Anomalies-Rich Floquet superconductivities induced by joint modulation of dynamic driving and static parameters

Current theoretical and experimental endeavors to realize an anomalous Floquet chiral topological superconductor (TSC), which is characterized by chiral Majorana edge modes independent of the Chern number, remain insufficient. Herein, we propose a new scheme that involves jointly tuning dynamic driving and static parameters within a magnetic topological insulator-superconductor sandwich structure to achieve this goal. The Josephson phase modulation induced by an applied bias voltage across the structure is utilized as a Floquet periodic drive. It is found that the interplay between the two kinds of tunings can bring about a lot more exotic Floquet TSC phases than those caused by only tuning the dynamic driving parameter (frequency ω or period τ). More importantly, just tuning static parameters (the chemical potential µ, Zeeman field gz , and proximity-induced superconducting energy gap Δb ) also can induce a series of novel topological phase transitions. Particularly, the features in the context of the three tunings are different from each other, originating from the combination of intrinsic and different extrinsic mechanisms. In addition, jointly tuning τ and µ (gz ) can have its own unique TSC phases. The proposed scheme should be readily accessible in experiments, and thus the family of anomalous Floquet TSC phases may be considerably enriched.

Unfortunately, there have been few efforts to address the corresponding superconducting counterpart except for only one very recent theoretical work [64].A 2D anomalous Floquet chiral TSC with Chern-number-independent chiral Majorana edge modes (MEMs) was predicted.It is considered that such a Floquet TSC is generated entirely by the intrinsic dynamics arising from the dc-biased Josephson effect.
On the other hand, the chemical potential µ, Zeeman field g z , and proximity-induced superconducting energy gap ∆ b have been found to exert significant influences on the realization of topological phases in lowand three-dimensional (3D) TSC structures.
A 1D Rashba nanowire, combined with proximity-induced s-wave superconductivity in the presence of Zeeman field g z , is characterized by midgap Majorana modes (either chiral or non-chiral) on edges along and perpendicular to the wires [65].At special values of g z and µ, a phase with a single chiral MEM (analogous to a p + ip superconductor [66,67]) can be realized, which is almost completely localized at the outmost wires.The TSC phase can cover a broader chemical potential window in the existence of expulsive interactions, even without requiring g z [68].By simply tuning µ, ones can access three distinct phases, i.e. topologically trivial s-wave, topologically nontrivial s ± -wave, and nodal superconducting phases under impurity subgap states [69].In the context of proximity-induced superconductivity induced by s ± -wave superconductor (SC), the evolution of the Majorana pair is caused by tuning g z , leading to that the SC undergoes topological phase transitions [70].Particularly, under a Floquet driving, regular 0-and anomalous π-Majorana end modes are generated by tuning µ and driving frequency ω [27].Furthermore, non-Hermitian Floquet TSCs with multiple MEMs were proposed, which are based on a Kitaev chain with periodically kicked superconducting pairings and gain/losses in the chemical potential µ [38,71].
An applied staggered Zeeman field g z on the stacked tunnel-coupled 2D electron-and hole-gas layers with Rashba spin-orbit interaction can generate a second-order TI phase.It is characterized by the emergence of zero-mass hinge interfaces hosting chiral gapless hinge states, which is stable up to relatively large values of g z [72].In addition, for a 3D TI with bulk s-wave superconductivity under a perpendicular g z , the Majorana states become more localized on a single surface with µ increased.However they spread into the bulk toward the opposite surface, and at µ being sufficiently high, the Majorana modes can tunnel between surfaces [73,74].
It is then natural to ask whether the three static parameters µ, g z , and ∆ b can exert significant influences on the realization of an anomalous Floquet chiral TSC in Floquet systems.So far, there has been a lack of systematic study on joint tuning of the dynamic driving and static parameters in Floquet TSC systems.And thus, the study is in need, which is the motivation of the present work.
In this work, we provide an affirmative answer to the above questions, based on a Floquet magnetic TI-based Josephson hybrid structure (see figure 1(a)).The static bias voltage applied across the top and bottom SC layers offers a driving protocol for the setup (the Josephson phase, i.e. the relative phase between the SCs as a periodic function of time).By jointly tuning the dynamic driving parameter (frequency ω or period τ ) and one of static parameters µ, g z , and ∆ b , (see figure 1(b)), a variety of anomalies-rich Floquet TSC phases are exhibited, most of which are not shown in [64].In the low frequency ω or large driving period τ region, for different µ, different novel Floquet topological phase transitions will occur sequentially as τ increases, also indicating a series of topological phase transitions at a fixed τ induced by µ.And the same features are found for different g z and ∆ b .Particularly, there always exist several special values for µ, g z , and ∆ b , respectively, at two any adjacent values of which, the corresponding topological phases appearing successively with the increase of τ are thoroughly different.This also indicates a series of topological phase transitions in the whole τ region tuned by µ, g z , and ∆ b , respectively.Remarkably, with the enhancement of µ and ∆ b , the numbers of the topological phases arising successively tuned by τ may both increase or decrease, while the number basically enhances with g z .More interestingly, for tuning both µ and g z , there are unique TSC phases, however, for tuning ∆ b , all the topological phases have appeared in those for tuning µ and g z .In addition, a total of up to 19 different topological phases are exhibited, which considerably enriches the family of anomalous Floquet TSC phases.A Zeeman field gz along the z direction is applied to make the TI become a 3D magnetic one.Two gate voltages Vg applied on the top and bottom surfaces of the TI can adjust the chemical potential µ.A periodic phase concerned with a bias voltage V0 between the two s-wave SC layers is used as a Floquet period drive, leading to the Josephson junction transformed into a Floquet TSC.(b) Tuning the driving period τ by varying the bias voltage V0 allows for dynamic modulation of the Floquet TSC phase.Meanwhile, the Floquet TSC phase is subject to the manipulation of the static parameters (chemical potential µ, Zeeman field gz, and proximity-induced superconducting energy gap ∆ b ).

