Transport signatures of symmetry protection in one-dimensional topological insulators
Topological insulators are identified as systems where a gapped bulk supports in-gap edge states, protected against symmetry-preserving local perturbations. In one dimension the robustness of the edge states shows up as a pinning of their energy levels in the middle of the bulk band gap. Here we propose a scheme for probing this intriguing feature by observing transport characteristics of a topologically nontrivial dimer array attached to external leads. We present predictions for transmission spectra using a numerical nonequilibrium Green’s function approach. Our analysis covers both time-independent and periodically driven (Floquet) arrays, with time-independent and periodically driven boundary perturbations which either preserve or break the protecting chiral symmetry.
pacs:71.10.Fd, 71.23.An, 73.20.At, 73.23.-b, 73.63.-b
Symmetry-protected topological phases of matter have been at the forefront of condensed matter research for more than a decade now, with topological insulators serving as a central paradigm in the case of noninteracting fermions Hasan (); Qi (). The hallmark of these systems is the presence of robust boundary states, originating from the nontrivial topology of the states in the bulk (“bulk-boundary correspondence” Rhim ()) and protected against local perturbations provided that these perturbations respect some relevant underlying symmetry.
While the bulk physics of topological insulators is by now well understood Chiu (), experimental confirmation of the robustness of their boundary states is still somewhat unfulfilled. A case in point is the quantum spin Hall insulator Maciejko () the very discovery of which triggered the avalanche of research in the field. Here experiments still fail to see the conductance quantization expected from the symmetry-protected ballistic edge states unless the conduction channel is extremely short EXP (). While the symmetry protection of the boundary states is not being challenged per se Novelli (), the case testifies to the difficulty of probing and unambiguously characterizing these states. The problem is aggravated by the fact that materials with the right electronic properties to firmly sustain a symmetry-protected topological phase are not easy to come by, making viable experimental case studies scarce. Driving a nontopological system into a topological phase by applying an external periodic field has expanded the range of candidate systems Kitagawa (); Lindner () with shaken optical lattices Hauke () and photo-induced states Rechtsman () being examples of “Floquet engineered” topological phases Bukov (). Intriguingly, this line of research has uncovered features not seen in ordinary static topological insulators Rudner (); Carpentier (), also as it comes to transport properties Kitagawa2 (); Gu (); Kundu (); Farrell ().
To experimentally validate the symmetry protection of boundary states in a topological insulator, be it static or periodically driven, one would want an easily controllable setup where there is a clearcut signal differentiating between the response to symmetry-preserving and symmetry-breaking local perturbations respectively, showing that the boundary levels are unperturbed (perturbed) in the former (latter) case. In this paper we try to contribute to this effort by suggesting a blueprint for such a setup.
In short, we suggest to use a 1D topologically insulating structure made out of a dimer array connected to external leads biased by an applied voltage (Fig. 1). In contrast to 2D and 3D topological insulators which host linearly dispersing boundary states which, ideally, support ballistic transport Hasan (); Qi (), a 1D topological insulator exhibits boundary states stuck at its two edges AsbothBook (). Still, in a finite system, the hybridization of the edge states may open a channel for coherent transport of electrons, making them share a feature akin to their higher-dimensional relatives. Here we take advantage of this property to propose a scheme where measuring a dc current across a finite topologically insulating dimer array is expected to yield distinct fingerprints of the symmetry protection of the edge states.
The paper is organized as follows: In Sec. II we describe the setup and the modeling of the dimer array by the Su-Schrieffer-Heeger (SSH) model SSH (). In this section we also predict the transmission spectra for undriven (time-independent) and periodically driven (Floquet) arrays, using Landauer-Büttiker theory Datta () within a nonequilibrium Green’s function formalism. In Sec. III we then turn to the central theme of the paper: the symmetry protection of the edge states. Our analysis, again employing a nonequilibrium Green’s function approach, covers both time-independent and periodically driven arrays, with time-independent and periodically driven boundary perturbations which either preserve or break the protecting underlying symmetry, here being a chiral symmetry. Sec. IV contains a discussion of some experimental issues, with Sec. V briefly summarizing our results. Technical details are expounded in four appendices.
