Quantized large-bias current in the anomalous Floquet-Anderson insulator

Arijit Kundu,1, 2 Mark Rudner,3 Erez Berg,4, 5 and Netanel H. Lindner2

1Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India2Physics Department, Technion, 320003, Haifa, Israel

3Niels Bohr International Academy and Center for Quantum Devices,University of Copenhagen, 2100 Copenhagen, Denmark

4Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel 761005Department of Physics, University of Chicago, Chicago, Illinois 60637, USA

We study two-terminal transport through two-dimensional periodically driven systems in whichall bulk Floquet eigenstates are localized by disorder. We focus on the Anomalous Floquet-AndersonInsulator (AFAI) phase, a topologically-nontrivial phase within this class, which hosts topologicallyprotected chiral edge modes coexisting with its fully localized bulk. We show that the uniqueproperties of the AFAI yield remarkable far-from-equilibrium transport signatures: for a large biasbetween leads, a quantized amount of charge is transported through the system each driving period.Upon increasing the bias, the chiral Floquet edge mode connecting source to drain becomes fullyoccupied and the current rapidly approaches its quantized value.

Topological phenomena, such as the quantum Hall ef-fect [1] and Thouless’ adiabatic pump [2], are charac-terized by the precise quantization of certain transportproperties. Recently, periodic driving has emerged as aversatile tool to control the topological characteristics ofquantum systems [3–20]. Such “Floquet” systems canbe realized in a wide variety of physical settings, includ-ing cold atomic, optical, and electronic systems [21–24].The extent to which Floquet systems may host quantizedtransport is an important direction of investigation.

Interestingly, periodically-driven quantum systemshost unique topological phases which cannot be re-alized by their static counterparts [5, 25–40]. Thericher topological classification of these systems is dueto their discrete (rather than continuous) time transla-tion symmetry, which is manifested as a periodicity of thequasienergy – the energylike variable that characterizesthe Floquet spectrum. Crucially, this structure providesthe basis for wholly new types of quantized transportphenomena, also without analogues in static systems.

The first example of a quantized transport phe-nomenon unique to periodically-driven systems was un-covered in Ref. [2]. There, Thouless showed thatthe charge transmitted through an insulating one-dimensional system is quantized as an integer multipleof the fundamental charge when the system is adiabati-cally driven through a cycle in parameter space.

More recently, in Ref. [30] it was shown that two-dimensional, disordered, periodically driven systems hosta unique topological phase called the Anomalous FloquetAnderson Insulator (AFAI). In the AFAI phase, all bulkFloquet eigenstates are localized, while chiral edge statesrun along the system’s boundaries. The AFAI’s chiraledge states exist at all quasienergies; each such chiraledge mode carries a quantized current when completelyfilled. In this work we show that, in a two-terminal trans-port setup, the AFAI carries a net quantized currentI = W2D/T in the limit of large source-drain bias (seeFig. 1). Here W2D is the winding number invariant thatcharacterizes 2D periodically driven systems [25, 30, 41].

0.0

1.0

L

R

FIG. 1. Quantized transport in the AFAI phase. (a) Two-terminal transport setup. A large source-drain bias ensuresthat the edge states running from source to drain are fullyfilled, while those running from drain to source are empty.(b) Bias (V ) dependence of the steady state current, I, forclean (light blue) and fully-localized (dark red) systems. Thecurrent saturates to the quantized value I = 1/T for V & 2Ω,where Ω = 2π/T is the driving frequency.

Associated with the quantized current, we find an in-homogeneous density profile in which the AFAI’s right-moving chiral edge state is fully occupied, while the left-moving chiral edge state is empty. Importantly, whilequantized pumping in the Thouless pump is found in theadiabatic limit, the large-bias quantized current carriedby a driven system in the AFAI phase occurs for inter-mediate driving frequencies (comparable to the system’snatural bandwidth).

The AFAI phase occurs in two-dimensional systems,whose dynamics are governed by a time-periodic Hamil-tonian HS(t) = HS(t+T ), where T is the driving period.The periodic driving gives rise to a unitary evolution

US(t) = T e−i∫ t0dt′HS(t′), where T denotes time order-

ing. The spectrum of the Floquet operator US(T ), givenby US(T )|ψn(0)〉 = e−iεnT |ψn(0)〉, defines the Floquetstates |ψn(t)〉 and their quasienergies εn.

