Time-induced second-order topological superconductors

Higher-order topological materials, characterized by the presence of topologically protected states at the boundaries of their boundaries (hinges or corners), have attracted attention in recent years. In this paper, we propose a means to transform an inherently trivial system into a second-order topological superconductor by introducing nontrivial winding of a quasimomentum in the time-domain. Unlike other proposals of Floquet second-order topological phases developed in recent years, the generation of Majorana corner modes in our scheme arises from the interplay of topology in both spatial and temporal dimensions, where the underlying static system lacks the mechanism to ever host them. Our scheme thus brings forward the idea of Floquet engineering to another level where periodic drive can itself be endowed with topology to potentially reduce the complexity of the underlying static systems for hosting exotic topological phases.


I. INTRODUCTION
Since their theoretical discoveries in early 1980s [1,2], followed by various experimental realizations since the last decade [3,4], topological phases of matter have remained an active field of research.Their main signature to host robust topologically protected states in the presence of systems' boundaries or defects is especially attractive with potential applications in designing robust electronic/spintronic devices [5] and fault-tolerant quantum computing [6,7].
In the last couple of years, a new direction within the area of topological matter emerges through the discovery of higher-order topological phases (HOTP) [8][9][10], which exhibit topologically protected states at the boundaries of the systems' boundaries.In particular, an n-th-order topological phase in D dimensions is characterized by the presence of topologically protected states at its D − n dimensional boundaries.In the following years, HOTP have been extensively studied  and experimentally observed in a variety of physical platforms, such as photonics [29,34], acoustics [35], electrical circuit devices [30], and solid-state systems [28].A common feature of these studies suggests that systems with at least four bands are necessary for the formation of HOTP.As a result, a construction of such HOTP inevitably requires a number of internal degrees of freedom and/or spatial variations in the system parameters, thus leading to a generally complex design.
In this paper, we present a possibility to create a particular class of HOTP, i.e., a second-order topological superconductor (SOTSC), by utilizing time as a resource to encode additional topology.To this end, by starting with a static system which lacks the required building-blocks to host any second-order topology, the key idea is to design specific time-periodic modulations to some system parameters such that nontrivial winding of a quasimo- * Raditya.Bomantara@sydney.edu.aumentum arises in the time domain.The resulting additional topology then fills in the missing ingredient for enabling topologically protected corner modes.While this work can be considered to lie within the timely area of Floquet topological phases , the idea of directly introducing topology in the time-domain (to the best of our knowledge) has never been explored in previous literature.Yet, in practice, such a topological structure can actually be very easily implemented by properly introducing two harmonic drive with the same frequency and a phase difference of π/2 (i.e., a sine and cosine terms), while its implementation offers an important advantage of significantly reducing the complexity in the design of the underlying static systems.
The mechanism behind such a construction can be understood as follows.In the enlarged Hilbert (Sambe) space [76,77], terms proportional to a sine and consine functions of time induce imaginary and real couplings between two copies of the static Hamiltonian (one of which is shifted in energy with respect to the other by that of a single photon).Not only such couplings may lead to gap openings hosting topologically protected boundary modes, as commonly studied in previous Floquet topological phases literature (see e.g.Refs.[38,40]), we now demonstrate that they may in fact also accommodate additional winding of a system parameter (quasimomentum) to enable a new topology which is otherwise absent in the underlying static system.The importance of this aspect becomes clearer by noting that nontrivial winding of two quasimomenta is essential for the emergence of second-order topology, i.e., one is to ensure that edge modes exist, whereas the other results in these edge modes possessing nontrivial topology so as to host topologically protected corner modes.Given that a single set of Pauli matrices can only accommodate one winding structure, it follows that a minimum of two set of Pauli matrices (hence four-band systems) are necessary in static systems.On the other hand, the above mechanism implies that the same can be achieved with only a two-band time-periodic system.
While the idea presented in this paper can be gen-eralized to other HOTP, we focus on the generation of SOTSC due to their ability to host non-chiral Majorana modes (MMs), which are capable of encoding nonlocal qubits for topological quantum computation [6,7].For many years, such non-chiral MMs are usually found at the ends of certain one-dimensional (1D) systems, i.e., first-order topological superconductors.As a result, the implementation of quantum gate operations, accomplished by moving some MMs around one another, generally requires the design of complex architectures based on these 1D systems so as to facilitate these braiding processes [78][79][80][81].On the other hand, two-dimensional (2D) first-order topological superconductors may only host chiral MMs at their edges [82,83], which are not directly relevant for quantum computing applications.
Though non-chiral MMs may also appear at the vortices in the bulk of certain 2D fractional quantum Hall systems [84], not only the latter is not straightforward to realize experimentally, such MMs are generally fixed in place and may not be readily manipulated to perform quantum gate operations.For these reasons, realizing non-chiral MMs in 2D SOTSC is especially advantageous not only because these MMs naturally exist without the introduction of vortices, but also that braiding of MMs can be more realistically implemented either through conductance-measurements [64] or adiabatic following [85][86][87][88].The geometry of a single SOTSC also implies its scalability towards designing topological qubit architectures or quantum error correction codes [89,90].
This paper is structured as follows.In Sec.II A, we present a minimal model demonstrating the possibility of encoding topology in the time-domain and briefly review Floquet theory.In Sec.II B, we invoke a two-photon sectors approximation of the Floquet Hamiltonian to elucidate the intuition behind time-induced topology, identify the relevant symmetries, and construct a bulk Z 2 invariant predicting the presence of Majorana corner modes.In Sec.II C, we show that the Majorana corner modes expected from the two-photon sectors picture also emerge in the exact time-periodic system.In Sec.III A, we show that the same symmetries identified under two-photon sectors approximation exist in the exact time-periodic system, thus allowing us to derive a more general bulk Z 2 invariant expression.In the same section, we investigate the presence of certain static perturbations to identify symmetries which actually play the role in protecting Majorana corner modes.In Sec.III B, we present another choice of time-periodicity that also leads to the timeinduced topology.In particular, the resulting system under consideration gives rise to a rare scenario whereby chiral and non-chiral MMs coexist, which can be potentially utilized to perform quantum state transfer as we briefly comment.We summarize this paper, compare our work with previous literature, and highlight possible future work in Sec.IV.

