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arXiv:2105.14842 · Physical review. B./Physical review. B

Relation between scattering matrix topological invariants and conductance in Floquet Majorana systems

Thomas Simons, Alessandro Romito, Dganit Meidan

Published Oct 15, 2021• 4 citations
arXivHTMLPDF

Abstract

We analyze the conductance of a one-dimensional topological superconductor periodically driven to host Floquet Majorana zero modes for different configurations of coupling to external leads. We compare the conductance of constantly coupled leads, as in standard transport experiments, with the stroboscopic conductance of pulsed coupling to leads used to identify a scattering matrix topological index for periodically driven systems. We find that the sum of the DC conductance at voltages corresponding to integer multiples of the driving frequency is quantitatively close to the stroboscopic conductance at all voltage biases. This is consistent with previous work which indicated that the summed conductance at $\mathrm{zero}/\ensuremath{\pi}$ resonances is quantized. We quantify the difference between the two in terms of the widths of their respective resonances and analyze that difference for two different stroboscopic driving protocols of the Kitaev chain. While the quantitative differences are protocol dependent, we find that generically the discrepancy is larger when the zero-mode weight at the end of the chain depends strongly on the offset time between the driving cycle and the pulsed coupling period.

DOI
10.1103/physrevb.104.155422

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cond-mat.mes-hallCross-listedarXiv:2605.13066•May 14, 2026
We analyze the conductance of a one-dimensional topological superconductor periodically driven to host Floquet Majorana zero modes for different configurations of coupling to external leads. We compare the conductance of constantly coupled leads, as in standard transport experiments, with the stroboscopic conductance of pulsed coupling to leads used to identify a scattering matrix topological index for periodically driven systems. We find that the sum of the DC conductance at voltages corresponding to integer multiples of the driving frequency is quantitatively close to the stroboscopic conductance at all voltage biases. This is consistent with previous work which indicated that the summed conductance at zero/\ensuremathπ\mathrm{zero}/\ensuremath{\pi}zero/\ensuremathπ resonances is quantized. We quantify the difference between the two in terms of the widths of their respective resonances and analyze that difference for two different stroboscopic driving protocols of the Kitaev chain. While the quantitative differences are protocol dependent, we find that generically the discrepancy is larger when the zero-mode weight at the end of the chain depends strongly on the offset time between the driving cycle and the pulsed coupling period.
Abstract One-dimensional (1D) Floquet topological superconductors possess two types of degenerate Majorana edge modes at zero andπquasienergies, leaving more room for the design of boundary time crystals and quantum computing schemes than their static counterparts. In this work, we discover Floquet superconducting phases with large topological invariants and arbitrarily many Majorana edge modes in periodically driven Kitaev chains (KCs). Topological winding numbers defined for the Floquet operator and Floquet entanglement Hamiltonian are found to generate consistent predictions about the phase diagram, bulk-edge correspondence and numbers of zero andπMajorana edge modes of the system under different driving protocols. The bipartite entanglement entropy further shows non-analytic behaviors around the topological transition point between different Floquet superconducting phases. These general features are demonstrated by investigating the KC with periodically kicked pairing or hopping amplitudes. Our discovery reveals the rich topological phases and many Majorana edge modes that could be brought about by periodic driving fields in 1D superconducting systems. It further introduces a unified description for a class of Floquet topological superconductors from their quasienergy bands and entanglement properties.
Floquet Majorana edge modes capture the topological features of periodically driven superconductors. We present a Kitaev chain with multiple time periodic driving and demonstrate how the avoidance of bands crossing is altered, which gives rise to new regions supporting Majorana edge modes. A one dimensional generalized method was proposed to predict Majorana edge modes via the Zak phase of the Floquet bands. We also study the time independent effective Hamiltonian at high frequency limit and introduce diverse index to characterize topological phases with different relative phase between the multiple driving. Our work enriches the physics of driven system and paves the way for locating Majorana edge modes in larger parameter space.
