How robust is quantized transport through edge states of finite length: Imaging current density in Floquet topological vs. quantum spin and anomalous Hall insulators

The theoretical analysis of topological insulators (TIs) has been traditionally focused on infinite homogeneous crystals with band gap in the bulk and nontrivial topology of their wavefunctions, or infinite wires whose boundaries host surface or edge metallic states. However, experimental devices contain finite-size topological region attached to normal metal (NM) leads, which poses a question about how precise is quantization of longitudinal conductance and how electrons transition from topologically trivial NM leads into the edge states. This is particularly pressing issues for recently conjectured two-dimensional (2D) Floquet TI where electrons flow from time-independent NM leads into time-dependent edge states---the very recent experimental realization of Floquet TI using graphene irradiated by circularly polarized light did not exhibit either quantized longitudinal or Hall conductance. Here we employ charge conserving solution for Floquet-nonequlibrium Green functions (NEGFs) of irradiated graphene nanoribbon to compute longitudinal two-terminal conductance, as well as spatial profiles of local current density as electrons propagate from NM leads into the Floquet TI. In the case of Floquet TI both bulk and edge local current densities contribute equally to total current, which leads to longitudinal conductance below the expected quantized plateau that is slightly reduced by edge vacancies. We propose two experimental schemes to detect coexistence of bulk and edge current densities within Floquet TI: (i) drilling a nanopore in the interior of irradiated region of graphene will induce backscattering of bulk current density, thereby reducing longitudinal conductance by $\sim 28$%; (ii) imaging of magnetic field produced by local current density using diamond NV centers.


I. INTRODUCTION
The defining property of topological insulators (TIs) is the band gap in the energy spectrum of the bulk material and gapless conducting boundary states.They are edge states in the case of two-dimensional (2D) systems or surfaces states in the case of three-dimensional ones [1].The paradigmatic cases which gave rise to the main concepts [2,3] in this field are: (i) quantum Hall insulator (QHI) in 2D electron gas which requires an external magnetic field to break the time-reversal invariance and whose edge states are chiral by allowing spin-unpolarized electron to propagate in only one direction; and (ii) quantum spin Hall insulator (QSHI) [4] which is time-reversal invariant but it requires strong spin-orbit coupling (SOC) effects, and whose edge states appear in pairs with different chirality and spin polarization.The last experimentally discovered member of 2D TI family is quantum anomalous Hall insulator (QAHI), which requires both nonzero magnetization to break the time reversal invariance and strong SOC effects, with edge states allowing only one spin species to flow unidirectionally [5].
In theoretical analysis, edge states are found as eigenfunctions Ψ kx (x, y) of the Hamiltonian of an infinite wire (periodic along the x-axis, so that eigenfunctions are labeled by the wavevector k x ) made of 2D TIs whose eigenenergies form subbands crossing the band gap [6,7].The width of the edge states is defined by the spatial region where the probability density is nonzero, |Ψ kx (x, y)| 2 = 0, while decaying exponentially fast towards the bulk of the wire.Interestingly, their width [8,9] can also depend on the arrangement of atoms along the edge, such as in the case of graphene wires where edge states of QHI and QAHI or QSHI are narrower in the case of zigzag arrangement of carbon atoms along the edge than in the case of their armchair arrangement [8][9][10].In paradigmatic 3D TIs like Bi 2 Se 3 , surface states actually have spatial extent of about ∼ 2 nm [11].
The zigzag edge, which is employed in devices in Fig. 1, can also introduce a kink in the subband of edge state [10], so that subband intersects with the Fermi energy E F at N R points with positive velocity and N L points with negative velocity, but only the difference N R − N L = |C| is topologically protected according to the bulk-boundary correspondence [2,3].Here C is an integer topological invariant (like the Chern number in the case of QHI and QAHI) associated with band structure in the bulk.This makes electronic transport through edge states of infinite length perfectly quantized in a robust way [12]-the zero-temperature two-terminal conductance is G(E F ) = G Q |C| for E F swept through the bulk band gap and insensitive to both magnetic and nonmagnetic disorder in the case of QHI and QAHI [10], or only nonmagnetic disorder in the case of QSHI.Although infinite ballistic wires, including those with topologically trivial edge states [15][16][17], also exhibit integer G(E F )/G Q , this is easily disrupted by disorder introduced around the edges or even within the bulk [17].Here h is the conductance quantum for spin-degenerate or spin-polarized edges states, respectively.