II. TIME-INDUCED TOPOLOGY A. Minimal model
To illustrate the main physics, we consider the following (2D) square lattice model describing a periodically driven p x + ip y superconductor, where c † i,j (c i,j ) is the fermionic creation (annihilation) operator at lattice site (i, j), µ represents the chemical potential, J x (t) = J 0,x cos(ωt) and J y (t) = J 0,y sin(ωt) are the time-periodic hopping amplitudes of period T = 2π ω in the x and y directions respectively, and ∆ ∈ R characterizes the p x + ip y pairing strength.While Eq. ( 1) looks like a toy model, its static version has actually been experimentally realized in Ref. [82] to detect the existence of chiral MMs, where effective p x + ip y is realized by proximitizing a quantum anomalous Hall insulator thin film, such as (Cr 0.12 Bi 0.26 Sb 0.62 ) 2 Te 3 , with a normal (swave) superconductor.
Under periodic boundary conditions (PBC), Eq. ( 1) can be recast in terms of quasimomenta k x and k y as where h BdG is the momentum space Bogoliubov-de- is the Nambu wave function, and σ i 's are Pauli matrices acting in this Nambu basis.The momentum space Floquet Hamiltonian associated with h BdG is then obtained as where σ 0 is the identity 2 × 2 matrix.