Ii Transport through Topologically Nontrivial Dimer Arrays
The setup we consider is schematically illustrated in Fig. 1. The system is composed of an array of dimers connected to two external leads by contacts represented by matrices and respectively. Electrons tunnel between the monomers of the dimers with intracell (intercell) amplitude . Applying a gate voltage allows for a shift of the electrostatic potential of the array with respect to the potentials of the leads, where the latter are biased by an applied voltage . Neglecting charging effects, the transport region made up of the dimer array can then be described by the single-particle device Hamiltonian
up to an overall constant determined by the gate voltage. Here runs over all cell indices, is a spin index, and and represent the bipartite structure of the array.
The Hamiltonian in Eq. (1) is that of the well-known Su-Schriefer-Heeger (SSH) model SSH (). The SSH model serves as a minimal model for a topological insulator and supports two topologically distinct phases, topologically trivial () and topological nontrivial () SSHtopo1 (); SSHtopo2 (); SSHtopo3 (). In the thermodynamic limit the topologically nontrivial phase is distinguished by the presence of zero-energy modes localized at the edges. When the length of the array is finite, these states hybridize and create additional transport channels across the structure. It follows that by attaching leads to the system one expects to observe nonzero transmission peaks even at biases within the bulk gap. Such peaks should gradually disappear with increasing length of the array, while being completely absent for topologically trivial arrays. In this way fingerprints of the topological edge states can be established.
ii.1 Transmission Through Undriven Dimer Arrays
Measurable transport characteristics, like the dc current, can be estimated by employing Landauer-Büttiker theory within a nonequilibrium Green’s function formalism (for details, see Appendix A). The dc current across the array is given by the Landauer-Büttiker formula Datta ()
Here and are Fermi-Dirac distribution functions with and the electrostatic potentials of the left and right leads respectively, with the electron charge. is the transmission spectrum, being a function of , and the electron density of states (cf. Appendix A). The factor of two is due to the electron spin. We here consider only a small applied bias so that the transmission can be considered to be independent of . In an experiment this corresponds to measurements of the linear conductance, equal to when at zero temperature. To probe for different energies one then measures the low-temperature linear conductance at different values of the applied gate voltage, in this way sweeping the experimentally accessible part of the spectrum of the dimer array.
The transmission spectra corresponding to two topologically distinct finite SSH chains, trivial and nontrivial, are illustrated in Fig. 2. In the topologically nontrivial case (blue color) one identifies a transmission peak at as expected from the discussion above, as well as related work in Ref. Dong, : In the topologically nontrivial phase the two degenerate zero-energy boundary states hybridize and open a conduction channel across the array. The width of the peak is proportional to the broadening of the zero-energy boundary levels due to their mutual hybridization as well as to the coupling to the external leads, here taken to be the same for left and right lead, .
It is important to know how the in-gap transmission peak changes with the number of dimers in the array. In this way one may predict the array size which yields the largest transmission due to the presence of the topological states. In Appendix C we develop a theory from which such scaling can be predicted. The idea is to project the transport problem onto the closest states with zero energy (i.e. the boundary states split by their mutual hybridization), and assume that the “high-energy” states of the bulk do not contribute to the transport at small applied bias. We then calculate the Green’s function projected onto the space spanned by these states. The transmission spectrum is then directly retrieved. For the detailed calculation scheme, see Appendix C. Here we present only the result: In Fig. 3 we show zero transmission vs. number of dimers used in the array. The analytic result (red) agrees well with the numerical data (blue), validating our prediction derived in Appendix C.
ii.2 Transmission Through Periodically Driven Dimer Arrays
An approach similar to the one above can be applied also for distinguishing topologically distinct phases in arrays of periodically driven dimers. A topologically nontrivial phase can be realized, for example, by applying a time-periodic voltage to a keyboard of additional gates on top of the dimer array in Fig. 1, each gate producing a local ac field which allows to modulate the local chemical potentials and hopping amplitudes between nearby monomers.
Periodically driven quantum systems are most easily described within the Floquet formalism Shirley (). This theory states that there is a complete set of solutions to the time-periodic Schrödinger equation of the form , where and are real constants called quasienergies. The quasienergies form bands very similar to energy bands of a time-independent spatially periodic system. However, the quasienergies are defined only modulo where is the period of the drive, and by this induce an additional repetition of the bands. Such quasienergy band structures can be described within topological band theory and topologically distinct phases can be identified similar to the case of time-independent systems Kitagawa (); Lindner ().