To study quantized transport in the AFAI phase, weconsider a finite region of AFAI connected to two wide-bandwidth (non-driven) leads, as shown in Fig. 1a. The

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leads are indexed by λ = L,R, standing for the leftand right leads, respectively. Dynamics of the combinedsystem-lead setup are described by the Hamiltonian

H(t) = HS(t) +∑λ=L,R

Hλ +∑λ=L,R

HSλ, (1)

where HS(t) = HS(t + T ) is the time-periodic Hamil-tonian of the AFAI system, Hλ is the Hamiltonian de-scribing lead λ, and HSλ describes the coupling betweenthe system and lead λ. We treat each lead as an idealFermi reservoir, with filling characterized by an equi-librium Fermi-Dirac distribution with chemical potentialµλ in lead λ. Specific forms for the Hamiltonian termsabove will be given below. Throughout this paper, weuse e, ~ = 1.

The AFAI phase can be realized by a variety of drivingprotocols and experimental platforms, including solid-state and cold atoms. For concreteness and simplicity,here we use the square lattice tight-binding model intro-duced in Ref. [30]. In this model, the AFAI is described

by the Hamiltonian HS(t) = HcleanS (t)+

∑i wic

†i ci, where

c†i (ci) is the fermionic creation (annihilation) operatorfor site i, and wi is a normally-distributed on-site disor-der potential with zero mean and standard deviation w.The clean (disorder-free) Hamiltonian is given by

HcleanS (t) =

∑〈ij〉

Jij(t)c†i cj +

∑i

Dnic†i ci, (2)

where Jij(t) are time-dependent nearest-neighbor hop-ping amplitudes. It is convenient to define two sublatticesA and B on the square lattice (see Fig. 2). The piecewise-constant amplitudes Jij(t) connecting the two sublatticesare modulated according to the five step cycle depictedin Fig. 2a, where each step has length T/5. Within eachstep, all nonzero hopping amplitudes (bold bonds) havestrength J = 5π

2T ; in the fifth interval, all Jij = 0. Theparameter D is a staggered potential on the A and B sub-lattices, with ni = +1 (−1) for the A(B) sub-lattice. Weemphasize that the quantization of the current at large-bias is universal and independent of the specific model;a cold atom realization based on Refs. [1, 43] is analyzedin the Appendix.

Within the AFAI phase, realized for nonzero w be-low a critical value [30], the system in an open geome-try exhibits chiral edge sates in coexistence with a fully-localized bulk. These chiral edge states are illustrated inthe example spectra for the clean system (w = 0) in aninfinite-strip geometry, shown in Fig. 2b.

We now study the steady-state current transportedthrough the system when it is coupled to leads. Tothis end, we consider the Heisenberg equations of motionfor the operators cj(t) = U(t)cj(t0)U†(t) and aλj (t) =

U(t)aj(t0)U†(t), where aλj is the fermionic annihilation

operator on site j of lead λ, and U(t) = T e−i∫ tt0dt′H(t′)

is the evolution operator for the full Hamiltonian, Eq. (1).To simplify notation we introduce the operator vectors

aλ = (· · · aλi · · · )T and c = (· · · ci · · · )T , and express

E/

FIG. 2. Model of the AFAI phase. (a) Driving proceeds in5 steps of equal length, T/5. In each step, the highlightedbonds are active with strength Jij = 5π/(2T ), Eq. (2), whileall others are set to 0. The sublattices A and B are denoted byblack and white circles, respectively. (b) Quasienergy spec-trum of Hclean

S , with D = π/(2T ). (c) Spectrum of the trun-cated extended zone (EZ) picture system Hamiltonian, HEZ

S

(with M = 3), see Eq. (9), in the absence of disorder. Whilethe bands near E = 0 have Chern number zero, close to thetruncation we find bands with Chern numbers ±1.

the system, lead, and system-lead coupling Hamiltonians

in Eq. (1) as HS(t) = c†HS(t) c, Hλ = a†λHλaλ and

HSλ = c†HSλaλ+h.c., respectively. We leave the specificforms of the matrices Hλ and HSλ unspecified for now.

The macroscopic leads are assumed to be attachedin the very long past, such that the system operatorsc(t) are completely determined by the distribution in theleads; i.e., there is no memory of any initial occupationsin the system. We then write a formal solution for theHeisenberg equation of motion, ic = HS c +

∑λ HSλaλ:

c(t) =

∫dt′G(t, t′)

[∑λ

HSλgλ(t′ − t0)aλ(t0)

], (3)

where gλ(t) = −i exp(−iHλt)θ(t) is the retarded prop-agator for lead λ and G(t, t′) is the full Green’s func-tion within the system. For the calculations below, it isconvenient to furthermore define the Fourier-transformedFloquet Green’s function,