B. Two-photon sectors approximation
To simplify our analysis, we first make an additional assumption that h c,BdG and h s,BdG are of sufficiently smaller energy scale as compared with that of h 0,BdG .It is to be emphasized however that such an assumption is not necessary for the generation of Majorana corner modes as demonstrated by our numerics in the next section.With the assumption in place, it is then justified to truncate H BdG so that only its zero and +1 (or −1) photon sectors, i.e., red colored blocks in Eq. ( 4), are included.In this case, another set of Pauli matrices η i may be introduced to represent the two Floquet photon sectors and write the truncated Floquet Hamiltonian H BdG as Ignoring the first two terms, it follows that H BdG takes the form equivalent to the minimal second-order topological insulator model studied in Refs.[9,10], where four corner modes associated with zero eigenvalue are expected when open boundaries are introduced in both x-and y-directions.Such corner modes are protected by Now including the first two terms into consideration, we first note that H BdG no longer respects the chiral symmetry defined above.However, it still satisfies two particle-hole symmetries about quasienergy ω 2 , i.e., where P 1 = σ x η x K and P 2 = σ y η y K (K being the complex conjugate).Moreover, diagonal and off-diagonal spatial symmetries (M D and M OD ) also exist [91], which are characterized by In this case, the particle-hole symmetries are responsible to protect the degeneracy of (Hermitian) zero and ω 2 quasienergy modes [92], whereas the two spatial symmetries guarantee that such zero and ω 2 modes, if exist, must be located at the system's corners.
If µ = 0, H BdG (k) can be easily diagonalized to yield the four eigenvalues where s 1 , s 2 = ±1.In particular, it follows that, taking into account that ε k,s1,s2 is defined modulo ω, there is no gap at zero quasienergy for any (nonzero) values of J 0,x , J 0,y , and ∆.Such a lack in zero quasienergy gap results in the absence of the four zero quasienergy corner modes when open boundaries are introduced.On the other hand, while introducing large µ = 0 may eventually reopen the gap around zero quasienergy, it leaves the system in the topologically trivial region since it can then be adiabatically connected to the static case (i.e., ∆ = 0) without closing the bulk gap.This argument implies that, similar to its static counterpart, our system cannot admit any zero quasienergy corner modes.This is not surprising since zero quasienergy modes (if any) originate from the topological structure of the underlying static system, which in our case cannot exist due to the lack of internal degrees of freedom (Pauli matrices) in the absence of any periodic-drive.By further making a simplifying assumption that J 0,x = J 0,y = 2∆ (in addition to again taking µ = 0), it can be easily checked that another bulk gap closing at ω 2 occurs at ∆ = ∆ c = √ 2 8 ω.Unlike the gap closing at zero quasienergy, however, the gap immediately reopens as ∆ is tuned away from ∆ c , thus signing a possible topological phase transition.In particular, ∆ < ∆ c is identified as a topologically trivial regime, since it is adiabatically equivalent to the static scenario (∆ = 0).The regime ∆ > ∆ c is thus expected to be topologically nontrivial and, together with the appropriate symmetry protections, four MPMs are expected to emerge at its corners when open boundaries are introduced.
Indeed, in Appendix A, we explicitly construct a bulk Z 2 invariant ν π characterizing the presence of MPMs, compatible with the two particle-hole and spatial symmetries defined above.In general, we find that the system is topologically nontrivial whenever In particular, under J 0,x = J 0,y = 2∆ and µ = 0, topological phase transition occurs at ∆ = ∆ c , where ν π becomes ill-defined, which is thus consistent with the quasienergy gap analysis above.
The Z 2 invariant ν π represents a quantity that signals the presence of quasienergy winding along either a diagonal or off-diagonal line in the 2D Brillouin zone (see Fig. 1).Due to the four symmetries M D , M OD , P 1 , and P 2 , observing such a feature is sufficient to determine the full topology of the system.To justify this argument, we note that potential topological phase transitions caused by quasienergy gap closing at a point not along the diagonal or off-diagonal line in the Brillouin zone can be smoothly deformed to those involving quasienergy gap closing at a point along either the diagonal or off-diagonal line.This can be understood from the fact a quasienergy gap closing at (k x,0 , k y,0 ) not along the diagonal and/or off-diagonal line must be accompanied by at least another quasienergy gap closing at (k x,1 , k y,1 ) due to M D and/or M OD .Moreover, these symmetries prevent the quasienergy gap closing points to reopen, unless by moving them to annihilate at a point along either the diagonal and/or off-diagonal line.
Due to P 1 and P 2 , however, these quasienergy gap closing points cannot actually annihilate after moving them at general locations along either the diagonal or off-diagonal line and must come at least in pairs within the line.In this case, such gap closing points must be further moved to either (0, 0), (±π, ±π), or (±π/2, ±π/2), i.e., points left invariant by either P 1 or P 2 , before they may properly annihilate and reopen.This results in a very symmetrical structure throughout the Brillouin zone, whereby considering the quasienergy bands along the 1D lines marked in either red or blue and either magenta or green in Fig. 1 is sufficient to capture properties of the system over the whole 2D Brillouin zone.In particular, the two topologies of the system, characterized by the presence and absence of the Majorana corner modes, are distinguished by the presence of winding in the quasienergy bands along one these lines (see the inset of Fig. 1), whose technical detail is elaborated in Appendix A.
The results presented thus far show that such MPMs are truly of dynamical origin, whose existence can be traced back from the presence of nontrivial quasienergy winding induced by time-periodicity.We have thus demonstrated another resourcefulness of Floquet engineering to create topological phases by embedding topological structure in the time domain.

