Interaction Driven Floquet Engineering of Topological Superconductivity in Rashba Nanowires

We analyze, analytically and numerically, a periodically driven Rashba nanowire proximity coupled to an $s$-wave superconductor using bosonization and renormalization group analysis in the regime of strong electron-electron interactions. Due to the repulsive interactions, the superconducting gap is suppressed, whereas the Floquet Zeeman gap is enhanced, resulting in a higher effective value of $g$-factor compared to the non-interacting case. The flow equations for different coupling constants, velocities, and Luttinger-liquid parameters explicitly establish that even for small initial values of the Floquet Zeeman gap compared to the superconducting proximity gap, the interactions drive the system into the topological phase and the interband interaction term helps to achieve larger regions of the topological phase in parameter space.

In this Letter, we present a comprehensive study of a setup consisting of an interacting Rashba nanowire (NW) proximity coupled to an s-wave superconductor and driven by a time-dependent magnetic field, see Fig. 1. This setup exhibits Floquet Majorana bound states (FMBSs) at each end of the NW if the Floquet Zeeman gap is larger than the superconducting gap [52]. However, high-amplitude time-dependent magnetic fields are not only difficult to apply but they also have detrimental effects on the superconductor, thus, the starting point of the current work is to focus on a low-amplitude magnetic field such that the Floquet Zeeman gap is small compared to the superconducting gap. Moreover, in lowdimensional systems, electron-electron interactions are important as even for weak interaction strength, the system changes drastically and cannot be described as Fermi liquid anymore. A generic interacting many-body system heats up to infinite temperature at sufficiently long times as described by the eigenstate thermalization hypothesis [61]. However, as demonstrated in Ref. [62], for a periodically-driven quantum many-body system, in a strong interaction regime, the prethermal Floquet state can be stabilized. Therefore, we consider prethermal region, which means that the time period for the Floquet FIG. 1. Sketch of the setup consisting of a one-dimensional Rashba nanowire (blue cylinder) aligned along x direction and brought into proximity to an s-wave superconductor (yellow slab). The uniform time-dependent magnetic field B(t) with frequency ω is applied along the NW axis and, thus, is applied perpendicular to the SOI vector αR.
term is short compared to the heating time-scale and study the setup in the presence of electron-electron interaction [12]. Using bosonization and renormalization group (RG) analysis [63][64][65][66][67], we show that even if the Floquet Zeeman gap is small compared to the superconducting gap, one can still obtain topological phases as the interaction renormalizes both superconducting and Floquet Zeeman terms. Repulsive electron-electron interactions renormalize the Floquet Zeeman gap, which is of Peierls type [68], and make it larger, whereas superconductivity gets suppressed such that the superconducting gap shrinks [12,67]. Thus, interactions helps to resolve the requirement of strong magnetic fields to obtain Floquet Majorana modes in the setup.
Model. We consider a one-band Rashba nanowire (NW) aligned along x direction, which is proximity coupled to an s-wave superconductor. The spin-orbit interaction (SOI) vector with strength α R defines the quantization axis along z direction. We apply an external timedependent periodic magnetic field B(t) = B cos(ω t)x, with magnitude B and frequency ω, along the axis of the NW. This choice ensures that B(t) is perpendicular to the Rashba SOI vector. The Hamiltonian contains three terms, namely, the kinetic energy term and SOI term as H kin , the superconducting pairing term H sc , and the time-dependent Zeeman term H Z (t): where Ψ σ (x) is the annihilation operator acting on an electron with spin σ = ±1 at position x, while the Pauli matrices σ x,y,z act on the spin space andk = −i ∂ x is the momentum operator. The amplitude of the time-dependent Floquet Zeeman term is given by t F = g µ B B/2, where g and µ B are the g-factor and Bohr magneton, respectively. The proximity induced superconducting pairing gap is of the size ∆ sc . The chemical potential µ is calculated from the SOI energy, E so = 2 k 2 so /2m 0 , where k so = m 0 α R / 2 is the SOI wavevector and m 0 is the mass.