Thus, it has been considered that the key experimental signature of topology in 2D condensed matter is conductance quantization in transport through edge states, which persists even in the presence of disorder as long as it does not break underlying symmetries of the topological phase or generates energy scales that are larger than the bulk band gap.However, for QHI, QAHI and QSHI of finite length, the zero-temperature longitudinal conductance G = I/V b , also denoted as 'two-terminal' since current I and small bias voltage V b are measured between the same normal metal (NM) leads, oscillates in Fig. 2 just below the quantized plateau at 2e 2 /h while remaining very close to it.We use zigzag graphene nanoribbon (ZGNR) of finite length [Fig.1(b)] with sufficiently large external magnetic field [13], or additional terms of the Haldane [14] or the Kane-Mele [4] models, to generate QHI, QAHI and QSHI, respectively.Their parameters are tuned so that all three examples of conventional timeindependent 2D TIs in Fig. 2(a) have identical topologically nontrivial bulk band gap ∆ g .Even though G(E F ) for E F within the bulk band gap ∆ g is not perfectly quantized in Fig. 2(a), its oscillations zoomed in Figs.2(b)-(g) are completely insensitive to nonmagnetic edge disorder (ED) introduced in the form of edge vacancies [as illustrated in Fig. 4].
It is worth mentioning that imperfectly quantized twoterminal G(E F ) was observed in early experiments on QSHI [18], provoking a lively search for exotic manybody inelastic effects [19][20][21][22][23][24][25] which can circumvent bandtopology constraints and introduce backscattering of electrons as they propagate through edge states.On the other hand, Fig. 2 demonstrates that imperfectly quantized G(E F ) can be due to a much simpler mechanismbackscattering at the NM-lead/TI-region interface.
Lacking perfectly quantized G(E F ) as the experimental signature of 2D TI phase, one can resort to direct imaging of spatial profiles of local current density that should confirm transport confined to narrow region defined by the edge states.Continuous experimental advances have made this possible, such as by using superconducting interferometry in Josephson junction setup [15,16] or scanning superconducting quantum interference device (SQUID) [26].In the latter case, one images magnetic field produced by the current from which one can reconstruct the local current density with ∼ µm spatial resolution [26].Even higher resolution, with reconstructed images having spatial resolution of ∼ 10 nm, has been achieved by using scanning tip based on electronic spin of a diamond nitrogen-vacancy (NV) centers [27][28][29].Particularly intriguing questions that such images can answer is how electron flux transitions from topologically trivial NM lead present in every experimental device into the region of 2D TI of finite length where the flux is confined within narrow edge currents, as well as how processes at the NM-lead/2D-TI interface affect the total current and the corresponding conductance.
Imaging of local current density could also offer new avenue for resolving a crucial issue for recently conjectured new class of 2D TIs-the so-called Floquet TI [30][31][32][33]-which is the connection between the Floquet quasi-energy spectrum and experimentally measurable dc transport properties.The Floquet TI phase arises in 2D electron systems driven out of equilibrium by strong light-matter interaction.For example, graphene [30][31][32][33], as well as other 2D materials with honeycomb lattice structure like transition-metal dichalcogenides [34], subject to a spatially uniform and circularly polarized (CP) light are predicted to transmute into Floquet TI with quasi-energy spectrum [35,36].Its multiple gaps share [31] the same topological properties as the band gap of QAHI described by the Haldane model [14].This means that the laser induced band gaps, such as ∆ 0 in Fig. 3(a) emerging at the charge neutral point (CNP) of graphene and ∆ 1 away from CNP, are crossed by subbands of chiral edge states [37,38].The eigenfunctions of these subbands decay exponentially towards the bulk with a decay length that depends only on the ratio of the laser frequency and its intensity.
The ∆ 1 gaps are called dynamical gaps [39] and they occur at energy ω/2 above/below the CNP.They can be reached using experimentally accessible parameters-for example, the very recent experiment [40] has been interpreted in terms of creation of a transient Floquet TI by driving graphene flake by 500 fs laser pulse at a frequency of ω = 46 THz, so that the photon energy is ω ≈ 191 meV and wavelength is λ ≈ 6.5 µm.However, the experiment of Ref. [40] did not observe either quantized longitudinal or transverse (Hall) conductance.Instead, they found that at the peak laser pulse fluence the transverse conductance within ∆ 0 gap saturated at plateau around G xy = (1.8 ± 0.4)e 2 /h, while no such plateau of G xy was observed within ∆ 1 gap.