Here we consider a dimer array described by Eq. (1) but with harmonically modulated hopping amplitudes and , with and constants and . Such arrays represent a periodically driven SSH model and in the topologically nontrivial regime they may support topological edge states not only at zero quasienergy but also at . It follows that in total there are four different topological phases distinguished by the presence or absence of edge states at quasienergies and . Both types of edge states create additional transport channels in finite arrays via hybridization and are expected to be seen in conductance measurements.
Transport properties can be obtained by again exploiting the Landauer-Büttiker formalism, now adapted to periodically driven systems. The calculations, numerical as well as analytical, are performed using the method developed by one of us in Ref. OB2018, . The idea is to derive all expressions within the frequency domain - sometimes called Floquet-Sambe space Sambe (). The advantage of this method compared to other equivalent techniques lies in its simplicity: Expressions for currents and densities essentially replicate well-known time-independent formulas but with time-independent objects replaced by their analogous within Floquet-Sambe theory. In this way the dc (time-averaged) current takes the form
where denotes the amplitude for a photon-assisted transmission of an electron from lead to lead at energy , and is the electrostatic potential of lead (). Note that this formula is in agreement with analogous ones derived using other methods, including the Floquet-Keldysh formalism Arrachea (). In contrast to the time-independent case in Eq. (2), Eq. (3) is not antisymmetric under exchange of the leads since for a driven system generally Kohler (). As a consequence, the differential conductance, , depends crucially on the voltage profile across the array. In the idealized case where the voltage drop is entirely at the contact between lead and the array, and , Eq. (3) implies that the zero-temperature linear (small bias ) conductance is given by Fruchart2 ()
with when the voltage drop is entirely at the contact between lead and the array. As before, to access the whole transmission range in one should perform the measurements at various gate biases. In the more realistic case, with an extended voltage profile across the array, the measured zero-temperature linear conductance will receive contributions from both transmission spectra and take a value between and . It follows that the topological fingerprints in both transmission spectra, to be uncovered below, should also be present in the realistic case of a linear conductance measurement. A short technical discussion on the computational approach including its numerical implementation is given in Appendix B. For a more detailed account we refer the reader to Ref. OB2018, . For simplicity, in what follows we display results only for , from now on simply denoted by , and skip the analogous discussion of .
The transmission spectra for four different sets of model parameters corresponding to the four distinct topological phases of the dimer array is illustrated in Fig. 4. Here, and in the following figures, the data was obtained from a numerical nonequilibrium Green’s function approach, using Floquet-Sambe matrices truncated to 7 rows and columns (for details, see Appendix B). All four cases in Fig. 4 agree perfectly with the topological invariants calculated in Refs. Lago, , OBHJ2017, : By the bulk-boundary correspondence for Floquet topological insulators FloquetBB (), the presence of topological edge states signalled by single peaks in the transmission gap around (“normal gap”) and/or (“anomalous gap”) corresponds to nonzero values of the topological index (normal gap) and (anomalous gap). In this way transport characteristics fingerprint distinct topological phases in a periodically driven system.
Similar to the undriven array, it is of obvious interest to understand how the transmission peaks change with the size of the array. Unfortunately, in the present case we have not been able to develop a fully analytic method to answer this question. Instead we follow the idea used in the time-independent case (cf. Appendix C) but now assisted by an approximate numerical diagonalization of the Hamiltonian in Floquet-Sambe space which allows us to obtain the required parameter values, cf. Appendix D. The result is then compared with the one obtained from the somewhat cumbersome numerical Green’s function method. The size dependence of the transmission at and , obtained using the two methods, is shown in Fig. 5. The data points from the two approaches are in very good agreement with each other, also for small system sizes (except for when ) where one does not expect such near-perfect agreement. The transmission at behaves similarly to the time-independent case, Fig. 3. On the other hand, the behavior of the transmission at is different. In Fig. 6 the transmission at is plotted as a function of not only array size but also of driving frequency . It is interesting to note that the transmission oscillates as the frequency is varied. The origin of such oscillations remains unknown to us and warrants a deeper investigation.
To conclude this section, let us stress that our analysis above does not rely on the particular choice of driving. Indeed, in an experiment one may look for other, maybe more easily attainable drives.