Model Hamiltonian and Floquet theory
Consider a chiral topological Josephson hybrid structure with a magnetic (Cr-doped [75]) 3D TI sandwiched in between the top and bottom s-wave SCs.In the structure, the Zeeman field along the z direction, bias voltage V 0 between the top and bottom s-wave SC layers, and gate voltage V g on both the top and bottom surfaces of the 3D magnetic TI, are applied as shown in figure 1(a).The BdG (Bogoliubov-de Gennes) Hamiltonian of the structure is written as [76][77][78] with the Hamiltonian of the magnetic TI Here σ and s are the the Pauli matrices denoting the surface layer and spin degrees of freedom, respectively, µ is the chemical potential, t(k) = t 0 − t 1 (cos k x + cos k y ) describes the hybridization between the top and bottom Dirac surface states of the 3D magnetic TI, and g z represents the Zeeman field.In equation ( 2), the pairing potentials of the top and bottom SCs respectively given by ∆ t,b , are assumed identical, and φ(t) stands for the Josephson periodic phase φ(t) = φ 0 + 2eV 0 t as a periodic driving with the frequency ω = 2eV 0 .The time-dependent Hamiltonian is needed to convert into a Floquet frequency-or period-dependent one, which is infinite-dimensional and truncated appropriately for numerical calculations.Its matrix elements are given by the following formula [79], where ) , (5) ω ) † , and σ ± = (σ 0 ± σ z )/2.For the matrix elements with |n − m| > 1, the presence of e i(n−m)ωt in equation ( 4) makes the integration result over the entire period τ naturally equal to 0.
The BdG Chern number C is defined by using the following formula [80] where all the quasi-energy bands lying between (0, ω/2] are involved and with the integral over the complete Brillouin zone.In equation (7), Ω n (k) the Berry curvature can be expressed as a summation of eigenstates Here, the obtained C for a 2D anomalous Floquet chiral TSC with Chern-number-independent chiral MEMs is, in general, clearly distinguished from that for the conventional Floquet chiral TSC.The latter is like the one in the high-frequency limit, which is adiabatically equivalent to a static chiral TSC with the same C. Besides, C is based on the Floquet theory, which converts the time-dependent Hamiltonian into a frequency-dependent one.We use a truncated Floquet Hamiltonian H F , and integrate over the entire Brillouin zone to obtain C.This process only requires static calculations without involving time integration.For the chiral MEMs in such a 2D anomalous Floquet chiral TSC, a relevant topological invariant, i.e. a homotopy-based winding number, is needed to deliberately describe them [18,20,22,[81][82][83]] with In equation ( 9), the time-evolution unitary with m representing the band index and P k,m (τ ) the projection matrix given by the eigenvector of U(k, τ ) for e −iεm .Here, e −iεm denotes the m-th eigenvalue of U(k, τ ), the summation in equation ( 10) extends over all eigenvalues, and ϵ serves as the branch cut of the logarithm by requiring i log ϵ (x) ∈ [ϵ, ϵ + 2π) for all x ∈ U(1).As we always set the branch cut to ϵ = −π, we have For the calculation of W , the integrations over not only the Brillouin zone but also time covering a complete driving period, are required, which is much different from the situation for calculation of C. Therefore, when calculating W, we use the original time-dependent Hamiltonian H(k, t), and a time evolution operator with periodic invariance U ϵ (k, t), which satisfies The values of both C and W determine the specific form of bulk-edge correspondence.In the quasi-one-dimensional spectrum, edge states can appear in the gaps at E = 0 and/or ω/2.The number of edge states in the gap at ω/2 is indicated by W, while the one in the gap at E = 0 is just the sum of C and W. Therefore, different combinations of C and W represent different topological phases.For the two topological phases with the same W, they also have the same number of edge states in the gap at ω/2.However, for the two topological phases with the same C but different W, they possess different numbers of edge states in the gaps at both E = 0 and ω/2.Specifically, in the gap at E = 0, for the driving period τ within high-frequency limit, the number of edge states is only dependent on C due to W = 0, while for τ beyond the limit, relies on both C and W. A detailed analysis will be performed by combining the specific edge states and topological phases in the subsequent sections.