The calculations of two-terminal [as in Figs. 2 and  3(b)] or multi-terminal conductance typically assume the Landauer-Büttiker setup [12,41] depicted in Fig. 1 where finite-size scattering region-time-dependent due to light irradiation in Fig. 1(a) or conventional time-independent in Fig. 1(b)-is attached to semi-infinite NM leads terminating at infinity into the macroscopic particle reservoirs.This is highly appropriate for Floquet TI since time-dependent potential applied in experiments [40] is confined to a finite region, either because of a finite laser spot or the screening inside metallic contacts.On the conceptual side, such setup ensures well-defined asymptotic states and their occupation far away from the irra-diated region, thereby evading technical difficulties when using time-dependent leads or reservoirs [42].It also ensures continuous energy spectrum of the whole system which plays a key role in both the Landauer-Büttiker and Kubo [43] formulation of quantum transport because it effectively introduces dissipation at infinity and thereby steady-state transport, while not requiring to explicitly model many-body inelastic scattering processes responsible for dissipation [41].
However, for the same two-terminal Landauer-Büttiker setup with irradiated scattering region a plethora of conflicting theoretical conclusions have been reached [33].For example, Refs.[45,46] predict quantization of longitudinal dc conductance within a few percent of 2e 2 /h, while Ref. [47] finds its anomalous suppression.To recover the quantized value, Ref. [48] proposed an ad hoc summation procedure over different energies in the lead.Both Refs.[49,50] confirm nonquantized G < 2e 2 /h within ∆ 0 gap and G < 4e 2 /h within ∆ 1 gap which, however, are largely insensitive to disorder like vacancies or on-site impurities.The precise quantization could be disrupted by dc component of pumping current [51,52], which appears [49] even at zero bias voltage due to timedependent potential in the Hamiltonian whenever the left-right symmetry of the device is broken statically or dynamically [53,54].
The absence of quantization is explained [33,49,50] by the mismatch between nonirradiated electronic states in the NM leads and edge states within the gaps of the Floquet TI.The mismatch between states in topologically trivial NM leads and TI scattering region exists also in conventional time-independent TI devices, but without significant disruption of quantized conductance in Fig. 2.However, specific to Floquet TIs is possibility of Floquet replicas to couple to bulk bands [49,55].That is, although edge states within the gap ∆ 0 are primarily built from states near the CNP of nonirradiated graphene, they also contain harmonic components near ± ω which open possibility for electronic photon-assisted tunneling into or out of states in the NM leads whose energies are far away from the CNP.
The quantization of conductance can be examined without resorting to time-independent Floquet formalism and by performing direct time-dependent quantum transport simulations [42].Due to high computational demand, such calculations are rarely pursued, but they yield longitudinal conductance reaching nearly quantized value after sufficiently long time [56].This then poses a question on the accuracy of truncation procedure that is inevitably done to reduce infinite matrices in the Floquet formalism where artifacts [57] can be introduced, such as dc current which is not conserved (i.e., different in the left and right lead) [46,58] or insufficient number of Floquet replicas is retained to achieve convergence.
In this study, we employ charge-conserving solution [57] for the Floquet-nonequilibrium Green functions (Floquet-NEGF) [46,57,58] which ensures that dc current in the left (L) and the right (R) lead are identical at each level of truncation of matrices in the Floquet formalism, i.e., the number of of "photon" excitations N ph retained.As an overtire, Fig. 3(d) demonstrates as well as that dc component of current converges at N ph = 7.Nevertheless, the conductance in Fig. 3(b) remains nonquantized in both ∆ 0 and ∆ 1 gaps.We then proceed to compare spatial profiles of local current density in conventional and Floquet TIs in Fig. 4 which offer detailed microscopic insight on how electrons propagate from one to another carbon atom as they transition from topologically trivial NM leads into the TI region, or within the TI region with possible edge or bulk vacancies introduced as disorder.
The paper is organized as follows.Section II A describes the Hamiltonian of Floquet TI defined on ZGNR, as well as charge-conserving Floquet-NEGF from Ref. [57], which is extended here to nonzero bias voltage and to obtain local current density.The same ZGNR is used in Sec.II B to define Hamiltonians for the conventional time-independent QHI, QAHI and QSHI, where we also provide steady-state NEGF expressions for local current density in these systems.Section III A presents results for two-terminal conductance of these four TIs, and Sec.III B compares spatial profiles of local current density as it flows from the NM leads into those four TIs.In Sec.III C we discuss experimental schemes to quantify bulk vs. edge contributions to total current within Floquet TI using either a nanopore [9,59,60] drilled in the interior of irradiated ZGNR, whose effect on the conductance is also explicitly calculated, or magnetic field imaging via diamond NV centers [29].We conclude in Sec.IV.