G(k)(E) =1

T

∫ T

0

dt

∫ ∞−∞

ds G(t, t− s)eiEseikΩt, (4)

and

ξλ(E) = HSλρλ(E)H†Sλ, (5)

where ρλ(E) =∑n δ(E −Eλn)|λn〉〈λn| captures the den-

sity of states of lead λ, with Hλ|λn〉 = Eλn|λn〉 [44].The net current flowing into the right lead, averaged

over one period, is given by

I =1

T

∫ T

0

dt i〈[H(t), NR(t)]〉, (6)

3

a) b)

k

Lead RLead L System

Fre

quen

cy

R

L

Lead RLead L System

L

Ener

gy

2M + 1

µ µ

Rµ + µE

T(k)RL(E) EZ(E = µ)

(3)

(1)

(0)

(1)

(2)

FIG. 3. Considerations leading to the sum rule in Eq. (10).(a) Transport through the driven system. A particle with en-ergy E enters the system from the left lead via the component

|φ(0)ε 〉 of a Floquet state |ψε(t)〉, with quasi-energy ε ≈ E . The

particle then scatters into a state with energy E + kΩ in the

right lead via its coupling to the component |φ(k)ε 〉. (b) Trans-

port in the static extended zone (EZ) system, see Eq. (9). TheEZ lead consists of 2M + 1 identical channels, shifted in en-ergy by integer multiples of Ω. A state in the lead with energy

µ and harmonic index n is coupled to the component |Φ(n)µ 〉

of the eigenstate |ψEZµ 〉 of HEZ

S with eigenvalue µ.

where NR(t) = a†R(t)aR(t) is the number operator for theright lead. Through Eq. (3) we express the system op-erators c(t) as linear combinations of the lead operatorsaλ(t0) in the distant past (we take t0 → −∞). Simi-larly, the lead operators aλ(t) can be written in terms ofaλ(t0). We assume that the state in each lead λ is givenby a Fermi distribution fλ with chemical potential µλ and

temperature Tλ: 〈a†λn(t0)aλm(t0)〉 = δnmfλ(ελn), where

a†λn creates an electron in eigenstate |λn〉 in lead λ (seeabove). Using Eqs. (3)-(5) and the Fermi distributionsfor the leads, a standard calculation gives [45]:

I = 2π

∫ ∞−∞

dE∑k

T

(k)RL (E)fL(E)− T (k)

LR (E)fR(E),(7)

T(k)λλ′(E) = Tr

[G(k)†(E)ξλ(E + kΩ)G(k)(E)ξλ′(E)

].

Here T(k)λλ′(E) is the probability for an electron at energy

E to be transmitted from lead λ′ to lead λ, along withthe absorption of k photons from the driving field.

As we now show, the steady-state time-averaged cur-rent carried by the AFAI, Eq. (7), is quantized in the limitof large bias, V → ∞, with µL = V/2, µR → −V/2. Inthis limit we may set fL(E) = 1 and fR(E) = 0, yielding

I =

∫ ∞−∞

dE σ(E), σ(E) = 2π∑k

T(k)RL (E). (8)

In the following, we show the quantization of the cur-rent by relating σ(E) to the differential conductance ofan associated static system. For illustration, we firstconsider the dominant processes contributing to σ(E),see Fig. 3a. In each process a particle in the left leadwith energy E scatters into a Floquet state of the sys-tem with quasienergy ε ≈ E + nΩ [46]. The integern is determined by our convention for Floquet states,

|ψε(t)〉 = e−iεt∑m |φ

(m)ε 〉e−iΩmt, with −Ω/2 ≤ ε < Ω/2.

The scattering process thus proceeds through the cou-

pling between the lead state and the component |φ(−n)ε 〉.

The particle then scatters into a state in right lead with

energy E+kΩ, via its coupling to the component |φ(k−n)ε 〉.

Thus, in the process of scattering from the left to theright lead the particle absorbs k photons from the time-periodic drive. The collection of processes involving suchchanges in the particle’s energy is captured by the sumappearing in the definition of σ(E), Eqs. (7) and (8).

We now re-express the current, Eq. (8), as I =∫ Ω/2

−Ω/2(dI/dε)dε, with dI/dε =

∑n σ(E+nΩ). The quan-

tity dI/dε can be related to the differential conductanceof a static system, which describes the periodically drivensystem in an “extended zone” (EZ) frequency-space pic-ture. The Hamiltonian of the static EZ system is given

by HEZ =∑Mm,nH

EZmn|m〉〈n|, where the sum runs over

−M ≤ n,m ≤M , and

HEZmn =− δmnnΩ +

∫ T

0

dt

Tei(m−n)ΩtH(t). (9)

The operator HEZ acts in enlarged Hilbert space, whichis a tensor product of the original Hilbert space and a(2M + 1)-dimensional auxiliary space, which we call theharmonic space.

