Download - arXiv:2002.04058v1 [cond-mat.dis-nn] 10 Feb 2020 · work model (CCN) is de ned by a unitary matrix that acts on a wavefunction sampled at discrete spatial lat-tice points in a continuum

Quantum Hall network models as Floquet topological insulators

Andrew C. Potter,1 J. T. Chalker,2 and Victor Gurarie3

1Department of Physics, University of Texas at Austin, Austin, TX 78712, USA2Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom

3Department of Physics and Center for Theory of Quantum Matter,University of Colorado, Boulder, Colorado 80309, USA

We use the (Chalker-Coddington) network model to establish a relationship between two dis-tinct topological phenomena: integer quantum Hall (IQH) phases of equilibrium systems, and non-equilibrium chiral Floquet (CF) topological insulators. The network model for an IQH system isrepresented by a unitary scattering matrix, which we map onto the Floquet operator of a period-ically driven system. Strikingly, the Floquet bands of this model have Chern number zero, andthe topological phase from this perspective is instead characterized by a chiral Floquet winding-number invariant. This raises a question: how can a model without Chern bands describe the IQH?We resolve this apparent paradox by showing that non-zero Chern number arises from the IQHperspective via the energy dependence of scattering network parameters. These results imply thatdespite having distinct topological invariants, IQH and CF systems share the same universal scalingproperties for their topology-changing transitions in disordered, but non-interacting settings.

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 vanishing

Hall 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 establish a relation between thesetwo distinct types of topological phenomena via a net-work model originally introduced by Chalker and Cod-dington [8] to describe the scattering dynamics of elec-trons near the quantum Hall plateau transition. The net-work model (CCN) is defined by a unitary matrix thatacts on a wavefunction sampled at discrete spatial lat-tice points in a continuum IQH system. We construct aperiodically time-dependent Hamiltonian, acting on thesame lattice, whose Floquet operator coincides with theCCN unitary, and show that this Floquet system hosts aCF phase, while having vanishing Chern number for itsFloquet bands at all 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 that thetopology changing phase transitions for disordered, non-interacting IQH and CF phases share the same universalscaling properties (though their fully non-equilibrium dy-namics away from criticality sharply differ [9, 10]).

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Network model definition – Consider the groundstate of a non-interacting electron gas in a uniform mag-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 ri on each link, and represented by acurrent amplitude ψi. Amplitudes on incoming and out-going links at a node are related by a scattering matrix,so that (referring to Fig. 1):(

ψ3

ψ4

)=

(eiϕ3 0

0 eiϕ4

)(cos θ sin θ− sin θ cos θ

)(ψ1

ψ2

). (1)

Here, θ parameterizes tunneling while ϕ3 and ϕ4 areAharonov-Bohm phases. A similar S-matrix with θ →θ ≡ π/2−θ describes scattering at the θ nodes. Disorderis modeled by taking ϕi to be an independent randomvariable on each link i, uniform in [0, 2π). The modelcan be specified for a closed system of N links by anN × N unitary matrix U , which is composed of 2 × 2sub-matrices, each having the form of Eq. 1 [11].

An eigenstate of the continuum Hamiltonian, sampledat the points ri, is represented by an eigenvector of Uwith eigenphase zero (mod 2π), 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 ϕi. From theshape of equipotentials near a saddle-point, one sees thatθ increases from 0 to π/2 as energy is increased acrossthe disorder-broadened Landau level. The accumulatedlink phase around a plaquette is (2π times) the numberof flux quanta passing through this plaquette. Random-ness in φi arises from small random variations in the areaof plaquettes. Additional energy-dependence of φi arisesdue to to the change in area of a plaquette with a changein the chemical 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 isolated

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FIG. 1. Coupled wire picture of network model. Upperpanel: Quasi-one dimensional strips with oppositely propa-gating edge states are indicated in gray. Red and blue scatter-ing matrices with scattering angles θ and θ = π/2−θ representinter- and intra- wire backscattering. Lower panels 1-3: Eachpanel depicts the local couplings of a stroboscopic Hamilto-nian that generates the network model dynamics. Solid blacklines represent coupling uij(π), dashed red and blue lines rep-resent uij(θ) and uij(π/2 − θ) respectively (see text for defi-nitions).

plaquettes 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/4.