A. Hamiltonian and quantum transport formalism for Floquet TI
The semi-infinite leads and the scattering region in Fig. 1 constitute, prior to introducing light or external magnetic field or SOC into the scattering region, an infinite homogeneous ZGNR described by the nearestneighbor (NN) tight-binding (TB) Hamiltonian where ij indicates sum over NN sites.Here ĉ † i (ĉ j ) creates (annihilates) an electron on site i of the honeycomb lattice hosting a single p z -orbital r|i = π(r − R i ) and γ ij = γ = 2.7 eV is the NN hopping from site i to j.The width of the ZGNR is chosen as W = 29a, where a is the distance between two NN carbon atoms in graphene.The ZGNR terminates at infinity into the macroscopic reservoirs of electrons whose chemical potentials are µ L = E F + eV b /2 and µ R = E F − eV b /2 for E F as the Fermi energy and V b as the applied dc bias voltage.Note that zero-temperature two-terminal conductance G(E F ) of an infinite homogeneous ZGNR described by Hamiltonian in Eq. ( 1) is plotted for comparison in Fig. 3(b) and labeled as nonirradiated (NIR).
In the case of Floquet TI, CP monochromatic laser light irradiates the scattering region (shaded blue) of finite length L = 30 √ 3a in Fig. 1(a).The electromagnetic field of light is introduced into the Hamiltonian via the vector potential A(t) = A 0 (e x cos ωt + e y sin ωt), where e x (e y ) is the unit vector along the +x-axis (+yaxis).The corresponding electric field generated by A(t) is E(t) = −∂A(t)/∂t.We neglect the relativistic magnetic field of light, so that electronic spin degree of freedom maintains its degeneracy and it is excluded from our analysis.The vector potential modifies the Hamiltonian in Eq. ( 1) via the Peierls substitution where z = eaA 0 /2 is a dimensionless measure of intensity of the CP light; ω is the frequency and r ij is the position vector connecting site i with site j.The new Hamiltonian Ĥ(t) with time-dependent hopping between sites i and j, γ ij (t) = γ exp i2z(e x cos ωt + e y sin ωt) • r ij , is time-periodic, Ĥ(t + T ) = Ĥ(t), with period T = 2π/ω.Any solution of the Schrödinger equation, i ∂Ψ(t)/∂t = Ĥ(t)Ψ(t), can be expressed as a linear combination, Ψ(t) = α c α φ F α (t), of the so-called Floquet functions [35,36] where ξ α QE is the Floquet quasi-energy and u α (t + T ) = u α (t) are periodic functions.They can be expanded into a Fourier series, n (r).The time-periodic Ĥ(t) can be expanded into a Fourier series as well where Ĥn is given in terms of the Bessel functions J m (z) of the first kind Using the matrix representation of the Fourier coefficients H n in Eq. ( 4) in the basis of orbitals |i , we construct the Floquet Hamiltonian [35,36] which is time-independent but infinite.The time-dependent NEGF formalism [42] operates with two fundamental quantities [61]-the retarded G r (t, t ) and the lesser G < (t, t ) Green functions (GF)which describe the density of available quantum states and how electrons occupy those states in nonequilibrium, respectively.
They depend on two times, but solutions can be sought in other representations, such as the double-time-Fourier-transformed [57,58] GFs, G r,< (E, E ).
In the case of periodic timedependent Hamiltonian, they must take the form [62 n (E), in accord with the Floquet theorem [35,36].The coupling of energies E and E + nω (n is integer) indicate "multiphoton" exchange processes.In the absence of many-body (electron-electron or electron-boson) interactions, currents can be expressed using solely the Floquet-retarded-GF Ǧr (E) which is composed of G r n (E) submatrices along the diagonal.Here we use and Σr (E) is the retarded Floquet self-energy matrix ) composed of the usual self-energies of the leads [63], Σ r (E) = p=L,R Σ r p (E), on the diagonal.All matrices labeled as Ǒ are representations of operators acting in the Floquet-Sambe [36] space, H F = H T ⊗ H e , where H e is the Hilbert space of electronic states spanned by localized orbitals |i and H T is the Hilbert space of periodic functions with period T = 2π/ω spanned by orthonormal Fourier vectors t|n = exp(inωt).