C. Exact system
In Sec.II B, we have elucidated the main mechanism underlying the presence of corner MPMs in the system described by Eq. (1).In doing so, we made a very crude analysis based only on two photon sectors in the infinitedimensional Floquet Hamiltonian.Here, we verify numerically that the predicted corner MPMs remain existing even when considering the full picture.To this end, we may directly construct the Floquet Hamiltonian associated with Eq. ( 1) under open boundary conditions (OBC) in both x-and y-directions, truncated up to a reasonably large maximum photon index n max to allow numerical processing, then diagonalize it and accept only FIG. 2. Quasienergy spectrum of Eq. ( 1) under (a) OBC and (b) PBC in both directions.In (a), the system size is taken to contain 15 × 15 lattice sites, and up to ±3 photon sectors of the Floquet Hamiltonian are included (i.e., nmax = 3).In both panels, a nonzero µ = 0.1 T has also been considered.
quasienergy solutions within (0, ω] [94].Alternatively, such quasienergy solutions can also be obtained by diagonalizing the one-period time evolution operator (obtained numerically e.g.via the use of split-operator method), which inherently takes into account all photon sectors in the Floquet Hamiltonian language.We have employed both approaches and similar results are obtained.As such, in the following we present our results based on the former approach only.
Figure 2 shows the calculated quasienergy solutions as the system parameters are varied according to J 0,x = J 0,y = 2∆ = 2m T , where m ∈ R is a dimensionless real variable.There, it is observed that at small to moderate values of m, the quasienergy spectrum qualitatively exhibits the same behavior as that predicted in Sec.II A, where the two quasienergy bands touch at m c ≈ 2π that separates the regime with and without MPMs, respectively associated with m < m c and m > m c .In particular, to distinguish corner and edge states from the rest of the bulk states, Figs.2(a) and (b) are plotted under OBC and PBC respectively.That the ω 2 quasienergy solutions are missing in Fig. 2(b) signifies that these MPMs are indeed localized near the corners.Moreover, as we have also predicted in Sec.II A, the lack of band gap around zero quasienergy leads to the absence of MZMs.
To further verify that MPMs observed in Fig. 2 are corner and not edge modes, we plot in Fig. 3 the system's quasienergy spectrum under PBC in one direction and OBC in the other, which shows that ω 2 solutions are indeed absent.Moreover, we explicitly calculate the support of each corner MPM on Majorana operators representing the system's lattice sites.To this end, we write each corner MPM as [64] , where c i,j = 1 2 (γ 2i,j − iγ 2i+1,j ).Given that the dominant contribution to γ c (t) comes from the zero photon sector, we plot in Fig. 4 the weights W (0) i,j associated with the four corner MPMs in our system, where they are clearly localized at one of the four corners.There, we have also introduced imperfections in the system parameter values, which amount to introducing a slight inhomogeneity of pairing strengths and hopping amplitudes in the x-and ydirections, i.e., ∆ x = ∆ + δ, ∆ y = ∆ − δ, and J 0,x = J 0,y .
At larger values of m, it is further observed that another gap closing occurs at ω 2 quasienergy, after which the system again becomes topologically trivial with no MPMs.That is, unlike the model considered in Ref. [66], gap closing in our model does not generate new Majorana corner modes.This is expected because MPMs in our system are protected by particle-hole instead of chiral symmetry as is the case in Ref. [66].It is known that in one-dimensional topological systems, particle-hole symmetry protection leads only to a Z 2 topological invariant (hence corresponding to the presence or absence of end states), whereas chiral symmetric topological phases may exhibit a Z topological invariant [93].The same reasoning is expected to hold in regards to second-order topological phases in 2D, thus explaining why only at most four MPMs (one at each corner) may emerge in our system.