Iii Symmetry Protection of The Edge States
The boundary states of topologically nontrivial insulators posses a variety of interesting properties, in particular, they are expected to be protected against symmetry-preserving local perturbations. For a one-dimensional system in the thermodynamic limit the corresponding symmetry-protected states remain exactly in the middle of the bulk gap as long as the applied perturbation preserves certain symmetries and does not close the gap AsbothBook (). Here we propose that symmetry protection can be present to some extent (to be spelled out precisely below) also in a finite-size system, and can be seen in transport measurements performed on a perturbed array of dimers.
iii.1 Symmetries of the SSH Model
To set the stage, let us briefly recall that the SSH Hamiltonian, Eq. (1), posses a sublattice (chiral) symmetry. This symmetry manifests the fact that there is no coupling between sites from the same sublattice. We define the chiral symmetry operator as the difference between the projectors onto the two sublattices and ,
from which one easily verifies that the SSH model indeed is chiral symmetric: . The SSH model is diagonal in the spin index and can be thought of as two independent copies of the spinless SSH model, implying that all results for the latter carry over directly to the spinful case AsbothBook (). In particular, chiral symmetry forces the eigenmodes with opposite energies to come in pairs and therefore, in the thermodynamic limit, it requires that the zero-energy edge modes stay put at zero energy as long as the chiral symmetry is preserved and the bulk gap is open. It follows that a spatial disordering of hopping amplitudes and maintains the protection. On the contrary, the edge states are not robust against a disorder in the chemical potential since the corresponding local operators couple sites on the same sublattice and hence break chiral symmetry. To perturb all edge states we here always consider summation over both spins.
The periodically driven SSH model with driving chosen as in Sec. II B ( and , with ) also exhibits chiral symmetry, now adapted to the Floquet formalism. The chiral symmetry within this theory is defined for the evolution operator over one full period of the drive and is explicitly given by the relation . This symmetry remains unbroken as long as is fulfilled OBHJ2017 (). One easily verifies that the chosen driving respects this condition. Moreover, in analogy to the time-independent case, the edge states are protected also against time-periodic perturbations which preserve the chiral symmetry. Examples of such perturbations include site-dependent disordering of the hopping amplitudes and that is even in time (which, trivially, includes static perturbations), as well as perturbations from an added local chemical potential that is odd in time FloquetBB (); OBHJ2017 (). It follows that for these perturbations the edge modes have to stay put exactly at or quasienergy. It is important to note that the zero reference time is defined with respect to the bulk driving ( in our case) and therefore the protection of the edge states crucially relies on the relative phase between the applied time-periodic perturbation and the bulk driving.
One can also establish protection of the edge states in time-independent topological insulators against time-periodic perturbations. This is so because time-independent systems are trivially periodic in time. As a case in point, the undriven SSH model possesses a Floquet chiral symmetry, , for every reference time . This implies that the edge states stay protected as long as the chiral symmetry is unbroken for at least one choice of reference time . Thus, the symmetry-protected edge modes will stay put at zero quasienergy for any harmonic disordering perturbation in , or that acts at the boundary. Higher harmonics or disorders in a few parameters will in general break the symmetry and result in a quasienergy shift of the edge states away from zero.
iii.2 Transport Properties: Time-independent Arrays Under Static Perturbations
Let us first focus on undriven arrays and then extend our study to the periodically driven ones. As we saw in the previous section, edge states in a finite-size array hybridize and by this build up transport channels. The hybridization destroys the protection since the boundary modes may overlap under perturbations, including symmetry-preserving ones, and for that reason may split in energy. The splitting will introduce an energy mismatch between distinct edge states and therefore threaten to kill the corresponding transmission peak. However, the edge modes overlap mainly in the middle of the array, with each mode quickly decaying when approaching the opposite side. In this way, any perturbation applied close enough to an edge has a very small effect on the state localized on the opposite edge of the array, and hence, its energy has to remain very close to zero. As for the effect of a perturbation on the state localized on the very same edge at which the perturbation acts, chiral symmetry ensures that its energy stays put at zero (or close to zero when hybridization with the opposite edge state is taken into account) if the perturbation respects chiral symmetry, otherwise the energy level will be shifted. This implies that under a symmetry-preserving boundary perturbation the transmission spectrum for a finite array will still contain a well-defined peak at zero energy. In contrast, if a symmetry-breaking perturbation is applied, the peak will disappear. This prediction can be formally justified by projecting the transport problem onto the space spanned by the edge states, similar to what we did in Sec. II A. It is then possible to analytically predict the behavior of the transmission peak and discuss to which extent it reflects the symmetry-protection of the edge states. The necessary calculations, being somewhat technical, can be found in Appendix C.