The charge current I p (t) in the lead p = L, R is timedependent due to Eq. ( 2), and it also has periodicity T = 2π/ω like the Hamiltonian itself.The dc component of current, either due to pumping by time-dependent potential [51][52][53][54] or due to applied bias voltage V b or both, is given by Such dc component injected into the lead p is obtained from the following NEGF expression (11) In our convention, I p > 0 indicates that charge current is flowing into the lead.Here fp is the Floquet Fermi matrix where f p (E) is the Fermi function of the macroscopic particle reservoir attached to lead p; ] is the Floquet level broadening matrix; Γ(E) = p=L,R Γp (E); the Floquetadvanced-GF is defined as Ǧa (E) = [ Ǧr (E)] † ; and 1 is the unit matrix in H e space.We note that Eq. ( 11) is generalization of the expression for charge current in Ref. [57] to include the applied bias voltage V b .The linear-response two-terminal conductance is then given by for small applied bias voltage eV b E F .While the space H e is finite-dimensional, with dimension equal to the number of sites N e within the scattering region, the space H T is infinite-dimensional and has to be truncated using |n| ≤ N ph .For truncation we employ the following convergence criterion where δ is the convergence tolerance.Since the operators acting in H e are represented by matrices of dimension N e × N e , the operators Ǒ acting on the truncated Floquet-Sambe space H F are represented by matrices of dimension (2N ph + 1)N e × (2N ph + 1)N e .Note that the trace in Eq. ( 11), Tr ≡ Tr e Tr T , is summing over contributions from different subspaces of H T so that the denominator includes 2N ph to avoid double counting.The part of the trace operating in H T space ensures that at each chosen truncation N ph of Floquet replicas charge current is conserved, I L = −I R , unlike other types of solutions [46,58] of the Floquet-NEGF equations where current conservation is ensured only in the limit N ph → ∞.
The bond current operator [64] between sites i and j is time-dependent due to Eq. ( 4) and it is given by We define the Floquet bond current matrix as which yields nonequilibrium part [64] of dc bond (or local) charge current flowing between site i and j as where Ǧ< (E) = p=L,R i Ǧr (E) Γp (E) fp (E) Ǧa (E).

B. Hamiltonian and quantum transport formalism
for QHI, QAHI and QSHI The two-terminal setup in Fig. 1(b) hosts one of the three conventional time-independent TIs as the scattering region (shaded green) of finite length L = 30 √ 3a.The QHI is realized by applying an external timeindependent magnetic field perpendicular to ZGNR.The magnetic field is described by a static vector potential A = (By, 0, 0) in the Landau gauge, which is then included into the Hamiltonian in Eq. ( 1) via the Peierls substitution Here (x i , y i ) indicates the position vector of a carbon atom at site i, and β = eBa 2 0 / √ 3 ≈ 0.07 is a dimensionless measure of the magnetic field strength.
The QAHI [5] is described by the Haldane model [1,14] on the honeycomb lattice Here ij indicates the sum over next-NN sites, and γ ij = − γ ji = i γ where we use γ = 0.14γ.The last two terms on the right hand side have different sign on the sublattice A and B of the honeycomb lattice, where we use m = 0.2γ.Finally, the QSHI is described by the Kane-Mele model [4] ) whose edge states crossing the topological nontrivial band gap are both chiral and spin-polarized [2,3].Here

is a row vector of creation operators ĉ †
iσ that create an electron on site i with spin σ =↑, ↓; c i is the corresponding column vector of annihilation operators; d ik is the unit vector pointing from site k to i; σ = (σ x , σy , σz ) is the vector of the Pauli matrices; and t SO is the strength of the intrinsic SOC [4,9].The zero-temperature two-terminal conductance G(E F ) = G Q T (E) of the setup in Fig. 1(b) is calculated using the Landauer transmission function [41,61] T where the conductance quantum is G Q = 2e 2 /h for QHI and QAHI and G Q = e 2 /h for QSHI.Here the retarded GF of the scattering region is given by G ] are the level-broadening matrices.To compute the nonequilibrium bond current between sites i and j we use [65] where J ij is the bond current operator in Eq. ( 15) but with time-independent hopping γ ij .

A. Conductance within the topological gap: FTI vs. conventional TIs
By diagonalizing ȞF in Eq. ( 6) for an infinite ZGNR that is periodic along the x-axis and irradiated by CP light over its whole length, we obtain the quasi-energy spectrum ξ QE (k x ) shown in Fig. 3(a).The chiral edge states crossing the light-induced gap ∆ 0 at ξ QE = 0 (shaded yellow) and ∆ 1 at ξ QE = ± ω/2 (shaded red) suggest naively that upon applying small bias voltage the zero-temperature linear-response two-terminal conductance in Eq. ( 13) should be quantized: G(E F ) = 2e 2 /h for E F within ∆ 0 gap; and G(E F ) = 4e 2 /h for E F within ∆ 1 gap due to one or two spin-degenerate conduction channels provided by the edge states, respectively.This is in analogy with chiral edge states of conventional time-independent TIs and their quantized conductance in Fig. 2. In contrast, the average conductance in Fig. 3(b) is G(E F ) ≈ 0.73 × 2e 2 /h within ∆ 0 gap and G(E F ) ≈ 1.87×2e 2 /h within ∆ 1 gap.We emphasize that these results are not an artifact of truncation of Floquet Hamiltonian ȞF in Eq. ( 6) since current in the L and R lead in Fig. 3(d) converge at N ph = 7 using δ = 1% criterion in Eq. ( 14).Also, our Floquet-NEGF formalism [57] ensures We additionally plot the total density of states (DOS) D(E) = j D j (E) in Fig. 3(c), which is nonzero within the gaps ∆ 0 and ∆ 1 due to contributions from the local DOS (LDOS) D j (E) originating [Fig.6(a)] from both edges and bulk of ZGNR.The LDOS is extracted from the Floquet-retarded-GF in Eq. ( 7) using where Tr T is the partial trace over states in H T .