As in Eq. (1), we write HEZ = HEZS +

∑λHEZ

Sλ +∑λHEZ

λ . An eigenstate of HEZS with energy E can be

expanded as |ψEZE 〉 =

∑n |Φ

(n)E 〉 ⊗ |n〉. The eigenval-

ues of HEZS in the range −Ω/2≤ E < Ω/2 approximate

the quasienergy spectrum of US(T ) = T e−i∫ T0dtHS(t), be-

coming exact for M →∞. Importantly, in this limit, foreach |ψEZ

E 〉 there is a corresponding partner Floquet statewith quasienergy ε = E + mΩ (with |ε| ≤ Ω/2) in the

original driven problem: |ψε(t)〉 = e−iεt∑n |φ

(n)ε 〉e−iΩnt,

with |φ(n−m)ε 〉 = |Φ(n)

E 〉.We now relate the relevant transport processes in the

static EZ and Floquet pictures (see Fig. 3). Considerthe differential conductance, σEZ(µ), of the EZ systemdescribed by HEZ. Since the lead is not driven, the spec-trum of HEZ

λ consists of 2M + 1 copies of that of Hλ,shifted by integer multiples of Ω; it can thus be viewedas a lead with many channels, labeled by the harmonicindex. We define σEZ(µ) by taking the Fermi level of theleft and the right EZ leads to be µ+ δµ and µ− δµ, andtake −Ω/2 ≤ µ < Ω/2 throughout [47].

Consider now the dominant processes contributing toσEZ(µ). The system-lead coupling HEZ

Sλ conserves theharmonic index. Therefore, a lead state with energy Eand harmonic index n (which corresponds to a state ofthe physical lead with energy E − nΩ) is coupled to the

state |ψEZE 〉 through the component |Φ(n)

E 〉. To obtainσEZ(µ), we sum the contributions of states with ener-gies close to µ from all harmonic-index channels in bothleads. Using the correspondence between |ψEZ

µ 〉 and|ψµ(t)〉, for M 1, we thus obtain (for details, see

4

FIG. 4. Steady state current vs. disorder strength w, forV Ω. As w is increased from zero, the steady state current(averaged over a period) rapidly approaches the quantizedvalue of 1/T [48]. The sample has dimensions L×W = 40×20sites. The leads are taken to have widths W0 = W/2. Inset:Bulk contribution to the steady state current, computed us-ing a cylindrical geometry with contacts on opposite edges ofthe cylinder, for w = 4.5/T . Exponential decay of the bulkcontribution with increasing L indicates that the system is inthe localized regime.

Appendix): ∑n

σ(µ+ nΩ) = σEZ(µ). (10)

Importantly, in the EZ picture, σEZ(µ) is just the two-terminal differential conductance of a disordered Cherninsulator, with µ lying in a mobility gap. To see why thisis the case, consider the spectrum of HEZ

S in the AFAIphase. In the spectral range −Ω/2 ≤ µ < Ω/2 it ex-hibits two important properties: (i) all bulk states arelocalized [49], and (ii) chiral edge states exist at all en-ergies within this range. These two properties of HEZ

Sare a direct consequence of the properties of US(T ) inthe AFAI phase. Since in the EZ picture the number ofedge states corresponds to the total Chern number of allbulk states below µ, the spectrum of HEZ

S must contain aband with nontrivial Chern number at an energy near theharmonic space truncation at n = −M . The quantizedtwo-terminal differential conductance of such a Chern in-sulator [50], σEZ(µ) =W2D, together with Eq. (8), yields

I =∑n

∫ Ω/2

−Ω/2dEσ(E + nΩ) =W2D/T .

For the model given in Fig. 2a, the above considera-tions are exemplified by inspecting the spectrum of thecorresponding HEZ

S (without disorder), given in Fig. 2c.Here we find a single chiral edge in the spectral range−Ω/2 ≤ E < Ω/2; in this spectral range, the Chernnumbers of the bands are all zero. However, the high-lighted bands near the bottom and top of the spectrum,which are strongly affected by the truncation, have Chernnumbers ±1.