We work in the Floquet formalism [48-51, and 69] to convert a time-dependent problem to a static one. The Hamiltonian is periodic in time, H(t) = H(t + T ), with period T = 2 π/ω. The frequency ω is chosen such that the Floquet Zeeman terms become resonant. As discussed in Ref. [52], this setup does not require the tuning of chemical potential to the SOI energy, instead the chemical potential µ has to be below the SOI energy such that the smallest Fermi wavevector in both effective bands coincide [see Fig. 2]. The eigenstates of Floquet operator, H F = H(t) − i ∂ t , are given by periodic functions e inωt , where the integer n labels different Floquet bands separated by an energy ω. This periodicity allows us to write H(t) = n H n e inωt , where H n = T 0 e −inωt H(t)/T . The Floquet Hamiltonian H F acquires a block diagonal form, written as where the static terms in H(t), i.e. H 0 = H kin + H sc , act on each Floquet band, while the oscillatory magnetic field couples different Floquet bands which are separated by an energy ω. We note that we work in the quasiequilibrium limit such that we have a well-defined quasi-Fermi energy. For simplicity, we consider only the single photon absorption processes, and thus focus only on the lowest two Floquet bands denoted by η = 1 and1. Thus, introducing the electronic operator Ψ ησ , which annihilates an FIG. 2. Energy spectrum of Rashba NW consisting of spin up (σ = 1, blue) and spin down (σ =1, red) branches. The chemical potential µ is set below the SOI energy and the driving frequency ω is chosen such that the smallest Fermi wavevector in the two Floquet bands, labeled by η = ±1, coincide. Close to the Fermi surfaces, the slowly varying right (left) fermionic field is denoted by Rησ (Lησ). electron of spin σ in band η, the Floquet Hamiltonian in the basis χ ησ =(Ψ 11 , Ψ 11 , Ψ † 11 , Ψ † 11 , Ψ1 1 , Ψ11, Ψ † 11 , Ψ † 11 ) is written as Here the Pauli matrices τ x,y,z and η x,y,z act on particlehole and Floquet band spaces, respectively. If the Floquet Zeeman gap exceeds the superconducting pairing gap, t F > ∆ sc > 0, the system hosts two zero-energy bound states protected by an effective time-reversal symmetry T = −iη z σ y K, where K is the complex conjugation operator [52]. Next, we include electron-electron interaction. First, we rewrite the kinetic energy in terms of the bosonic fields [see Supplemental Material (SM) [70]] as where the indices of the bosonic fields φ ηβ and θ ηβ refer to β = c (s) for charge (spin) sectors of the η-Floquet band. We set = 1 here as well as in the following calculation. Further, u ηβ and K ηβ corresponds to the velocity and Luttinger Liquid (LL) parameter, respectively. The cross-term, characterized by the velocityũ, describes the repulsive interaction between the Floquet bands. To simplify the problem, we fix K 1β = K1 β = K β and u 1β = u1 β = u β . For an ideal LL, where v F is the Fermi velocity [71,72]. Notably, we consider only the charge density interaction between the Floquet bands, as we are interested in the parameter regime in which interactions are spinrotational symmetric with K s ∼ 1. The superconducting pairing term and the Floquet Zeeman term are rewritten where analytical results predict that the system is topological [tF (l1) > ∆sc(l1)], however, due to the second-order correction, the system is in the trivial phase. Other parameters used in the second-order RG numerics are as whereφ η = φ ηc − θ ηc − φ ηs + θ ηs + φη c + θη c + φη s + θη s and α is the renormalized lattice constant of the NW, which grows under RG. As a result, the total effective Hamiltonian is given by H = H kin + H sc + H Z .
RG equations and analysis. Next, we derive the RG equations for different coupling constants, velocities, and LL parameters in H . Most importantly, we are interested in finding out whether it is possible to reach the topological phase even if the initial (non-renormalized) value of t F is smaller than ∆ sc and, thus, one expects the system to be in the trivial phase in the absence of interactions. To compare different competing terms, we work in dimensionless units and define∆ sc = α ∆ sc /u c andt F = α t F /u c . Further, performing an RG analysis, our goal is to determine whent F dominates over∆ sc . In RG language, this means finding the parameter regime when the system reaches the strong coupling limit, i.e.