Results and discussions
With the gradual diminishment of V 0 from the high-frequency limit, Floquet TSC states with chiral MEMs across the E = ω/2 energy gap are generally expected to produce, accompanied by Floquet topological phase transitions.The Floquet TSC states are generated entirely by the intrinsic dynamics in the present Josephson hybrid structure, even with the starting from the topologically trivial state (C = 0), as shown in the following.The corresponding topological phase transitions are modulated by Josephson phase through simply changing the bias voltage V 0 .The dynamic driving (tuning frequency ω or period τ ) is necessary to produce the topological phase transitions, which is intrinsic.However, the possible results of topological phase transitions are also sensitively dependent on static parameters (e.g.µ, g z , and ∆ b , etc).Specifically, only for proper static parameters, can the Floquet TSC states be induced, which is extrinsic.In the following, therefore, we investigate the Floquet TSC phases and corresponding phase transitions tuned jointly by dynamic driving (ω or τ ) and static (µ, g z , and ∆ b ) parameters.

Topological phase transitions tuned jointly by chemical potential and driving period
In this section, we present anomalies-rich Floquet chiral TSC phases with Chern-number-independent chiral MEMs, which are tuned simultaneously by chemical potential µ and driving period τ .
Table 1 shows the topological phase diagram, the topological phase transitions as a function of µ and τ .It is divided into areas 1, 2, and 3 according to the different topological phases in the high-frequency limit region.At the high-frequency limit, the phase transitions from µ = 0.3 to 0.4 and from µ = 0.9 to 1 are due to the closures of the bulk energy bands at X(π, 0) and M(π, π), respectively.
In area 1, when µ is slightly increased, e.g.µ = 0.1, the topological phases appearing sequentially with the increase of driving period τ keep unchanged, including the corresponding driving period regions.This also indicates that there exists no topological phase transition tuned by µ at any fixed τ .It follows that for joint tuning of the dynamic driving and static parameters, the former thoroughly predominates over the latter.With the further increase of µ, e.g.µ = 0.2, the topological phases arising successively with τ still keep constant in the range for small τ (from 0 to 2.2), while are thoroughly different in the range for big τ (from 2.3 to 2.6).Specifically, for the latter range, the order of the topological phases occurring successively with τ is the three phases (0, 1), (2, −1), and (3, −2), replacing the two phases (3, −2) and (2, −1) at µ = 0.1.This indicates the thoroughly different topological phase transitions with τ and several ones at fixed periods τ tuned by µ.As µ is continuously enhanced, e.g.µ = 0.3, the order of the topological phase appearing sequentially with τ in the range for small τ (from 0 to 2.3) keeps unchanged.Yet the period region corresponding to each phase is considerably different from that at µ = 0.2, which means possible topological phase transitions with µ at some τ .Particularly, in the range for big τ (from 2.3 to 2.6), the order of (0, 1), (2, −1), and (3, −2) is replaced by the one of (0, 1), (3, −2), and (4, −2), where (4, −2) is a new phase.And thus, the thoroughly different topological phase transitions with τ emerge.It also follows that in the region for the big τ , the topological phase transitions at a fixed τ tuned by µ from 0.2 to 0.3 appear.Therefore, tuning µ in the range of small µ exerts a significant influence on producing the new topological phases only at the range of big τ , as seen in table 1.There are six kinds of phases, i.e. (1, 0), (0, 1), (−1, 2), (2, −1), (3, −2), and (4, −2) in this area.
When µ is increased into area 2, a variety of new topological phases thoroughly different from those in area 1, start to emerge.For example, at µ = 0.4, the order of (−1, 0), (−2, 1), (0, −1), (−1, 0), (−2, 1), (−1, 0), and (0, 0) is presented.This implies the entirely new phase topological transitions modulated by not only τ but also µ from µ = 0.3 to 0.4 (between the adjacent areas) at any fixed τ .Similarly, in this area, when µ is slightly increased, e.g.µ = 0.5, the topological phases emerging successively with the increase of τ keep unchanged.However, each corresponding driving period region varies except for the high-frequency limit region, indicating that the topological phase transitions with µ from 0.4 to 0.5 can appear at most fixed τ .With the further increase of µ, e.g.