Numerical simulations.—To support the argumentsabove, we now numerically study the steady state cur-rent. We simulate the model described above, Eq. (1),

Den

sity

,n

iB

ond

curr

ent,

j ij

(1/T

)

x

y

x

y

W0

FIG. 5. (a) Map of the steady-state period-averaged density,ni, for w = 4.5/T . The large bias between the leads, V Ω,ensures that the edge state running from source to drain isfully occupied, while that running from drain to source isempty. (b) The period-averaged bond currents jij (see text).The current density is concentrated at the interface betweenfully occupied and empty regions.

for a range of system sizes and disorder strengths w, seeFig. 4. We take D = π/(2T ), and the leads to haveconstant density of states, ρ0λ = 1/J . The lead-systemcoupling HSλ is taken to yield ξλ(E) =

∑r∈W0

ρ0λ|r〉〈r|,where the sum runs over W0 system sites directly adja-cent to lead λ (see Fig. 5a).

In the presence of disorder, all bulk states are local-ized and the current through the bulk vanishes exponen-tially with the distance between the leads. To probe this,we computed the current in a cylindrical geometry, withleads attached at opposite ends of the cylinder such thatthere were no edge states connecting the source and drain(shown in the inset of Fig. 4). As shown in Fig. 1b andthe main panel of Fig. 4, for the Hall bar geometry ofFig. 1a the total current through the system saturates tothe quantized value I = 1/T in the insulating regime, forlarge (finite) bias.

As explained above, a quantized current is expectedto flow in the AFAI when the edge-states exiting fromthe left lead are completely filled, while those exiting theright lead are empty. We confirm this picture (for a typi-cal disorder realization) by mapping out the steady state

time-averaged local density, ni = 1T

∫ T0dt 〈c†i (t)ci(t)〉, in

Fig. 5a. This situation is realized for “good” contacts,with appropriately strong couplings ξλ and large enoughcontact width W0 (see Fig. 5 and the Appendix).

To further investigate the spatial distribution of thecurrent, we map out the period-averaged bond current

density, jij = 1T

∫ T0dt 2Im[Jij(t)〈c†i (t)cj(t)〉], see Eq. (2).

As shown in Fig. 5b, the current density is concentrated

5

at the boundary of the filled and empty regions. This re-sult may at first seem counterintuitive since a) all statesin the bulk are localized and b) we expect the quantizedcurrent to be carried by the chiral edge states. However,it is crucial to remember that the local current densityjij(t) includes contributions of both transport currentsand magnetization current [41, 51]. The quantized trans-port current is indeed carried by the chiral edge states, asthey are the only delocalized states in the system (alsosee the Appendix).

Summary.—In this work we demonstrated theoreti-cally a new topological quantized transport phenomenon,occurring in disordered two-dimensional periodicallydriven systems. In contrast to the equilibrium quantizedHall conductivity, in the AFAI phase, which occurs in asystem far from equilibrium, we find a quantized currentin the limit of large bias [57]. Looking ahead, disorder-induced localization may provide a route for stabilizinginteracting Floquet phases of matter by suppressing en-ergy absorption from the periodic drive. Recently, severalworks proposed interacting analogues of the AFAI [52–

56]. Determining whether quantized transport and otherresponse functions can be used to probe these interactingphases will be crucial for further progress in the field.Acknowledgements.—A. K. acknowledges the support

from the Indian Institute of Technology - Kanpur andwas supported in part at the Technion by a fellowship ofthe Israel Council for Higher Education. N. L. acknowl-edges support from the People Programme (Marie CurieActions) of the European Union’s Seventh FrameworkProgramme (No. FP7/2007–2013) under REA GrantAgreement No. 631696, from the Israeli Center of Re-search Excellence (I-CORE) “Circle of Light.”. N. L. andE. B. acknowledge support from the European ResearchCouncil (ERC) under the European Union Horizon 2020Research and Innovation Programme (Grant AgreementNo. 639172). M. R. gratefully acknowledges the sup-port of the European Research Council (ERC) under theEuropean Union Horizon 2020 Research and InnovationProgramme (Grant Agreement No. 678862), and the Vil-lum Foundation. M. R. and E. B. acknowledge supportfrom CRC 183 of the Deutsche Forschungsgemeinschaft.

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[42] A. Quelle, C. Weitenberg, K. Sengstock and C. MoriasSmith, New Journal of Phys., 19, 113010 (2017).

[43] S. Krinner, D. Stadler, D. Husmann, J. Brantut and T.Esslinger, Nature 517, 64 (2015).

[44] In the λn basis, we have [ρλ(E)]nn =−12πi

Im [gλ(E)]nn.[45] S. Kohler, J. Lehmann, and P. Hanggi, Phys. Rep. 406,

379-443 (2005).[46] Due to finite coupling strength, the energy conservation

condition is broadened on the scale of the tunneling rate.[47] The states of HEZ

L are populated according to a Fermi-Dirac distribution with chemical potential µ+ δµ regard-less of their harmonic index (and similarly for the rightlead).