A. Symmetries protection
While the emergence of corner MPMs is naturally explained by invoking two-photon approximations, we have demonstrated in Sec.II C that they also persist in the exact time-periodic scenario.This observation can be first understood from the fact that all the symmetries we identified under two-photon approximations also exist in the actual system, which can be obtained by replacing the Pauli matrices η x , η y , and η z with their infinite dimensional counterparts, i.e., where we take sgn(0) = +1.It can be easily checked that η x , η y , and η z remain mutually anticommuting, and they transform as η i η j = δ i,j + i ijk η k similar to 2 × 2 Pauli matrices.As such, the Z 2 invariant derived based on these symmetries under two-photon approximations can also be generalized to the exact system, which is detailed in Appendix A 2 and obtained as which looks very similar to ν π , but it now captures other topological phase transitions happening at larger param- eter values.
To compare the two Z 2 invariants, we first note that according to ν π , the system is topologically nontrivial as long as | ω 2 − µ| < √ 8∆, suggesting that MPMs always exist as long as ∆ is very large.However, as seen from Fig. 2, there is clearly another gap closing at sufficiently large ∆ after which MPMs disappear.The modified invariant ν π on the other hand captures this additional trivial regime by noting that there are two terms in Eq. (10) with −1 values, resulting in an overall ν π = +1.Generally, it can be observed that as ∆ further increases, the system alternately switches between topologically trivial and nontrivial.In particular, the system is in the topologically trivial (nontrivial) regime whenever − µ| with n being even (odd).
While the Z 2 invariant above is derived by utilizing all P 1 , P 2 , M D , and M OD symmetries, not all of them are actually necessary to preserve the existing MPMs.To demonstrate this, under the system parameters for which MPMs are previously observed, i.e., those used to obtain Fig. 4, we add a static hopping term of the form to the original Hamiltonian of Eq. (1).It is easily checked that such a term breaks the second particle-hole symmetry (P 2 ).Yet, as Fig. 5 shows, MPMs remain existing even at moderate values of J s,x = J s,y = J and disappear only after quasienergy bulk gap closing occurs.The fact that only the three symmetries P 1 , M D , and M OD protect our MPMs is relevant to further justify that such MPMs indeed exist due to a new topology induced in the time-domain, and not due to the emer-gence of additional symmetries induced by time-periodic drive [73][74][75].Indeed, in the absence of any time-periodic drive, these symmetries are also present, now described by the static operators