To derive the RG equations, first we calculate differ-ent correlation functions between φ ηβ and θ ηβ for the kinetic part H kin using Green functions [70]. Subsequently, utilizing the operator product expansion (OPE) method [63], we compute the RG equations up to the first-order, Here, we define The dimensionless RG flow parameter is l = ln(α/α 0 ), where α 0 is the bare value of the lattice constant. Notably, the velocities and LL parameters do not flow under first-order RG. We are interested in the gapped regime where∆ sc andt F are RG relevant (terms growing as a function of l). Moreover, the superconducting pairing To estimate the relevant parameter regime for K c , we consider the limiting caseũ = 0 and K s = 1, which results in K c > 1/3 (K c + 1/K c < 6) for∆ sc (t F ) to be RG relevant, thus providing us the lower bound. Therefore, in what follows, we focus on the repulsive interaction regime with 1/3 < K c < 1.
We solve the first-order RG [see Eq. (7)] for∆ sc andt F considering that the latter reaches the strong coupling limit i.e. t F (l 1 ) = 1, where l 1 is the dimensionless RG flow parameter. Therefore, one obtains . For K s = 1, the values of the physical superconducting pairing gap and Floquet Zeeman gap are given by In the presence of strong repulsive interaction, the Floquet Zeeman gap tends to exceed the superconducting pairing gap even if this was not the case for the initial values, see Fig. 3(a). Thus, interaction drives the system into the topological phase by satisfying the criterion t F > ∆ sc . Generally, there is a crossover between t F and ∆ sc depending upon the ratio of their initial values. When t F (0)/∆ sc (0) = 1, the crossover happens at K c = 1 and thus for K c < 1 (K c > 1), the system is in topological (trivial) phase. As the ratio t F (0)/∆ sc (0) decreases, the crossover point for K c , which can be calculated by putting ∆ sc (l 1 ) = t F (l 1 ) in Eq. (8), shifts to smaller values, indicating that one requires stronger repulsive interaction in the NW to reach the topological phase. The LL parameters stay close to the initial value before reaching the blue vertical arrow, which justifies the assumption of first order RG. The smaller the ratio tF (0)/∆sc(0) < 1, the stronger interactions are required to reach the topological phase. Other initial conditions are chosen as Kc(0) = 0.5, 0.6, 0.7, 0.8, 0.9 from left to right, To check that the flow of LL parameters and velocities do not affect the analytical results obtained in the first-order, we recompute the RG equations up to the second-order [70]. As these are involved coupled differential equations, we solve them numerically. Generally, the second-order RG eqs. give small corrections to the firstorder result, see Fig. 3(b). However, close to K c = 1, the corrections are more relevant rendering the system trivial. Examining the RG flow of the physical gaps in second-order [see Fig. (4)], we observe that, up to the strong coupling limit point, K c stays close to the initial value and hardly flows. This justifies our assumption of focusing only on the first-order RG equations. Also the effect of the ratio t F (0)/∆ sc (0) on the physical gaps in both RG orders matches exactly [see Figs. 3(a) and 4)]. Generally, the repulsive interaction suppresses the superconducting gap [12,67]. In contrast to that, the Floquet Zeeman gap is enhanced [68], which results in a higher value of the effective g-factor compared to the noninteracting case. One can understand the enhancement in a simple way, the Floquet Zeeeman term has a form similar to a spin-flip backscattering term. As discussed in Ref. [65], the backscattering amplitude increases as interactions get stronger, thus resulting larger gaps in comparison to the non-interacting case. This allows one to satisfy the topological criterion t F (l 1 ) > ∆ sc (l 1 ) even if the non-interacting bare value of t F (l 0 ) is smaller than ∆ sc (l 0 ).
Finally, we also compute the phase diagram as a function of the ratio ∆ sc (0)/t F (0) and the initial LL parameter value K c (0) [see Fig. 5]. For a non-interacting system (K c = 1), if ∆ sc (0)/t F (0) ≥ 1, system is always in the trivial phase. However, when K c < 1, the topological phase emerges due to the presence of interactions. As the ratio ∆ sc (0)/t F (0) increases, we require lower values of K c , i.e. stronger interaction strength to reach the topological phase. We also would like to emphasize the role played by the interband interaction cross-termũ. If u is absent [see Fig. 5 (b)], the phase boundary between the topological and trivial phase is shifted to lower values of K c (0) compared to the case of finiteũ [ Fig. [5 (a)]. Thus, the interband interaction term results in larger parameter space corresponding to the topological phase.