Using our nonequilibrium Green’s function approach, in Fig. 7 we present numerical results for a finite-size dimer array in the nontrivial topological phase, subject to a boundary perturbation in the chemical potential , and in the hopping amplitudes and . In perfect agreement with a state-space projection analysis and the discussion above, one observes a dramatic drop in the zero bias transmission under a boundary disorder in the chemical potential (symmetry-breaking) while the peak survives under boundary perturbations in hopping amplitudes (symmetry-preserving). As indicated in the panels of Fig. 7, the perturbations of the hopping amplitudes are taken to be complex when doing the computations, in this way breaking particle-hole symmetry and leaving only the chiral symmetry to protect the edge states.
Considering the symmetry-breaking perturbation, it is quite striking that the presence of the transmission peak can be suppressed by perturbing just one (or a few) sites in the dimer array, in this way switching the transport from conducting () to insulating (). Let us also point out that if one applies a sufficiently strong symmetry-preserving boundary perturbation, the peak will first split into two and then gradually disappear. This is so because the edge state wavefunction carries some small but not negligible weight on the opposite side of the finite array, and therefore the levels of the states may split under a large boundary perturbation, even if the symmetry is preserved.
iii.3 Transport Properties: Periodically Driven Arrays Under Static Perturbations
Following closely the discussion in Sec. III B, we now generalize the analysis to periodically driven arrays. The robustness of the transmission peaks at and against a boundary perturbation also for these systems requires that the perturbation respects chiral symmetry. Analogous to the time-independent case, this may be explained formally by viewing the problem within a projected state space. The technical details can be found in Appendix D.
In Fig. 8 we present numerical results for periodically driven dimer arrays in the topological phase with , subject to various static boundary perturbations in each of the parameters (chemical potential), and , (hopping amplitudes). The behavior of the “normal” and “anomalous” midgap transmission peaks agree with the symmetry arguments: Both flavors of peaks (at and respectively) survive perturbations in the hopping amplitudes while vanishing rapidly under perturbations in the chemical potential. As for the time-independent case in Sec. III B, the perturbations in hopping amplitudes were taken complex-valued in order to break the particle-hole symmetry which would otherwise have protected the edge states also when chiral symmetry is broken.
iii.4 Transport Properties: Periodically Driven Arrays Under Time-Periodic Perturbations
The midgap edge states are protected also against symmetry-preserving time-periodic perturbations OBHJ2017 (). The time-dependence here grants an extra degree of flexibility compared to the perturbations discussed earlier and therefore these types of perturbations are probably of higher practical interest. In Fig. 9 we disturb arrays in a topological phase by a harmonic boundary perturbation in a single parameter (hopping amplitude , or chemical potential ) and vary the phase of the perturbation. Depending on the phase, the chiral symmetry may either be preserved or broken (cf. Sec. III A). The obtained transmission spectra faithfully signal this fact: The peaks at and only survive perturbations in hopping amplitudes (chemical potential) that are even (odd) in time. Note that in order to clearly see the anticipated effect we here had to increase the disordering amplitudes compared to the time-independent case. The most distinct transport signature of the symmetry protection is the behavior of the in-gap transmission spectra under the perturbations in the same parameter and amplitude but with a phase mismatch in time. In Fig. 9 this qualitative difference is strikingly manifest. One also notes in Fig. 9 that the transmission peaks in the anomalous gap (at ) are more sensitive to a breaking of the chiral symmetry than the peaks in the normal gap (at ).