µ = 0.6, there still exist five topological phases, and the order of the former four ones keeps the same, but the corresponding driving period regions are much different.And then, with each increment of 0.1 in µ from 0.6 to 0.9, one more topological phase emerges as τ grows.The Table 1.The topological phase diagram tuned jointly by dynamic driving and static parameters (τ and µ), where each phase is labeled by two integer-valued topological invariants, BdG-Chern number and winding number, written as (C, W).The phase diagram is divided into areas 1 (blue), 2 (green), and 3 (orange) according to the different topological phases in the high-frequency ω (small τ ) limit.Here, gz = 0.6 and ∆ b = 0.4.former several phases are always identical and have the same order.Nevertheless, their corresponding driving period regions are much different for different µ.Particularly, the two new topological phases (1, −1) at µ = 0.8 and 0.9 as well as (−4, 2) at µ = 0.9 are exhibited.All these mean that a variety of new topological phase transitions manipulated jointly by µ and τ take place in area 2. In this area, there are seven kinds of phases, i.e. (−1, 0), (−2, 1), (0, −1), (0, 0), (1, −1), (−1, 1), and (−4, 2), which are thoroughly from those in area 1, in particular, the range of µ corresponding to area 2 is comparatively wide.
From what has been observed above, it is concluded that just tuning static parameter µ always induces a series of novel topological phase transitions at any fixed τ between the adjacent areas.This also suggests that as long as the topological phase transitions at a fixed high-frequency ω (small τ ) limit tuned by µ appear, the ones at a fixed big τ will take place.The reason can be explained by the combination of intrinsic and extrinsic mechanisms as follows.As τ increases, the reason for the topological phase transition is that the energy gap of the Floquet bulk energy band at E = ω/2 is in turn closed.The point for the energy gap closure lies at the high symmetry point of the Brillouin zone (see figure 2).The variable spacing between the gap closure points leads to a wide or narrow range of driving periods for different topological phases.We only consider the gap at the high-symmetry point because it is the crucial indicator of the topological phase transition.More specifically, when the topological phase transition takes place, the gap at the high-symmetry point closes first, and then the closing point opens again with the increase of τ , forming a new topological phase.Until the gap at the next high-symmetry point closes, the topological phase will not change again.
When τ is relatively large, the Floquet bulk energy band is also deformed by adjusting µ, and this deformation changes the order in which the high symmetry points intersect with the E = ω/2 line.For example, the third topological phase from µ = 0.1 to 0.2 undergoes a band deformation, causing the topological phase transition from (−1, 2) to (2, −1) at a fixed τ .This results in that intersecting of E = ω/2 line with energy band of h (0) ω changes from at the high-symmetry X point first (see figure 2(a)) to at the M point first (see figure 2(b)).
In addition, with the change of µ, there is another kind of topological phase transitions due to the closure of the high symmetry point of the bulk energy band near E = 0.For example, in the range of µ from 0.3 to 0.42 (see figures 3(a)-(c)), with the increase of µ, the bulk energy band at the high symmetry point X(π, 0) first opens, then closes and lastly opens.Resultantly, the system is driven into the topological phase transition, making the Chern number C of the system change from 1 to −1.At the boundaries between different areas, these two types of transitions often interact together.That is, the energy gap closes at both E = 0 and ω/2, which leads to completely different topological phases under modulation of µ at the adjacent boundaries of two areas.The general relation between the bulk topological indices (C, W) and edge Majorana physics can be understood as follows.Let us first denote the numbers of chiral MEMs within the E = γ gap as n edge (γ) for γ = 0, ω/2, where the positive (negative) n edge (γ) stands for the number of right-(left-) shifted chiral MEMs.Then the bulk-edge correspondence is given by [49,64] The bulk-edge correspondence can be exemplified by the bulk quasi-energy bands or the edge state of the anomalous Floquet chiral TSC phase (C, W) = (1, −1) at τ = 1.9 and µ = 1 as shown in figure 4. It is found that no edge state is generated at E = 0 (see figure 4(a)), but there exist two edge modes or a pair of modes (near the two edges n = 0 and 100) at E = ω/2 (see figure 4(b)), which are respectively given by the red and black lines.For each mode at E = ω/2, there is left-or right-shifted chirality, e.g. the edge mode corresponding to the red line, belongs to the former.Obviously, the total number of edge modes agrees with equation (12).The distributed probabilities at the blue circle on the left-shifted chiral edge state at E = ω/2 are shown in figure 4(c), from which we can demonstrate that the chiral MEM indeed locates near the boundary with n = 100.