Even though Floquet TI in irradiated ZGNR does not exhibit quantized conductance plateau in Fig. 3(b), its conductance is largely insensitive to ED.For example, G(E F ) is reduced by ∼ 2% within ∆ 0 gap and by ∼ 15% within ∆ 1 gap upon introducing edge vacancies.Nevertheless, this is still less robust than conventional timeindependent TIs whose conductance within the topologically nontrivial band gap is completely insensitive to ED, as shown in Figs.2(e)-(g).The disorder is introduced by removing carbon atoms on the top and bottom edges of the scattering region, as illustrated in Fig. 4(b), while imposing the following conditions: (i) ED introduced in NIR ZGNR leads to complete conductance supression G(E F ) → 0 within the same energy interval defined by ∆ 0 gap; (ii) ED preserves the left-right symmetry of the device, so that charge pumping is absent when the ED ZGNR is irradiated with CP light [49,53,54] in the absence of dc bias voltage V b = 0.

B. Spatial profiles of local current density: Floquet TI vs. QHI, QAHI and QSHI
The spatial profiles of local current density, i.e., bond current J neq ij defined in Eqs. ( 17) and ( 22) for Floquet TI and conventional time-independent TIs, respectively, allows us to visualize how electrons transition from topologically trivial NM leads into chiral edge states within the TI region.Figures 4(c), 4(e) and 4(g) shows that bulk states contribute to current density within the leads, but current density becomes confined to narrow flux near the edges of QHI, QAHI and QSHI.The width of the flux corresponds to spatial extent of the edge state.As expected due to chirality of edge states, current flows only along the top edge in QHI and QAHI, while in QSHI it flows on both the top and bottom edges [26].This is because boundaries of QSHI host a pair of spin-polarized edge states [4], so that on the top edge electrons with spin σ =↑ move from left to right while at the bottom edge electrons with spin σ =↓ move from left to right.Upon introducing ED in Figs.4(d), 4(f) and 4(h), topological protection and quantized transport through edge states manifest by local current density circumventing the disorder since any backscattering would require to cross over to the other edge which is forbidden due to the absence of bulk states [12].
In contrast, local current density is nonzero within the whole Floquet TI in Fig. 4(a), with larger flux near the edges.Upon introducing ED, the edge flux circumvents the disorder but due to general nonlocality of quantum transport bulk flux is also reduced which explains slight reduction of conductance in Fig. 3(b) within gaps ∆ 0 and ∆ 1 .Interestingly, SQUID-based imaging of QSHI made from HgTe quantum wells has found that gate tuning of bulk conductivity can lead to transport regime where edge and bulk local current densities coexist [26].The trace of local current density is scanned by detecting its magnetic field produced according to the Biot-Savart law, which is possible even through the top gate employed to tune the carrier density.In this regime, experimental images were analyzed to quantify contribution of edge and bulk local currents to the total current.We perform similar analysis in Fig. 5 which shows that in pristine Floquet TI from Fig. 4(a), edge current contributes 44% and bulk current contributes 56% to the total current over the transverse cross section [marked by dashed line in Fig. 4(a)].Conversely, in the presence of edge disorder in Fig. 4(b), edge current contributes 52% and bulk current contributes 48% to the total current over the same transverse cross section.

