t ˜ U (cid:104) ˜ U † ∂ k x ˜ U , ˜ U † ∂ k y ˜ U (cid:105)(cid:17) ˜ U ( t, k ) = (cid:40) U (2 t, k ) 0 < t ≤ T / e +2 iH F ( t − T/ U ( T, k ) T / < t ≤ T . (A1)Here, ˜ U ( t ) is the Floquet unitary time evolution U ( t, k ),supplemented by backwards time-evolution under thetime-independent Floquet Hamiltonian U ( T, k ) = e − iH F T , to deform U ( t ) into a unitary loop: ˜ U (0) =˜ U ( T ) = 1. One can confirm: χ ( U CCN ( θ )) = (cid:40) ≤ θ < π/ π/ < θ ≤ π/ θ = 0 , π/
2, where the network model unitary ¯ ✓
Coupled wire picture of network model.
Quasi-one dimensional strips with oppositely propagatingedge states are indicated in gray. Red and blue scattering ma-trices with scattering angles θ and ¯ θ = π/ − θ represent inter-and intra- wire backscattering. Right panels 1-3: Each paneldepicts the local couplings of a stroboscopic Hamiltonian thatgenerates the network model dynamics. Solid black lines rep-resent coupling u ij ( π ), dashed red and blue lines represent u ij ( θ ) and u ij ( π/ − θ ) respectively (see text for definitions). raised to the fourth power is already a unitary loop: U ( T, k ) = −
1. Appealing to the topological invari-ance under perturbations that do not close gaps at quasi-energies 0 , π , we can then extend this result to genericvalues of θ (cid:54) = π/ χ is the con- struction of a closed, non-trivial loop in the space ofunitaries, passing through the identity and the networkmodel U . That construction is achieved here by the intro-duction of H ( tt