Topological phase transitions regulated jointly by Zeeman field and driving period
In this section, we illustrate the anomalies-rich Floquet chiral TSC phases with Chern-number-independent chiral MEMs tuned jointly by Zeeman field g z and driving period τ .
Table 2 shows the topological phase diagram, the topological phase transitions as a function of g z and τ .It is divided into four areas 1, 2, 3, and 4 according to the topological phase of the high-frequency limit region.At the high-frequency limit, the phase transitions tuned by g z from 0.3 to 0.4 and from 0.7 to 0.8 are both due to the closure of the bulk energy band at Γ(0, 0), while the one from 0.8 to 0.9 stems from the closure at X(π, 0).
In area 1, when g z is slightly increased, the topological phases emerging sequentially with the increase of τ have a great change.Specifically, there exist four topological phases appearing successively at g z = 0.1 but three ones at g z = 0.2.Although the former two phases are the same for both cases, the corresponding period regions are much different from each other.Particularly, the latter two phases at g z = 0.1 are (0, 0) and (2, −2), which are thoroughly from the latter one at g z = 0.2.These indicate not only the new different phase transitions tuned by τ in the range for big τ between each other but also the phase transitions tuned by g z from 0.1 to 0.2 at most fixed τ .Thus, the features are much different from those with varied µ from the beginning in table 1.With the further increase of g z from 0.2 to 0.3, the topological phases arising successively with τ keep unchanged, but the corresponding driving period regions change greatly, which indicates that there exist topological phase transitions tuned by g z at some fixed τ .In this area, there exist four kinds of different topological phases, i.e. (0, 0), (1, −1), (−1, 1), and (2, −2).
As in table 1 for tuning µ, with g z increased into area 2, a variety of new topological phases thoroughly different from those in area 1, start to emerge, e.g. at g z = 0.4, in the order of the five phases (1, 0), (0, 1), (−1, 2), (1, 0), and (2, −1).This indicates not only the entirely new topological phase transitions tuned by τ but also the topological phase ones induced by g z from 0.3 to 0.4 at any fixed τ between the adjacent two areas.With g z increased to 0.5, the former four topological phases occurring sequentially with the increase of τ , including the order, keep unchanged, and only the last one is (3, −2) replacing (2, −1) at g z = 0.4.But the corresponding driving period regions change greatly except for the high-frequency limit.Thus, not only is the last phase transition tuned by τ different, but also the phase transition modulated by g z can take place at most fixed τ .And then, as g z increases gradually from 0.5 to 0.7, the number of topological phases that appear in sequence with the rising τ also increases by one.The former five phases are always identical and Table 2.The same as in table 1 except that µ is replaced by gz and the topological diagram is divided into areas 1 (blue), 2 (green), 3(orange), and 4 (purple).Here, µ = 0 and ∆ b = 0.4.
have the same order, however, their corresponding driving period regions are much different for different g z .Particularly, the two new topological phases (2, −1) at g z = 0.6 and 0.7 as well as (3, −1) at g z = 0.7 are exhibited.All these mean that a variety of new topological phase transitions tuned by both g z and τ take place in area 2. There are six kinds of phases in this area, i.e. (1, 0), (0, 1), (−1, 2), (2, −1), (3, −2), and (3, −1), are thoroughly different from those in area 1.With g z increased into area 3, e.g.g z = 0.8, there are still seven topological phases occurring sequentially, but they are thoroughly different from those in the former two areas.As a result, different novel topological phase transitions tuned by τ are produced and there exist the topological phase transitions manipulated by g z at almost all τ .In this area, there are four kinds of phases, i.e. (2, 0), (1, 1), (0, 2), and (4, −2), which are not exhibited in the former two areas.
More interestingly, when g z is slightly increased, e.g.g z = 0.9, area 4 just turns up.There still exist seven topological phases arising successively, but they are much different from those at g z = 0.8.There exist eight topological phases appearing sequentially at both g z = 1 and 1.1.Only the third phase (−2, 2) at g z = 1 and the last phase (2, 0) at g z = 1.1 are added, respectively, but the order for other phases is the same as the one at g z = 0.9.Thus two new phase transitions tuned by τ from (−1, 1) to (−2, 2) and from (−2, 2) to (0, 0) at g z = 1, replace the one (−1, 1) to (0, 0) at g z = 0.9.Another new one tuned by τ from (1, 0) to (2, 0) at g z = 1.1, is produced.However, similarly due to the corresponding driving period region for each phase varying with g z , the phase transitions tuned by g z at a fixed τ can appear.In this area, there is a new phase (−2, 2), which is never exhibited in the former areas.
From the above, we observe that not only the six new topological phases (1, 1), (0, 2), (2, 0), (2, −2), (−2, 2) and (3, −1) but also some exotic topological phase transitions in table 2 are never exhibited in table 1.Now, take the phase (C, W) = (0, 0) in table 2 with τ = 1.6 and g z = 1 as an instance to illustrate the bulk energy bands or the edge state of the anomalous Floquet chiral TSC phase in figure 5.It is found that there exist four (two pairs of) edge modes (near the two edges n = 0 and 100) in the energy gap at E = 0, which are respectively given by the red and black lines.One pair lies near k y = 0, the other lies near k y = π.Each mode has a left-or right-shifted chirality, e.g. the one (the red line) near k y = 0 has the right-shifted chirality, while the one (the red line) near k y = π belongs to the left-shifted chirality.The similar features are shared by the edge modes in the energy gap at E = ω/2.The total number of edge modes obviously satisfies equation (12).The distributed probabilities of chiral modes at E = 0 and ω/2 for the edge modes (the red lines) at the blue circles are exhibited in figures 5(c)-(f), respectively, demonstrating they locate near the one with n = 100.
The anomalies-rich Floquet topological phases and corresponding phase transitions induced by g z have the same origination as the ones by µ.The Floquet bulk energy band closes at E = ω/2 with the increase of τ , leading to a new topological phase transition.Similarly, the increase of g z also deforms the bulk energy band like µ, thus bringing about the topological phase transition at a fixed τ .However, the different features between the two tunings of static parameters (µ and g z ) observed above, originate from the combination of intrinsic (tuning τ ) and different extrinsic (tuning µ and g z ) mechanisms.µ and g z occupy different positions in the spin-dependent Hamiltonian.The former is determined by spin-independent term  1 with τ = 1.9, gz = 0.6, ∆ b = 0.4, µ = 1, E = 0 (a) and ω/2 (b).This phase features no chiral MEM penetrating the quasi-energy gap at E = 0, but a pair of ones with left-shifted and right-shifted chiralities at E = ω/2.The red line is the chiral modes from the edges of n = 100.(c) The profiles or distributed probabilities of chiral mode at the energy marked by the blue circle in (b).The momentum for the blue circle in (b) is ky = 3. µσ 0 ⊗ s 0 , while the latter is thoroughly dependent on spin, determined by the term g z σ 0 ⊗ s z .Resultantly, the difference in energy bands induced by tuning µ and g z , is produced.