Abstract Floquet Majorana edge modes capture the topological features of periodically driven p-wave superconductors. We present a Kitaev chain with multiple time periodic driving terms. Our results demonstrate how multiple driving will affect Floquet bands in frequency space, leading to more robust Floquet Majorana edge modes against driving frequency ω in comparison with the single driving scenario. Meanwhile, we have proposed how to predict Majorana edge modes via the Zak phase of Floquet bands. Besides, in contrast to the cases with single driving term, where the constant phase can be gauged out by properly choosing the initial time, we have shown the relative phase between multiple driving can not be gauged out and will play a dominant role in deciding topological phase transitions. For the sake of completeness, we also investigate the high frequency limit. Analytical results on effective Hamiltonian can be obtained via Magnus expansion and relative phase induced topological transitions can be shown explicitly.
The topic of Majorana zero modes has been a focal point of the condensed matter physicscommunity in recent times due in part to their potential applications in quantum computation.Most notably, the exotic exchange statistics of Majorana zero modes form the basis oftopologically protected quantum gates. The physical exchange, or braiding, of Majorana zeromodes is often modeled in two-dimensional systems or by using networks of wires. Recentwork has established that Majorana zero modes can be braided in a purely one-dimensionalsetting with a periodic, or Floquet, drive. Specifically, the work of Bauer et al. in Ref.[1] demonstrates that a working braiding protocol can be simulated on a Kitaev chain withperiodically modulated parameters. This Floquet system gives rise to both Majorana zeromodes and Majorana pi modes, the latter of which serves as an additional degree of freedomin the braiding protocol.We reproduce the results of Ref. [1] through numerical simulation and extend theirbraiding protocol such that it includes both disorder and interactions. We demonstrate thatthe protocol remains successful when the disorder is weak compared to the parameters of thesystem. In the case of strong disorder, we demonstrate that the protocol fails. In addition,we show that the protocol is quite sensitive to the presence of interactions. We illustratethat interactions cause a severe delocalization of the Majorana zero modes leading to thefailure of the protocol
In one-dimensional topological superconductors driven periodically with the frequency omega, two types of topological edge modes may appear, the well-known Majorana zero mode and a Floquet Majorana mode located at the quasienergy PLANCK CONSTANT OVER TWO PI omega/2. We investigate two Josephson-coupled topological quantum wires in the presence of Coulomb interactions, forming a so-called Majorana box qubit. An oscillating gate voltage can induce Floquet Majorana modes in both wires. This allows for the encoding of three qubits in a sector with fixed electron parity. If such a system is prepared by increasing the amplitude of oscillations adiabatically, it is inherently unstable as interactions resonantly create quasiparticles. This can be avoided by using instead a protocol where the oscillation frequency is increased slowly. In this case, one can find a parameter regime where the system remains stable.
We study quantum transport in a periodically driven (Floquet) topological system coupled to static fermionic reservoirs. Using the Floquet nonequilibrium Green's-function (NEGF) formalism we show, from exact numerics for a strip geometry, that the two-terminal (longitudinal) conductance is quantized as ∣Wε∣ e2/h|W_{\varepsilon}|\,e^2/h∣Wε​∣e2/h, while the Hall (transverse) conductance is quantized as Wε e2/hW_{\varepsilon}\,e^2/hWε​e2/h, where WεW_{\varepsilon}Wε​ is the Floquet winding invariant associated with the quasienergy gap at ε=0\varepsilon = 0ε=0 or ε=Ω/2\varepsilon = Ω/2ε=Ω/2. Quantization is achieved only after summing over the contribution of all Floquet sidebands. We provide an analytic understanding of this Floquet conductance sum rule, by considering the Hall conductance in the weak coupling limit. In that limit, we show that the Floquet Hall conductance gets contributions from the Floquet sidebands, which includes the signs of the velocities of the edge modes. Their sum yields exact quantization, as predicted by the Floquet sum rule. We find that in a wide range of parameter regime, the convergence is fast, making observation of the sum rule and Floquet winding numbers accessible to experiments.