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) = Uψ(t), (2)

where ψ(t) is the vector of all the link amplitudes atthe time step t [12]. Writing the eigenvalues of U ase−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, Cn, defined separately for eachFloquet band, n, and

2. 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].

3

For the network model, we observe that the Chernnumbers for the Floquet-bands of U are identically zero.This claim can be verified by inspection at the trivial andtopological limiting points, θ = 0 and θ = π/2. At thesepoints, the bulk motion consists of short loops aroundthe elementary plaquettes of the network model (Fig. 1),indicating that the Floquet bands admit a strictly local-ized Wannier basis of orbitals, each having support onlyon the four links of a single plaquette. Such a localizedWannier basis implies vanishing Chern number [13], im-mediately implying that each individual eigenstate haszero Chern number. This conclusion extends to all val-ues of θ 6= π/4, since Chern number may only changeat a delocalization transition, which in this model occursonly at θ = π/4. The regions either side of this crit-ical point inherit the vanishing Chern number of theirlimiting points, at θ = 0, π/2 respectively.

This argument, valid for arbitrary disorder, can be di-rectly verified for the clean version of the model by ex-plicit computation of the Floquet band-structure. Intro-ducing crystalline momenta k, the matrix U is block-diagonal with blocks of the form

U(θ,k) =

0 0 sin θe−ikx cos θe−iky

0 0 − cos θeiky sin θeikx

cos θ sin θ 0 0− sin θ cos θ 0 0

. (3)

We denote the eigenvalues of U(θ,k) by εn(θ,k) forn ∈ {0, 1, 2, 3}. At θ = 0 and θ = π/2, εn are inde-pendent of k, taking the values εn = π/4 + πn/2. As θdeviates from these extreme values, the bands disperse,until they touch at θ = π/4 in a sequence of Dirac-pointsat wave-vectors k = (0, 0) or k = (π, π) (Fig. 2). 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 6= 0. As a first step, we compare behavior atθ = 0 and θ = π/2 for a system with open boundaries,where χ can be computed simply by counting the numberof chiral edge states wrapping the quasi-energy Brillouinzone. There is no edge state in the first case, and one inthe second case. The eigenvectors of U corresponding tothe edge state can be given explicitly. Let integer j la-bel links in order along the boundary, and for simplicityset all ϕj = 0. Then the vector with ψj = eikj on edgelinks and ψi = 0 on all other links is an (unnormalized)eigenvector of U with quasi-energy εedge = k.

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FIG. 2. Floquet band structure – The four bands ε(k)of the clean network model evolution operator at θ = π/4,plotted as a function of kx and ky across the Brillouin zone.

This mode indeed wraps the quasi-energy Brillouinzone as k varies, and is manifestly absent in the otherphase of the model (θ ≈ 0), demonstrating that

χ(UCCN(θ)) =

{0 0 ≤ θ < π/4

1 π/4 < θ ≤ π/2 (4)

and hence that the network model describes a Floquetchiral unitary index changing phase transition.

Although inspection of the stroboscopic edge motionis sufficient to give the value of bulk invariant 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 a known obstacle: tocompute χ from bulk behavior alone, it is not sufficient toexamine stroboscopic times (for which the bulk motion isalways trivial when all Cn = 0). Instead, one must exam-ine the micro-motion within a single period. The networkmodel as formulated is blind to this micro-motion, andadditional choices are needed to define it (though the fi-nal result will be independent of these choices).