B. Other time-periodic functions
The time-periodicity introduced in Eq. ( 1) is not the only one capable of introducing time-induced topology.It is expected that a class of other time-periodic functions will also work, provided that non-trivial winding mechanism elucidated above remains.In particular, given that any time-periodic function f (t) can be Fourier decomposed as f (t) = n f (s,n) sin(nωt) + f (c,n) cos(nωt) , it is generally sufficient to choose the two time-periodic functions J x (t) and J y (t) to be even and odd in t respectively, so that only J (c,n) x and J (s,n) y are nonzero, which include J x (t) and J y (t) terms introduced in Sec.II A.
To further verify the above argument, we may now take J x (t) and J y (t) to comprise as series of Dirac delta functions, so as to include all higher-harmonics in their Fourier decomposition, but J x (t) (J y (t)) contains only cosine (sine) contributions.In this case, diagonalizing the truncated Floquet Hamiltonian associated with Eq. ( 1) no longer represents a feasible way to numerically obtain its quasienergy spectrum as the presence of higher-harmonic terms necessarily requires keeping a large number of Floquet photon sectors to achieve a reasonable accuracy.On the other hand, the one-period time evolution operator associated with Eq. ( 1) is easily obtained as products of six exponentials In this case, quasienergies can be equivalently obtained by diagonalizing U T and taking the phase of its eigenvalues exp (−iεT / ).In both panels, nonzero µ = 0.1 T is taken.
In Fig. 6, we plot the new quasienergy spectrum of Eq. ( 1) under the modified time-periodic modulations defined by Eq. ( 12) by varying the parameters 2∆T = J 0,x = J 0,y .As expected, quasienergy ω 2 solutions associated with MPMs can be clearly identified for a range of parameter values.In addition, it is further observed that at small parameter values, a significant bulk gap around zero quasienergy emerges (see Fig. 6(b)), which hosts chiral edge states when OBC are applied (notice the additional series of quasienergy solutions filling the zero quasienergy gap in Fig. 6(a)).Consequently, at some appropriate parameter values, we find a rare possibility of coexisting chiral and non-chiral MMs in the system.It is expected that such a scenario may have a promising application in quantum information processing, particularly for the task of quantum state transfer [95][96][97][98][99].That is, one may imagine of encoding quantum information in some non-chiral MMs localized at corners of the one side of the system, transferring it to the chiral MMs, and retrieving it on the other side of the system by utilizing non-chiral MMs localized at its corners.The detail and feasibility of this procedure will be left for future work.

IV. CONCLUDING REMARKS
In this paper, we proposed the construction of corner MPMs without internal (pseudo-spin or orbital) degrees of freedom or spatially modulating any system parameters.In this case, the interplay between topological superconductivity and topological structure in the timedomain provides the necessary ingredient for the emergence of truly dynamical Majorana modes at the system's corners with no static analogues.
Before ending this paper, it is necessary to compare the results presented above to earlier work on Floquet higher-order topological phases [67][68][69][70][71][72][73][74][75].As mentioned at the beginning, the main feature of our work is the implementation of topology in the time-domain, thus creating a scenario in which topological phenomena are observed in an otherwise inherently trivial system.While some of the earlier work [67][68][69][70][71][72][73][74][75] also demonstrate the generation of topologically nontrivial phases by applying certain time-periodic drive to a static topologically trivial system, the static system under consideration may also potentially exhibit a topologically nontrivial phase by either tuning some system parameters or adding appropriate mass terms.In this case, the time-periodic drive simply plays the role of either system parameters renormalization or mass terms simulation, whose topology may thus (in principle) be traced back from the underlying static system.
To more concretely establish the above argument, we shall compare our construction with those of Refs.[73][74][75], which at first glance might look similar to ours (i.e., the use of monochromatic time-periodic drive).In Refs.[73][74][75], the time-periodic drive is designed such that the resulting Hamiltonian obeys a time-glide symmetry, which can then be viewed as an effective reflection symmetry in the enlarged Hilbert (Sambe) space.In this case, the role of the time-periodic drive is to effectively create a symmetry necessary for the formation of second-order topological phases, whereas the underlying static Hamiltonian already contains the necessary topological structure.This is further evidenced by the fact that four-band models are used in these references, i.e., the minimum number of bands expected for the formation of second-order topology in static systems.By contrast, the static system considered in this paper corresponds to a two-band (first-order) chiral topological superconductor, whose second-order topology is always trivial under any circumstances due to the lack of mass terms (with only one set of Pauli matrices available) to open the edge states' gap.In this case, the necessary symmetries capable of protecting second-order topology (were it allowed to exist) are already present in the underlying static system; and the proposed time-periodic drive genuinely plays the role of creating new topology in the system that leads to the formation of Majorana corner modes.
Various directions for potential future studies can be envisioned following the above discovery of time-induced topology.In the area of Floquet engineering, other realizations of existing (first-or higher-)order topological phases with significantly simpler systems may be possible through the application of several appropriate time-periodic potential following the above construction.In the area of quantum computing, the relatively less demanding system's complexity for hosting such timeinduced MMs may offer an opportunity to take a step forward towards the physical realizations of large-scale Majorana qubit architectures.Moreover, the possibility of time-induced topological superconductors to host chiral and non-chiral MMs simultaneously may allow the design of Majorana-based quantum state transfer scheme, as briefly commented in Sec.III B. Finally, we expect that the idea of time-induced topology may open up opportunities for the discovery of novel Floquet topological phases.
where we have defined The quasienergy winding along (without loss of generality) the blue dashed line in Fig. 1 can then be calculated as (employing Cauchy residue theorem) The same result is also obtained when calculating the quasienergy winding along one of the off-diagonal lines, i.e., n od = n d .We may finally define two Z 2 invariants as which respectively determine the presence of Majorana corner modes at the diagonal and off-diagonal corners.Since both expressions are identical, we can define a single Z 2 invariant ν π = ν d = ν od , such that the system under consideration supports four MPMs at its corners or none at all whenever ν π = −1 or ν π = 1 respectively.