iii.5 Transport Properties: Time-Independent Arrays Under Time-Periodic Perturbations
It is also interesting and possibly of higher relevance for future experiments to look at undriven arrays of dimers subject to periodic boundary perturbations. One expects such setups to be easier to implement in practice because they require much less control. In the top six panels of Fig. 10 we present transmission spectra for topologically nontrivial undriven arrays perturbed by the same time-periodic perturbations as for the driven arrays considered earlier (cf. Fig. 9). In agreement with theory, the midgap transmission peaks are only slightly affected by such perturbations and this is because in each of these cases there is always a reference time for which the Floquet chiral symmetry is preserved OBHJ2017 (). However, adding a higher out-of-phase harmonic to the perturbation breaks the chiral symmetry for all reference times and the transmission peak disappears much faster than if the perturbation is symmetry-preserving for only one reference time (see Fig. 10, bottom two panels). To see the difference in behavior we have to go to very large disordering amplitudes since the quasienergy shift induced by such symmetry-breaking perturbation is expected to be extremely small OBHJ2017 (). In fact, the quasienergy splitting may be even smaller than the quasienergy difference due to the finite-size effect. Therefore, in both symmetry-preserving and symmetry-breaking cases the transmission goes down first at roughly the same rate. At larger perturbation amplitudes, however, the different behaviors of the mid-gap transmissions become distinguishable: A small but not negligible bump near is present in symmetry-preserving but not in symmetry-breaking cases.
Iv Towards Experimental Tests
The setup studied in this work, Fig. 1, should be possible to realize experimentally by an array of quantum dots weakly attached to two leads. A number of experiments have recently been carried out on various multidot nanostructures Hensgens (); Mukhopadhyay (); Puddy (), with efficient control over 14 quantum dots reported in Ref. Puddy, . Thus, experimental studies of dimer arrays of the size probed in our numerical computations do appear to be within reach. It is here important to remember, however, that we have completely neglected electron-electron interactions in our analysis, and by this, the strong charging effects that come with quantum dots. These will reduce and smoothen the transmission peaks and could possibly wipe them out completely. Still, their reduction is expected to be gradual, allowing for detectable peaks to remain in some range of charging energies. To estimate this range is an important yet challenging task to be addressed.
Alternatively, one may look for other solid state systems that effectively behave as arrays of dimers and in which charging effects are strongly suppressed. Here we point to two very recent experimental setups where the electron-electron interaction seems to play a less noticeable role. In Ref. Drost, the SSH dimer arrays were realized by exploiting a vacancy lattice in a chlorine monolayer on a Cu(100) surface. The presence of midgap edge states was explicitly demonstrated by studying local density of states with scanning tunneling spectroscopy. Another realization, in Ref. Belopolski, , exploited a heterostructure made of alternating layers of topologically trivial and nontrivial insulators. The edge states were there identified using angle-resolved photoemission spectroscopy measurements. Both these experiments suggest other ways of realizing SSH arrays in the solid state, opening for possibilities to study transport signatures of symmetry protection discussed in this work.
We have proposed a setup for obtaining transport signatures of the robustness of topological edge states in a finite dimer array in the presence of local boundary perturbations. Representing the array by the Su-Schrieffer-Heeger model and using a numerical nonequilibrium Green’s function approach with the array contacted to two leads biased by a small voltage, we have obtained the transmission spectra encoding the zero-temperature linear conductances for different types of perturbations. The results which cover time-independent as well as periodically driven (Floquet) arrays, with time-independent and periodically driven local boundary perturbations which either respect or violate the protecting chiral symmetry boost our proposal for experimental tests of symmetry protection of topological edge states: A perturbation which respects (violates) the symmetry correlates perfectly with the presence (absence) of a distinct mid-gap peak in the transmission spectrum. Recent advances in the precise control and study of multidot nanostructures Hensgens (); Mukhopadhyay (); Puddy (), vacancy lattices Drost (), and topological multilayer insulator structures Belopolski (), suggest that an experimental test could soon become possible.
This work was supported by the Swedish Research Council through Grant No. 621-2014-5972.
Appendix A Details on the Green’s function formalism: Time-independent
Here we provide details on the non-equilibrium Green’s function theory used in this work. Transport properties can be extracted from the (retarded) Green’s function defined as
where is the device Hamiltonian in Eq. (1), () are self-energies representing the leads, and with infinitesimal. Within the Landauer-Büttiker formalismDatta (), the transmission spectrum defined in Eq. (2) can be obtained from the relation
with broadening functions . We shall simplify the problem by applying the so-called wide-band limit approximation WBL (). This assumes that the density of states in the leads does not vary much near the Fermi energy and therefore can be taken constant. It follows that can be considered independent of . Also, within this approximation we do not take into account shifts of the Green’s function poles: is set to zero and the self-energies take the simple form , where are energy-independent real symmetric matrices. The wide-band limit approximation well represents metallic leads WBL () and drastically simplifies computations.