C. Proposed experimental schemes for probing edge vs. bulk transport within Floquet TI: Graphene nanopore and magnetic field imaging
The spatial profiles of local current density of conventional time-independent TIs in Figs.4(c)-(h) indicate that any disorder introduced in the interior of ZGNR will have no effect on the two-terminal conductance in Fig. 2.This was explicitly demonstrated in Ref. [9] for the case of QSHI (based on graphene plus heavy adatoms).Therefore, we propose to employ a nanopore in the ZGNR interior as the simplest technique that can detect the presence of bulk current density in Figs.4(a) and 4(b) in the case of Floquet TI.We introduce nanopore in Figs.6(b) and 6(c) in such a way that it preserves the left-right symmetry of the device in order to avoid any charge pumping by time-dependent potential of light [49,53,54].In experiments, nanopores are routinely drilled, without disrupting the surrounding honeycomb lattice of graphene, for applications like DNA sequencing [59], and they could also be deployed to block phonon transport in thermoelectric applications [9,60].Figures 6(a) and 6(b) confirms that nanopore does not impair high LDOS [Eq.(23)] near the edges of the Floquet TI, which correspond to chiral edge states from Fig. 3(a).Figure 6(c) shows that local transport in the presence of nanopore utilizes both left-to-right moving chiral edge states and bulk states.Since electrons flowing through the bulk are backscattered by the nanopore, presence of nanopore reduces conductance by about 28% in Fig. 6(d A more detailed probing of edge vs. bulk transport in ∼ µm-sized devices, such as those employed in recent experiments [40] to convert graphene into Floquet TI, could be achieved using diamond NV centers.The device can be fabricated on a diamond containing high-density, nearsurface NV ensemble [27,29], along with a graphite top gate separated by hexagonal boron nitride to tune the carrier density [29].The spin state of NV centers, which serve as the sensor of magnetic field produced by local current density, can be optically initialized and readout via imaging the NV photoluminescence onto a camera.Such a setup has the advantages of being able to operate over a wide range of temperatures, from cryogenic to room temperature (e.g., the experiment in Ref. [40] was done at 80 K); it can be readily integrated with an optical cryostat necessary for experiments involving THz radiation; and it has less stringent vibrational requirement compared to scanning setups.We note that THz radiation is far detuned from any of the NV orbital/spin transitions and hence it will not affect NV centers at all.In Ref. [40], a constant DC bias generates a current I 125 µA, and THz pulses drive the system into Floquet TI state for about 3 ps at ∼ 210 kHz repetition rate.Hence, one has a time-averaged typical current density JF ∼ 80 pA/µm in a 1-µm-wide device.This corresponds to a typical stray magnetic field µ 0 J ∼ 0.1 nT, where µ 0 is the permeability of free space.While it is a small field, its measurement is attainable with existing NV sensing technologies.For example, a single NV can sense ∼ nT field with a Hahnecho sequence over 100 seconds signal averaging at room temperature [66].Detection of ∼ 0.1 nT field is attainable in combination with entanglement-assisted repetitive readout [67,68], which is available for NV ensemble [69], as well as with enhanced coherence at cryogenic temperatures and with dynamical decoupling se-  17)] in the same irradiated ZGNR with a nanpore as in (b).(d) Zero-temperature two-terminal conductance G(EF ) of irradiated ZGNR with a nanopore (orange line) vs. conductance of irradiated pristine ZGNR (blue line) within the gap ∆1 in Fig. 3(a).The former is reduced by ∼ 28% with respect to the latter.In all panels we use ω = 3γ, z = 0.5 and N ph = 7.
quences [70].One can measure the differential current density ∆J(x, y) ≡ J FTI (x, y) − J NIR (x, y), where J FTI (x, y) [J NIR (x, y)] is current density within the Floquet TI [nonirradiated normal phase], by pulsing on the THz radiation during one free precession time of the Hahn-echo and keeping the THz drive off during the other free precession.The current density J NIR (x, y) can be measured separately in a Hahn-echo measurement without any THz pulses to enable one to extract J FTI (x, y).Diffraction-limited optical imaging of magnetic field has resolution ∼ 400 nm [29], which is enough to resolve edge currents separated by a width of 1 µm.If further improvement in spatial resolution is needed, ∼ 10 nm resolution can be achieved via Fourier gradient imaging [71].

IV. CONCLUSIONS
In conclusion, using steady-state NEGF formalism applied to two-terminal Landauer-Büttiker setup [Fig.1(b)] with scattering region consisting of conventional time-independent TIs-such as QHI, QAHI and QSHIdefined on graphene nanoribbon in order to generate chiral edge states of finite length, we demonstrate that their conductance is never perfectly quantized [Fig.2] due to backscattering at the NM-lead/2D-TI interface.Nevertheless, it remains very close to perfect plateau at 2e 2 /h within the topologically nontrivial band gap, and it is completely insensitive to edge disorder.The spatial profiles of local current density visualize how electrons flow from bulk states within topologically trivial NM leads into the narrow flux defined by edge states within the TI region, while circumventing any edge disorder within the TI region.