Conclusions. We studied the effects of electronelectron interactions on a driven Rashba NW with proximity gap and analyzed the interplay between the Floquet Zeeman term and superconducting pairing term using bosonization techniques and RG analysis. The repulsive Coloumb interaction drives the system into the topological phase even if the initial (bare) value of the Floquet Zeeman gap is smaller than the superconducting proximity gap. Under RG flow, the physical Floquet Zeeman gap is enhanced whereas the proximity gap is suppressed, pushing the system into the topological phase. The proposed setup is important as it does not require the tuning of the chemical potential close to the spin-orbit energy and it exhibits topological superconductivity, and thus Floquet Majorana modes, even for weak strengths of the driving magnetic field due to the presence of electronelectron interactions.
Acknowledgments -This work was supported by the Swiss National Science Foundation (SNSF) and NCCR QSIT. This project received funding from the European Unions Horizon 2020 research and innovation program (ERC Starting Grant, grant agreement No 757725).

Bosonization
In this section, we first linearize the spectrum close to the Fermi momenta and subsequently bosonize the Hamiltonian in order to include electron-electron interactions in the analysis [63-67, and 69]. The Fermi points k F,ησr of the Floquet band η with the spin σ have the form k F,1σ± = σ k so ± k so 1 + µ/E so and k F,1σ± = σ k so ± k so 1 + (µ + ω)/E so . The resonance condition is satisfied, if k F,11+ = k F,11− and k F,11+ = k F,11− . We write the fermionic fields in terms of the right mover and left mover fields as Ψ 11 = R 11 e ik F,11+ + L 11 e ik F,11− , (S1) Here we denote the slowly-varying right and left moving field by R(x) and L(x). Further, we linearize the sum of kinetic energy and SOI terms, which takes the following form where v F is the Fermi velocity in the NW. The linearized form of s-wave pairing term has the following form Finally, in terms of fermionic right and left movers, the Floquet Zeeman term that couples the lower and upper Floquet bands is given by To include the electron-electron interactions, we consider only the low-lying excitations close to the Fermi level. As the particle-hole excitation are bosonic in nature, we bosonize the Hamiltonian by defining left and right moving fermions in terms of the charge (φ ηc , θ ηc ) and spin (φ ηs , θ ηs ) bosonic fields described by following definition where α is the short-distance cut-off of the theory and we assume it to be the lattice constant of the NW. The bosonic fields satisfy the commutation relation [φ ηβ (x), θ η β (x )] = i π δ ηη sgn(x − x). The field φ ηβ and θ ηβ relate to the β = c (charge) and β = s (spin) density and current in the η-band, respectively. Thus we rewrite the linearized Hamiltonian in bosonized fields and obtain different terms in the Hamiltonian H defined in Eqs. (4)-(6) of the main text. Notably, we also include interband interaction terms [see Eq. (4) of the main text]. In addition, there is a backscattering term involving spin up and spin down electrons for each Floquet band separately of the form where g η is the coupling strength. However, in the regime of K s ≥ 1, this term is either marginal or irrelevant [67]. Thus, we drop this term in the main text.

Green functions of unperturbed Hamiltonian H kin in the presence of interband cross term
In this Appendix, we derive first the Matsubara Green functions and later different correlation functions of φ − θ fields in the presence of the cross term for the unperturbed Hamiltonian in H kin defined in Eq. (4) of the main text. The quadratic part of the Matsubara action is written as S = S c + S s . Here we split the action into two parts S c and S s , corresponding to the charge and spin-sectors, respectively, with the following form The correlation functions for the spin sector are unchanged as the cross-term between different Floquet band appears in the charge sector only and has the following form where the expectation value · · · 0 is taken with respect to the LL action S s defined in Eq. (S7). For the charge sector, we calculate the action in Fourier space which then becomes where Φ c (q, ω) = [φ 1c (q, ω), θ 1c (q, ω), φ1 c (q, ω), θ1 c (q, ω)].