In general, the self-energies describe tunneling between the leads and the transport region. Here we assume that the coupling is only between the leads and the first/last sites of the array. Therefore, the matrices are zero everywhere except for the first () or last () diagonal entries. Thus, from Eq. (7), the transmission is given by
with spin index either or . and are the only non-zero elements of the matrices and respectively, and is the corresponding matrix entry of the Green ’s function .
Appendix B Details on the Green’s function formalism: Periodically-driven
Transport through periodically driven systems can be addressed in a similar way as for the undriven case. The idea is to go to the so-called Floquet-Sambe space Sambe (), transforming the periodically driven problem into an equivalent time-independent one. Within such a representation the time-periodic Hamiltonian turns into a Floquet-Sambe Hamiltonian , defined by
where and is the identity operator. The notation here represents a time-periodic state with period , in particular . The inner product between the time-periodic states is obtained by time-averaging the conventional inner product over one period. Explicitly, the Floquet-Sambe Hamiltonian reads as follows:
Here we focus on the dimer array, Eq. (1), driven by time-periodic external gates and subject to a time-independent voltage (from the bias between the electrochemical potentials of the two leads contacted to the array). Within this scenario the dc current may be obtained through Eq. (3) with photon-assisted transmissions () defined by OB2018 ()
We have here used the notation in Ref. OB2018, : is the (retarded) Floquet-Sambe Green’s function corresponding to a time-periodic Hamiltonian (obtained from Eq. (1) by making one or several of the parameters in time-periodic), are Floquet-Sambe broadening functions with Floquet-Sambe self-energies. Within the considered class of systems the Floquet-Sambe self-energies are given by
with conventional time-independent self-energies (). Finally, in Eq. (11) represents the Floquet-Sambe zero matrix with the block being replaced by the time-independent broadening function . For more details we refer the reader to Ref. OB2018, .
In analogy to the time-independent case we simplify the description by employing the wide-band limit approximation, now extended to the Floquet formalism: It is assumed that is -independent and is equal to . Clearly, this assumption is valid as long as does not vary much near the energy window with equilibrium Fermi energy and cutoff of the Floquet-Sambe space. Usually, (with counting the number of blocks in the Floquet-Sambe matrix in Eq. (10)) is taken as a small number OB2018 (); Rudner (), yielding an accurate approximation for most metallic leads.
By considering symmetric couplings between the leads and the first/last sites in the array, we arrive at a final expression for the total photon-assisted transmissions
Here the trace is taken over the Floquet-Sambe diagonal blocks, with the spin either or . The indices on and are the same as the indices in Eq. (8). Note that in general , even when the left and right broadening functions are the same, .
Appendix C The projected space representation: Time-independent
The transport properties for applied biases much smaller than the width of the bulk band gap can be efficiently addressed by projecting the transport problem onto the states closest to zero energy, i.e., the boundary states. Since the spin degree of freedom does not play an important role in the model (it merely produces a factor of two in all relevant formulas), we can consider electrons with a fixed spin and suppress the spin index in the analysis that follows.
At finite array sizes, the boundary states are modified by the boundary confinement and split in energy. Moreover, the chiral symmetry (cf. Sec. III. A) restricts them to be of the form with energy and with energy . Now, we assume that the high-energy bulk states do not contribute to the transport properties at biases much smaller than the width of the bulk band gap. We may then treat the problem in the projected space spanned by . By the structure of these states, is also spanned by and . Note that in the thermodynamic limit the states and are equal to the symmetry-protected boundary states because each of them lives on a different sublattice. We may also include perturbations in the discussion: If the resulting level splitting is small in comparison to the bulk gap, then we may still perform all calculations in and get essentially exact results.
Unperturbed system: The Hamiltonian (1) projected onto the space is given by
where . The diagonal terms are zero because couples only sites from different sublattices. Clearly, has eigenstates with energies .
The (retarded) Green’s function in is given by
where are self-energies in . Note that the self-energy () that gets projected onto couples only state () to lead L (R).
Now, by projecting Eq. (8) onto , one obtains, in obvious notation:
We are interested in the scaling of the transmission at zero bias:
where and within the wide-band limit approximation. The projected self-energies and are proportional to the wavefunction amplitudes on the first and last sites of the array. The wavefunctions and energy needed for computing and may be retrieved from a numerical diagonalization of the Hamiltonian , Eq. (1).