[48] The negative overshoot at low disorder is due to the as-pect ratio of the system, see the Appendix. Note that atvery strong disorder, the system undergoes a transition toa trivial insulator, where the current vanishes. This tran-sition was studied in Ref. [30].

[49] Note that delocalized states exist at energies close to thetruncation. However, for sufficiently large M their energiesdo not lie in the window Ω/2 < µ < Ω/2, and thereforedo not contribute to transport.

[50] The quantized value is obtained for sufficiently large con-tact length.

[51] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82,1959 (2010).

[52] H. C. Po, L. Fidkowski, T. Morimoto, A. C. Potter, andA. Vishwanath Phys. Rev. X 6, 041070 (2016)

[53] H. Chun Po, L. Fidkowski, A. Vishwanath, and A. C.Potter, arXiv:1701.01440.

[54] L. Fidkowski, H. Chun Po, A. C. Potter, and A. Vish-wanath, arXiv:1703.07360.

[55] A. C. Potter, A. Vishwanath, and L. Fidkowski,arXiv:1706.01888

[56] F. Harper and R. Roy, Phys. Rev. Lett. 118, 115301(2017).

[57] The I-V characteristic of quantum Hall systems is gener-ically non-linear at large source-drain biases, but the cur-rent does not saturate to a universal quantized value as itdoes for the AFAI.

Appendix A: Sum rule

In this section, we derive the sum rule appearing in

Eq. (10) of the main text. Consider |φ(k)±α 〉, the kth

Fourier component of the time-periodic Floquet state|φ±α (t)〉, which satisfies(

HS(t)∓ iΓ− i ddt

)|φ∓α (t)〉 = (εα ∓ iγα)|φ∓α (t)〉. (A1)

In the above Γ =∑λ ξλ, where we have assumed a

constant density of states in each lead. The ∓ sym-bol in |φ∓α (t)〉 labels the right and left eigenstates ofthe non-Hermitian operator

(HS(t)− iΓ− i ddt

), respec-

tively. The Floquet states satisfy the completeness andorthogonality relations,∑

α

|φ−α (t)〉〈φ+α (t)| = I,

∫ T

0

dt

T〈φ−α (t)|φ+

β (t)〉 = δαβ .

The Green’s function satisfies(id

dt+ E −HS(t) + iΓ

)G(t, E) = I, (A2)

with G(t, E) =∑k e−ikΩtG(k)(E). The Floquet Green’s

function can thus be written as:

G(k)(E) =∑αn∈Z

|φ(n+k)−α 〉〈φ(n)+

α |E − εα − nΩ + iγα

. (A3)

As written in Eq. (8) of the main text, the current isgiven by:

I = 2π∑k

∫ ∞−∞

dE Tr[G(k)†(E)ξR G

(k)(E)ξL

], (A4)

and we define

σ(E) = 2π∑k

Tr[G(k)†(E)ξR G

(k)(E)ξL

]. (A5)

With this definition, the current is I =∫∞−∞ dE σ(E), or

equivalently:

I =

∫ Ω/2

−Ω/2

dε dI/dε, dI/dε =∑n∈Z

σ(ε+ nΩ). (A6)

From Eq. (A3), we obtain:

dI/dε = 2π∑n,k

Tr[G(k)†(ε+ nΩ)ξRG

(k)(ε+ nΩ)ξL

]

= 2π∑n,kαβm,q

〈φ(m)+α |ξL|φ(q)+

β 〉ε− εα + (n−m)Ω + iγα

×

×〈φ(q+k)−β |ξR|φ(m+k)−

α 〉ε− εβ + (n− q)Ω− iγβ

. (A7)

Recall that our goal is to relate dI/dε to the differen-tial conductance of the static extended-zone (EZ) system,and thereby derive Eq. (10) of the main text. The EZHamiltonian, HEZ, is defined in the main text, Eq. (9).Considering that the system-lead coupling is time inde-pendent, and assuming a constant density of states, i.e,(HEZ

Sλ )mn = δmnHSλ, the EZ eigenstates satisfy:(HS ∓ iΓEZ − i d

dt

)|Φ∓α,p〉 = (Eα,p ∓ iγEZ

α,p)|Φ∓α,p〉,(A8)

7

with ΓEZmn = δmnΓ, and where −M ≤ p ≤ M . To-

gether, α and Ψ provide a complete labeling of theEZ eigenstates. The left and right eigenstates form acomplete set, with the relations:

∑α,p |Φ−α,p〉〈Φ+

α,p| =

I, 〈Φ−α,p|Φ+β,p′〉 = δαβδpp′ . Correspondingly, the Green’s

function of the EZ system satisfies:(E −HEZ

S + iΓEZ)GEZ(E) = I. (A9)