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. Thatis, we require U(t) = T e−i

∫ t0H(t1)dt1 (where T denotes

time ordering) such that U(T ) = U , the CCN unitary.This relation does not uniquely fix H(t). However, if wedemand that H(t) is spatially local, any such choice ofH(t) will produce the same value of χ. The completenessof the Floquet classification for lattice models [6, 14, 15]also implies that any such local generating H(t) mustbe explicitly time-dependent (in contrast to the staticHamiltonian for continuum Landau levels).

To construct a generating H(t), we adopt a coupled-wire perspective [16] by grouping the chiral segments of

4

the model into pairs of alternating strips (Fig. 1). WebuildH(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 hαβ = Jσy, where σy is the Paulimatrix acting on the two amplitudes (ψα, ψβ)T . Whenapplied for time t = λ/J this produces the unitary evolu-tion uαβ(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, uαβ(λ = π), on the horizontal and di-agonal bonds, shown on lower panels 2 and 3 of Fig. 1.Scattering is introduced via an initial partial swap step,by applying uαβ(θ) and uαβ

(π2 − θ

)at the inter- or intra-

wire scattering nodes respectively (panel 1 of Fig. 1).

With a local generating Hamiltonian in hand, one cancompute χ directly via the formula [6]

χ[U ] =1

8π2

∫dtdkxdkyTr

(U†∂tU

[U†∂kxU , U

†∂ky U])

U(t,k) =

{U(2t,k) 0 < t ≤ T/2e+2iHF (t−T/2)U(T,k) T/2 < t ≤ T . (5)

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−iHFT , to deform U(t) into a unitary loop: U(0) =U(T ) = 1. One can confirm Eq. 4 directly using Eq. 5.This computation is simplest to do for θ = 0, π/2, wherethe network model unitary raised to the fourth poweris already a unitary loop: U4(T,k) = −1. Appealing tothe topological invariance under perturbations that donot close gaps at quasi-energies 0, π, we can then extendthis result to generic values of θ 6= π/4.

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 ϕi = 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 � 1. In-cluding ϕ(E), the quasi-energies of U are εn(k, θ)−ϕ(E).Eigenenergies of the continuum problem form bands de-fined by:

ϕ(E) = εn(k, θ) + 2πm (6)

for integer m. For large Φ there are many such bands.

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FIG. 3. 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

4(tanh [E + π/4] + 1)

and ϕ(E) = 4E. The horizontal axis is placed at the criticalenergy E = Ec for which θ(Ec) = 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.

Recalling that θ ≡ θ(E) is a (slowly varying) func-tion of E, the solution to Eq. (6) for each m,n defines asurface θm,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, there isone exceptional pair (m,n) for which θm,n(k) < π/4 atk = (0, 0) but θm,n(k) > π/4 at k = (π, π) (or vice-versa,depending on the parity of n). For this exceptional pair,oppositely charged monopoles lie on opposite sides of thesurface, which therefore has a full unit of Berry flux pass-ing though it. The associated band of eigenstates hencehas unit Chern number, and summing over all bands werecover unit Chern number for the Landau level. A nu-merical demonstration for particular energy dependenceis shown in Fig. 3.

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-

5

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 [9], but instead spread sub-diffusively in IQH sys-tems due to overlap with critically extended states [10].

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 2d 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 [17–19]. Moreover, analogous 1d scatteringnetwork constructions can be obtained by compactifying2d examples to extend these results to all 1d classes [20],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 1d where phase tran-sitions are characterized by RG-flow to infinite random-ness and have equivalent interacting and non-interactingscaling properties [21] By this route the 1d Floquet modelstudied in Ref. [22, 23] is related to an Ising model, whichis known to have a network model representation [24].However, the correspondence will presumably fail for in-teracting 2d systems, since in that context CF phasesare governed by rational-fractional-valued topological in-variants with a completely distinct structure from Chernnumber [14, 15], and because the non-interacting CF-to-trivial critical-point will broaden into an interveningthermal phase upon including interactions [25].

Acknowledgements – We thank Yang-Zhi Chou,Matthew Foster, Itamar Kimchi, Sid Parameswaran, andRomain Vasseur for insightful discussions. This work was

supported 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).

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