Exact system
The above construction can be readily generalized to obtain the modified Z 2 invariant for the actual timeperiodic system described by the infinite-dimensional Floquet Hamiltonian.To this end, it is first noted that all the previously considered symmetries, i.e., M D , M OD , P 1 , and P 2 , exist also in the exact system by modifying η x and η y from Pauli to infinite dimensional matrices as elucidated in Sec.III A of the main text.As a result, similar basis transformation leading to a block off-diagonal structure in the previous section can also be found.In this case, instead of being a 4×4 Hamiltonian, an infinite dimensional block off-diagonal matrix is obtained of the form  (A9)

FIG. 1 .
FIG. 1.The nontrivial winding of the quasienergy bands along the blue dashed line arises due to gap closing and reopening (different colors in the inset).Due to P2 and P1 quasienergy bands along the blue (green) and red (purple) dashed lines have identical structures.

FIG. 3 .
FIG. 3. Quasienergy spectrum of Eq. (1) under (a) OBC in the y-direction and PBC in the x-direction, (b) OBC in the x-direction and PBC in the y-direction.In both panels, 40 sites are taken in the direction where OBC are applied, up to ±3 photon sectors of the Floquet Hamiltonian are included (i.e., nmax = 3), a nonzero µ = 0.1 T is considered, and the other parameters are set as J0,x = J0,y = 2∆ = 4 T .

FIG. 4 .
FIG. 4. Supports of each corner MPMs on Majorana operators representing the system's 20 × 20 lattice sites.Only zero photon sector contributions are shown, where the corner MPM solutions are obtained by diagonalizing the truncated Floquet Hamiltonian containing up to ±3 photon sectors.System parameters are chosen as ∆ = 2.0 T , δ = 0.1 T , J0,x = 3.4 T , J0,y = 3.6 T , and µ = 0.1 T .

FIG. 5 .
FIG. 5. Quasienergy spectrum in the presence of P2 breaking term Hs under OBC as Js,x = Js,y = J is varied.The rest of the parameter values are the same as those chosen in Fig. 4.

FIG. 6 .
FIG.6.Quasienergy spectrum of Eq. (1) under the modified time-periodicity of Jx(t) and Jy(t) described in Sec.III B where (a) OBC and (b) PBC are applied in both directions.In both panels, nonzero µ = 0.1 T is taken.