We represent GEZ(E) in the basis of eigenstates above as

GEZ(E) =∑α,p

|Φ−α,p〉〈Φ+α,p|

E − Eα,p + iγEZα,p

. (A10)

The differential conductance of the EZ system at en-ergy µ is given by

σEZ(µ) =dIEZ(µ)

dµ= 2πTr

[GEZ †(µ)ξEZ

R GEZ(µ)ξEZL

],

(A11)

where (ξEZλ )mn = δmnξλ. Substituting Eq. (A10) into

Eq. (A11), and writing |Φ±α,p〉 =∑n |Φ

(n)±α,p 〉 ⊗ |n〉, gives

σEZ(µ) = 2π∑

α,p,β,p′m,n

〈Φ(m)+α,p |ξL|Φ(m)+

β,p′ 〉µ− Eα,p + iγEZ

α,p

〈Φ(n)−β,p′ |ξR|Φ(n)−

α,p 〉µ− Eβ,p′ − iγEZ

β,p′.

(A12)

We focus on value of µ between −Ω/2 and Ω/2. Cru-cially, for µ in this range, the main contribution to σEZ(µ)comes from states with |Eα,p − µ| . γα,p. For M [theparameter controlling the truncation of the EZ Hamilto-nian, see main text around Eq. (9)] much greater than 1,

we have limM→∞ |Φ(m)±α,p 〉 = |φ(m+p)±

α 〉; the correspond-ing eigenvalues also satisfy limM→∞ Eα,p = εα + pΩ andlimM→∞ γEZ

α,p = γα. Thus,

σEZ(µ) = 2π∑α,β

m,n,p,p′

〈φ(m+p)+α |ξL|φ(m+p′)+

β 〉µ− εα − pΩ + iγα

〈φ(n+p′)−β |ξR|φ(n+p)−

α 〉µ− εβ − p′Ω− iγβ

. (A13)

After a relabeling of indices, the expression abovematches that in Eq. (A7). Thus, comparing withEq. (A6), we have shown

∑n σ(µ+ nΩ) = σEZ(µ).

Appendix B: Cold Atom Realization

In this section we demonstrate the quantized two ter-minal transport in a model which is amenable to directrealization in a cold atom system. The model relies ona recent work [1], which considers a driven honeycomboptical lattice for cold atoms. Ref. [1] proposes a driv-ing protocol in which the driving period T is dividedinto three time steps of length T/3. In each step thelattice is shaken with a high frequency along one of thethree mirror plane directions of the honeycomb lattice.With appropriately chosen driving amplitude, the shak-ing can fully suppress the static hopping along bondswith components parallel to the shaking direction, whileleaving the perpendicular bonds unaffected. By choos-ing the driving period T to obey JT/3 ≈ π/2, whereJ is the hopping amplitude (for the unsuppressed bondsperpendicular to the shaking direction), an anomalousFloquet band structure with chiral edge states and zerobulk Chern numbers is obtained.

Building on that proposal, we consider a tight bindingmodel on a honeycomb lattice subjected to a three-step

piecewise constant driving protocol, see Fig. 6a. In eachstep of the driving period, the only non-zero hoppingamplitudes are those parallel to one of the three mir-ror planes. In Fig. 6b, we show the quasienergy band-structure for this model in a cylinder geometry (periodicboundary conditions in one direction), where the driv-ing period T and the hopping strength J are related viaΩ = 2π/T ≈ 1.88J . The band structure features twonearly flat bands and chiral edge states along the edgesof the cylinder. Introducing static onsite disorder real-izes an AFAI phase. We numerically calculate the currentaveraged over a driving period in a two terminal setup,see Eq. (6) in the main text. The results are plottedin Fig. 6c. For the cylinder geometry, the current de-creases exponentially with increasing disorder, indicatingthat the system is indeed in the (localized) AFAI phase.When computed in the Hall bar geometry, the systemexhibits a quantized current for disorder strength largeenough such that the localization length is smaller thanthe system size.

Appendix C: Dependence on contact width

In order to achieve quantized two-terminal transport,it is necessary that the chiral edge states fully equili-brate with the leads in the contact region. For example,

8

(a) (c)(b)

FIG. 6. (a) Three step driving protocol for the non-disordered version of the AFAI. For each time step of duration T/3, onlyhoppings in a direction parallel to one of the three mirror plane directions of the hexagonal lattice are nonzero, and havestrength J > 0. b) The quasienergy spectrum for Ω = 2π/T ≈ 1.88J in a ribbon geometry shows two nearly flat bands andchiral edge states. An AFAI can be realized by introducing static on-site disorder. (c) In a two terminal setup at large bias,a quantized time-averaged current is obtained in the Hall bar geometry (open boundary conditions in the y direction), forsufficiently strong disorder strength w such that the localization length is larger than system size. In a cylindrical geometry(closed periodic boundary conditions in the y direction), the current decays exponentially with increasing disorder strength.