In contrast, when the scattering region is converted into the Floquet TI by irradiating graphene nanoribbon by circularly polarized light, conductance within lightinduced topologically nontrivial band gaps is not quantized, but it changes little with edge disorder.This results confirm previous findings in the literature [49,50] while ensuring proper convergence and charge current conservation in the solution of Floquet-NEGF equations [57].Furthermore, we use Floquet-NEGF formalism to compute spatial profiles of local current density which is high along the edges, following high LDOS near the edges, but it remains nonzero in the interior of the Floquet TI as well.Thus, observing quantized transport in Floquet TI would require to minimize coupling to bulk states.We propose a very simple technique to detect presence or absence of bulk states in quantum transport through Floquet TI-conductance measurements under laser irradiation should be performed using uniform graphene nanoribbon, as in the very recent experiments [40], as well as repeated after a nanopore [59] is drilled in the interior of such nanoribbon.If local current density is nonzero in the bulk, it will be scattered by the nanopore which leads to 28% reduction [Fig.6(d)] of the two-terminal conductance when compared to graphene nanoribbon without the nanopore.Finally, we delineate possible experimental scheme for direct imaging [29] of magnetic field produced by edge and bulk local current density based on diamond NV centers whose orbital/spin transitions are far detuned from THz radiation employed [40] in recent experiments to convert graphene into Floquet TI.

FIG. 1 .
FIG. 1.(a) Schematic view of a two-terminal device where an infinite ZGNR is attached to two macroscopic reservoirs with chemical potentials µ L and µ R on the left and right, respectively, so that µ L − µ R = eV b is externally applied dc bias voltage.The scattering region (blue shaded) in the middle of ZGNR of length L and width W is Floquet TI generated by irradiation [30] by circularly polarized light of intensity z and frequency ω [Eq.(2)].The scattering region (shaded green) in panel (b) is quantum Hall, quantum anomalous Hall or quantum spin Hall insulator with their parameters tuned to produce the same topologically nontrivial band gap ∆g [Fig.2(a)] in the bulk of all three conventional time-independent TIs.
FIG. 2. (a) The zero-temperature two-terminal conductance vs. the Fermi energy EF for ZGNR-based pristine QHI, QAHI and QSHI; as well as for these three 2D TIs with edge disorder (denoted by ED) in the form of vacancies illustrated in Figs.4(d), 4(f) and 4(h).The gap in the bulk of all three 2D TIs is tuned to ∆g = 0.54γ and marked in panel (a).The zoom in of conductance values within the rectangle in panel (a) is shown in: (b)-(d) for pristine ZGNR; and (e)-(g) for edge-disordered ZGNR.

FIG. 3 .
FIG. 3. (a)Quasi-energy spectrum ξ QE (kx) for an infinite ZGNR that is irradiated by CP monochromatic laser light of frequency ω = 3γ and intensity z = 0.5 over its whole length.The spectrum is obtained by diagonalizing the corresponding Floquet Hamiltonian [Eq.(6)] truncated to −N ph < n < N ph Floquet replicas where N ph = 7 is chosen.The yellow shaded region marks the topological gap ∆0 around ξ QE = 0 corresponding to CNP, while the red shaded region marks the dynamical topological gap ∆1 around ξ QE = ± ω/2.(b) The zero-temperature two-terminal conductance vs. EF (computed using N ph = 7) of two-terminal device in Fig.1(a) whose scattering region is Floquet TI of finite length due to irradiation by CP light.The pristine irradiated ZGNR is marked by FTI and irradiated edge-disordered ZGNR is marked by FTI-ED.The conductance of an infinite nonirradiated (NIR) pristine ZGNR is also show as a reference.(c) Total DOS for the same device marked by FTI in panel (b).(d) Convergence of lead currents I L and I R vs. N ph at EF = ω/2 .

FIG. 5 .
FIG. 5. Spatial profile of local current density in Figs.4(a) and 4(b) over the transverse cross section within: (a) pristine Floquet TI; or (b) Floquet TI with edge disorder.The position of the transverse cross section is marked by dashed vertical line in Figs.4(a) and 4(b), respectively.The horizontal dashed line in both panels marks the extent of the edge state.

FIG. 6 .
FIG. 6. (a)The LDOS [Eq.(23)] evaluated at E = ω/2 in the center of ∆1 gap in Fig.3(a) for irradiated pristine ZGNR.(b) The LDOS evaluated at E = ω/2 for irradiated ZGNR with a nanopore drilled in the interior of the nanoribbon.(c) Time-averaged local bond current [Eq.(17)] in the same irradiated ZGNR with a nanpore as in (b).(d) Zero-temperature two-terminal conductance G(EF ) of irradiated ZGNR with a nanopore (orange line) vs. conductance of irradiated pristine ZGNR (blue line) within the gap ∆1 in Fig.3(a).The former is reduced by ∼ 28% with respect to the latter.In all panels we use ω = 3γ, z = 0.5 and N ph = 7.