Alternatively, one may exploit the fact that it is straightforward to analytically find the two zero-energy edge modes of in the thermodynamic limit. These modes may then be used to approximate the states , simply by truncating them at the applicable size of the array. It follows that the energy is obtained from
and where are the hopping amplitudes defined in Eq. (1), with the number of dimers in the array.
Perturbed system: As our first case study, we add a symmetry-breaking perturbation in the chemical potential to the first site in the array. For this case the projected Green’s function reads as follows:
where is the projected amplitude of the perturbation, . Therefore, within the wide-band limit approximation we get
which can be expressed as
with given in Eq. (18). As expected, the transmission is seen to drop with .
Next, we add a symmetry-preserving perturbation in the hopping amplitude between the first and second sites of the array, call it . The corresponding projected Green’s function is given by
where is the projected perturbation, . This term is extremely small and vanishes completely in the thermodynamic limit. It is small because . Thus, one does not expect a significant change in the transmission unless the perturbation is extremely large, in case employing the projected space becomes inappropriate.
Appendix D The projected space representation: Periodically-driven
In periodically driven chiral systems the driving may open an additional gap, a so-called dynamical (or “anomalous”) gap, which may also host symmetry-protected edge states Rudner (); Carpentier (). The chiral symmetry in the periodically driven case, with the chiral symmetry operator defined in Eq. (5), obligates the Floquet steady states Sambe () to come in pairs: is a steady state with quasienergy if and only if is also a steady state but with quasienergy .
Here we follow a very similar strategy as in the time-independent case: We project the transport problem onto the space spanned by a pair of gapless states. There are two different gaps in this case and we have to consider them separately. As before, the spin degree of freedom is omitted in this discussion.
d.1 The pair of gapless edge states near zero quasienergy
The chiral symmetry requires Floquet boundary states to form pairs of the form and with quasienergies and respectively. Accordingly, the space spanned by these modes is also spanned by the two states . These states satisfy .
Unperturbed system: The Floquet-Sambe Hamiltonian projected onto the states is given by
where because and , and where we have defined , implying that . Now, has support on a single sublattice of the array because . Thus, at the initial time they correspond to two edge states localized on opposite edges of the array. The states are expected to stay localized on separate edges throughout the whole time evolution with rapidly decaying amplitudes as one moves away from an edge because they are gapped out from the bulk. This statement can also be verified numerically. However, the edge states are not anticipated to remain nonzero on only one of the sublattices. This is because the chiral symmetry relation is fulfilled only for the evolution operator at zero reference time. Formally, we write the following expression for the projected Floquet-Sambe Green’s function:
where are projected Floquet-Sambe self-energies representing the coupling to the leads. Here () only couples the () state to the left (right) lead since we assume that are localized at separate edges at all times . Within the wide-band limit approximation, and . It follows that the total photon-assisted transmission within the reduced space is given by
where is projected on the left edge state . At the expression for the total transmission reduces to
The reduced projected broadening function is defined through with , where is the time-independent part of the steady-state .
In the present case there is no straightforward way to analytically obtain the quasienergy and/or states . To predict the size dependence of the transmission with array size one must resort to a numerical diagonalization of the Floquet-Sambe Hamiltonian from which and can then be obtained.
Perturbed system: Let us consider a general local time-periodic perturbation located at the left edge of the array. In the following we describe how the the symmetries present in influence the transmission spectrum.
Any perturbation modifies the projected Floquet-Sambe Green’s function as follows:
where and . Clearly, is very small because is a local perturbation acting at the left edge and is the edge state at the right boundary. The term may be larger but is still small because of the same reason. The only term that can potentially modify the projected transmission in a significant way is with both and being localized at the same edge. However, perturbations satisfying the chiral relation exactly nullify . This is because with the property . In this way the transmission peak reflects a resilience of the edge states against symmetry-preserving perturbations, as was anticipated in Sec. III C.
d.2 The pair of gapless edge states near quasienergy
In a finite system, the levels of two edge states that live in the anomalous gap around quasienergy are split due to finite-size hybridization. Let us focus on one state from the pair, call it with quasienergy . As dictated by chiral symmetry, its complementary state then has quasienergy . Shifting the quasienergy of by