Topological phase transitions manipulated jointly by superconducting energy gap and driving period
In this section, we display a 2D anomalous Floquet chiral TSC with Chern-number-independent chiral MEMs tuned commonly by the superconducting energy gap ∆ b and driving period τ .
Table 3 illustrates the topological phase diagram, the topological phase transition as a function of ∆ b and τ .The diagram is divided into areas 1, 2, and 3 according to the different topological phases in the high-frequency limit region.At the high-frequency limit, the phase transitions induced by ∆ b from 0.1 to 0.2 and from to 1.1 are respectively due to the closure of the bulk energy band gap at Γ(0, 0) and X(π, 0).
When ∆ b is slightly increased, e.g.∆ b = 0.2, area 2 just emerges.Similarly, a variety of new topological phases thoroughly different from those in area 1, start to emerge in the order of the five phases (1, 0), (0, 1), (−1, 2), (1, 0), and (3, −2).This indicates the entirely new topological phase transitions tuned by τ and the topological transitions manipulated by ∆ b from 0.1 to 0.2 at any fixed τ between the adjacent areas.The range of ∆ b corresponding to area 2 is considerably wide.It is found that the number of topological phases appearing successively with the increase of τ can be different at different ∆ b , which could increase or decrease with the enhancement of ∆ b .This is similar to the situation for increasing µ but not for enhancing g z .The number almost always increases with the enhancement of g z .At different fixed ∆ b from 0.2 to 1, the former four topological phases emerging successively with the increase of τ keep unchanged, but the corresponding driving period region for each phase has a great change.The latter several topological phases appearing sequentially with the increase of τ at different fixed ∆ b are much different.In addition, the topological phase transitions modulated by ∆ b at a fixed τ including the high-frequency limit region, can appear.In this area, there exist seven new kinds of topological phases, i.e. (0, 1), (1, 0), (−1, 2), (3, −2), (2, −1), (0, 0) and (−1, 0), which are thoroughly different from those of area 1.
With the further increase of ∆ b from 1 to 1.1, area 3 starts to appear.The former three topological phases appearing sequentially thoroughly turn into new ones.Different phase transitions by τ and the phase transitions by ∆ b at any fixed τ between the adjacent two areas, are resultantly induced.However, the latter two topological phases arising successively keep unchanged, thus no new phase transitions with τ are produced and the ones at few τ between the adjacent parts by ∆ b from 1 to 1.1 are exhibited.In this area, there exist only two kinds of new topological phases, i.e. (−2, 1) and (0, −1), which are never exhibited in the former two areas.
Although all the topological phases in table 3 are found to exhibit in tables 1 and 2, there still exist the exotic topological phase transitions, which are never exhibited in tables 1 and 2.
The above different features by the modulation of ∆ b from those by µ and g z can be attributed to the different Hamiltonian.The location of ∆ b in it differs from those of µ and g z , i.e. ∆ b is an off-diagonal term and closely linked to the periodic phase e iφ (t) , which leads to the difference in energy bands.
Similarly, take the anomalous Floquet chiral TSC phase (C, W) = (0, 2) at τ = 1.7, g z = 0.6, ∆ b = 0.1, and µ = 0 shown in the table 3 as an example to illustrate the bulk energy bands or the edge state of the anomalous Floquet chiral TSC phase in figure 6.
Remarkably, it is found that there exist two pairs of chiral edge modes at E = 0.The two modes with a left-shifted chirality from the boundary n = 0 are given by the black lines, while the other two modes with the right-shifted chirality from the boundary n = 100 are respectively presented by the red and blue lines for subsequent analyses.The same situations are for E = ω/2.Similarly, equation ( 12) is also fulfilled by the total number of edge modes.The distributed probabilities of chiral modes at E = 0 and E = ω/2 for the edge modes (the red and blue lines) with k y = 0.063 and 2 are respectively exhibited in figures 6(c)-(f), which demonstrate they locate near the one with n = 100.
From the above three sections, we observe that different kinds of topological phases with τ amount to 19 by tuning µ, g z , and ∆ b .The kind of topological phases (−3, 1) by tuning µ and the three ones (−2, 2), (2, −2), and (3, −1) by tuning g z are respectively not exhibited in the other two situations.However, for tuning ∆ b , there are no such new topological phases.