FIG. 7. Convergence of current to the quantized value as afunction of contact width W0, for three values of the system-lead coupling JSλ. Here we used L × W = 40 × 20 sites,disorder strength w× T = 1.5, and constant density of statesof the leads, ρλ = 1/J .

consider the current impinging on the drain contact. Itis essential that each particle reaching the drain is ab-sorbed into the drain lead with unit probability, to avoidit returning back to the source and diminishing the nettransmitted current. The probability for the particle tobe absorbed into the lead approaches one, exponentiallywith the width W0 of the contact; the lengthscale asso-ciated with this exponential is controlled by the matrixelements JSλ governing hopping between the system andthe lead.

In Fig. 7 we demonstrate the convergence of the trans-mitted current as a function of W0 and JSλ. We takeJSλ to be constant for all sites within the contact region.Here we focus on the weak coupling regime, where JSλ is

FIG. 8. Steady state current vs. disorder strength w, forV Ω, for a “short and wide” AFAI. As w is increasedfrom zero, the steady state current (averaged over a period)rapidly approaches the quantized value of 1/T from above.The sample has dimensions L×W = 20× 30 sites. The leadsare taken to have widths W0 = W/2.

small compared with the hopping matrix elements withinthe system, J . As expected in this limit, the absorptionlength decreases with increasing JSλ/J .

Appendix D: Dependence on aspect ratio

In Fig. 4 of the main text, we studied the steady statecurrent averaged over a period, as a function of disorderstrength w. For low values of w, the localization lengthof the bulk Floquet states is larger than the system size.In this regime, a transport current between the two leadscan be carried by the bulk states. Therefore, for low val-ues of w, we expect the current to be larger than the

9

FIG. 9. Density (a) and bond current (b) maps for an AFAI inthe cylindrical geometry. The system has periodic boundaryconditions along the y direction, and is connected to leads atx=0 and x = Lx, with Fermi levels +V

2and −V

2. A single

disorder realization is used.

quantized value of 1/T . This expectation is indeed con-firmed by our numerical simulation, as shown in Fig 4.In addition, Fig. 4 shows an interesting feature: as w isincreased further, the deviation from the quantized valueof 1/T becomes negative, and with increasing values ofw, the current approaches the quantized value from be-low. This feature can be understood as follows: since thesystem studied numerically in Fig. 4 is “long and nar-row” (L = 40,W = 20, where the leads are attachedto the narrow edges), for intermediate values of disorderthe localization length is smaller than the length of thesystem but larger than its width. Therefore, the bulkcannot transport particles between the leads, and at thesame time there is still significant backscattering between

the counterpropagating edge states on opposite sides ofthe sample. To confirm this explanation, we numericallysimulated the transport experiment in a system which is“wide and short” (L = 20,W = 30, where the leads areattached to the wide edges). The results, shown in Fig. 8,indeed do not show any negative overshoot.

Appendix E: Magnetization and transport current

The bond current is the sum of the magnetization cur-rent and the transport current. The bulk of the AFAIis fully localized and is therefore insulating and cannotcarry any transport current. It can, however, host mag-netization currents in regions where the magnetization isinhomogeneous (recall that in a steady state the magne-tization current is defined as jm = ∇×m). Note that dueto the aforementioned relation, steady-state magnetiza-tion currents (which are by nature circulating currents)do not contribute to the total current transported acrossany cut all the way through the system, and thereforecannot contribute to transport between the leads. For theAFAI in a Hall bar geometry, the only delocalized statesthat permit particles to propagate from one lead to theother are the AFAIs topological edge states. Thereforeany transport current between source and drain must becarried by the edge states.

Since the AFAI bulk cannot carry transport current,the density profile in a cylindrical geometry should onlyexhibit a non-vanishing density near the lead with thelarger Fermi energy. Likewise, the bond current can onlybe significant in a small region close to the lead where theparticles are located (far away sites, being empty, cannothost currents). To numerically confirm this expected be-havior, we computed the density and bond current pro-files in the steady state in the cylindrical geometry. Theresults, shown in Fig. 9, directly confirm the above ex-pectations demonstrating that the bulk does not carryany transport current between the two leads.

[1] A. Quelle, C. Weitenberg, K. Sengstock and C. MoriasSmith, New Journal of Phys., 19, 113010 (2017).