Experimental feasibility
Finally, we briefly discuss the experimental feasibility.Pb(Nb) is a suitable choice for the s-wave SC, as it has a critical temperature T c of 7.2 (9.5) K.By the proximity between the SC and TI, the superconductivity on the surface of the TI can be experimentally induced.The monocrystalline Bi 2 Te 3 (Sb 2 Te 3 ), which has been reported experimentally [75,85,86], is a possible candidate for the 3D TI, with a relatively small bulk energy gap of about 300 (100) meV.The magnetic TI can be achieved by doping Cr into Bi 2 Te 3 or Sb 2 Te 3 [87].The magnetic TI-superconducting structures have already been realized experimentally, so the proposed magnetic TI sandwich Josephson structures are feasible to fabricate.In experiments, the static parameters µ, g z , and ∆ b are tunable by varying the gate voltage V g , the external magnetic field, and the temperature, respectively.The dynamic parameter (ω or τ ) is modulated by simply changing the bias voltage V 0 .The proposed Floquet setup and corresponding Floquet chiral TSC phases can be therefore experimentally realizable.3 with τ = 1.7, gz = 0.6, ∆ b = 0.1, µ = 0, E = 0 (a) and ω/2 (b).This phase features two pairs of chiral edge modes at both E = 0 and ω/2, near the boundaries with n = 0 (the black line) and 100 (the red and blue ones).The profiles or distributed probabilities (c) and (e) of chiral modes at E = 0 and ky = 0.063 corresponding to the circles of the blue and red lines in (a), respectively.For (d) and (f), the same as in (c) and (e), respectively, except for E = ω/2 and ky = 2 in (b).

Summary
In summary, we study the anomalous Floquet chiral TSC phases by jointly tuning dynamic driving and static parameters, based on a magnetic TIs-superconductor sandwich Josephson hybrid structure.A static bias voltage applied across the top and bottom superconducting layers provides a periodic driving.By jointly tuning dynamic driving (τ ) and static (µ, g z , and ∆ b ) parameters, a variety of anomalies-rich Floquet TSC phases are exhibited.There exist four features as follows.(1) In the low frequency ω region, for different µ, different novel Floquet topological phase transitions can occur sequentially with τ increased.The processes are accompanied by the closure of the energy gap at the high-symmetry point E = ω/2 of the bulk energy bands.The same features are for g z and ∆ b .(2) For µ, there always exist several special values, at two any adjacent values of which, the corresponding topological phases appearing successively with the increase of τ could be thoroughly different.A series of topological phase transitions in the whole τ region tuned by µ are indicated as well.The same features exhibit for g z and ∆ b .During all the phase transitions, the closure of the bulk band gap at E = 0 occurs simultaneously at the high-symmetry points of Brillouin zone.(3) The numbers of the topological phases arising successively tuned by τ may both increase or decrease with the enhancement of the two parameters µ and ∆ b , while the number is basically increased with g z .(4) Jointly tuning τ and µ (g z ) can have its own unique TSC phases.
The features among the tunings of three static parameters are different, originating from the combination of intrinsic and different extrinsic mechanisms.The family of anomalous Floquet TSC phases induced by the tunings is considerably enriched.
Here, it is worth noting that in the phase diagram, the computation time for a single point is approximately one hour.To achieve a continuous phase diagram, the order of magnitude for the interval values of both the periodic driving and static parameters is 10 −2 .Take table 1 as an instance, a complete phase diagram would necessitate 25 351 points, indicating more than 1000 days of computational time, an unfeasible duration for us.A practical approach to obtain more detailed topological phase information is to select a region of interest in the parameter space based on the present topological phase diagram (e.g.table 1).Taking the phase transition boundary region with τ = (1.6,1.7) and µ = (0.3, 0.4) as an example, the complete calculation would only require 144 points, which could be completed in 6 days.If the calculation is deployed on a high-performance work cluster or supercomputer, the time will be further reduced.Such an operation not only saves time but also ensures the accuracy and reliability of the results.Moreover, it does not affect our qualitative conclusions and characteristics, which are thoroughly determined by the properties of the topological invariants.

Figure 1 .
Figure 1.(a) Schematic diagram of a Floquet sandwich-like TI Josephson hybrid structure, where the middle layer is a TI, and the top and bottom layers are s-wave SCs.A Zeeman field gz along the z direction is applied to make the TI become a 3D magnetic one.Two gate voltages Vg applied on the top and bottom surfaces of the TI can adjust the chemical potential µ.A periodic phase concerned with a bias voltage V0 between the two s-wave SC layers is used as a Floquet period drive, leading to the Josephson junction transformed into a Floquet TSC.(b) Tuning the driving period τ by varying the bias voltage V0 allows for dynamic modulation of the Floquet TSC phase.Meanwhile, the Floquet TSC phase is subject to the manipulation of the static parameters (chemical potential µ, Zeeman field gz, and proximity-induced superconducting energy gap ∆ b ).

Figure 2 .
Figure 2. Different chemicals µ = 0.1 (a) and 0.2 (b) lead to intersecting of the E = ω/2 lines (e.g. the two lines marked by the dash-dot and dash ones) with the static bands h (0) ω at different high-symmetry points.Here, only the upper half spectrum is presented for simplicity and the other parameters are ∆ b = 0.4 and gz = 0.6.

Figure 3 .
Figure 3.The evolution of the bulk energy band near E = 0 induced by µ, which is exemplified by µ = 0.35 (a), 0.394 (b), and 0.42 (c).The other parameters are the same as in figure 2.

Figure 4 .
Figure 4. Edge spectrums of the anomalous Floquet chiral TSC phase (C, W) = (1, −1) of table 1 with τ = 1.9, gz = 0.6, ∆ b = 0.4, µ = 1, E = 0 (a) and ω/2 (b).This phase features no chiral MEM penetrating the quasi-energy gap at E = 0, but a pair of ones with left-shifted and right-shifted chiralities at E = ω/2.The red line is the chiral modes from the edges of n = 100.(c) The profiles or distributed probabilities of chiral mode at the energy marked by the blue circle in (b).The momentum for the blue circle in (b) is ky = 3.

Figure 6 .
Figure 6.Edge spectrums of the anomalous Floquet chiral TSC phase (C, W) = (0, 2) of table3with τ = 1.7, gz = 0.6, ∆ b = 0.1, µ = 0, E = 0 (a) and ω/2 (b).This phase features two pairs of chiral edge modes at both E = 0 and ω/2, near the boundaries with n = 0 (the black line) and 100 (the red and blue ones).The profiles or distributed probabilities (c) and (e) of chiral modes at E = 0 and ky = 0.063 corresponding to the circles of the blue and red lines in (a), respectively.For (d) and (f), the same as in (c) and (e), respectively, except for E = ω/2 and ky = 2 in (b).

Table 3 .
The same as in table 1 except that µ is replaced by ∆ b .Here, µ = 0 and gz = 0.6.