Floquet higher-order topological insulators and superconductors with space-time symmetries

Floquet higher-order topological insulators and superconductors (HOTI/SCs) with an order-two space-time symmetry or antisymmetry are classified. This is achieved by considering unitary loops, whose nontrivial topology leads to the anomalous Floquet topological phases, subject to a space-time symmetry/antisymmetry. By mapping these unitary loops to static Hamiltonians with an order-two crystalline symmetry/antisymmetry, one is able to obtain the $K$ groups for the unitary loops and thus complete the classification of Floquet HOTI/SCs. Interestingly, we found that for every order-two nontrivial space-time symmetry/antisymmetry involving a half-period time translation, there exists a unique order-two static crystalline symmetry/antisymmetry, such that the two symmetries/antisymmetries give rise to the same topological classification. Moreover, by exploiting the frequency-domain formulation of the Floquet problem, a general recipe that constructs model Hamiltonians for Floquet HOTI/SCs is provided, which can be used to understand the classification of Floquet HOTI/SCs from an intuitive and complimentary perspective.


I. INTRODUCTION
The interplay between symmetry and topology leads to various of topological phases. For a translationally invariant noninteracting gapped system, the topological phase is characterized by the band structure topology, as well as the symmetries the system respects. Along with these thoughts, a classification was obtained for topological insulators and superconductors (TI/SC) [1][2][3] in the ten Altland-Zirnbauer (AZ) symmetry classes [4][5][6][7][8], which is determined by the presence or absence of three types of nonspatial symmetries, i.e. the time-reversal, particle-hole and chiral symmetries.
One nice feature of these tenfold-way phases is the bulk-boundary correspondence, namely, a topologically nontrival bulk band structure implies the existence of codimension-one gapless boundary modes on the surface, irrespective the surface orientation. (The codimension is defined as the difference between the bulk dimension and the dimension of the boundary where the gapless mode propagates).
When considering more symmetries beyond the nonspatial ones, the topological classification is enriched. Topological crystalline insulators [9][10][11][12][13] are such systems protected by crystalline symmetries. They are able to host codimension-one gapless boundary modes only when the boundary is invariant under the crystalline symmetry operation. For example, topological crystalline insulators protected by reflection symmetry [10] can support gapless modes only on the reflection invariant boundary. On the other hand, inversion symmetric topological crystalline insulators do not necessarily give rise to codimension-one gapless boundary modes [14,15], because no boundary * yang.peng@csun.edu is invariant under inversion.
Remarkably, it was recently demonstrated that a crystal with a crystalline-symmetry compatible bulk topology may manifest itself through protected boundary modes of codimension greater than one [16][17][18][19][20][21][22][23][24][25][26]. Such insulating and superconducting phases are called higher-order topological insulators and superconductors (HOTI/SCs). Particularly, an nth order TI/SC can support codimension-n boundary modes. (The strong TI/SCs in the tenfold-way phases with protected boundary modes at codimension one can be called as first-order TI/SCs according to this definition.) A higher-order bulk-boundary correspondence between the bulk topology and gapless boundary modes at different codimensions was derived in Ref. [26] based on K-theory.
Beyond equilibrium or static conditions, it is known that topological phases also exist, and one of the famous examples is the Floquet topological insulator, which is proposed to be brought from a static band insulator by applying a periodic drive, such as a circularly polarized radiation or an alternating Zeeman field [27][28][29][30][31]. A complete classification of the Floquet topological insulators (as well as superconductors) in the AZ symmetry classes has been obtained in Ref. [32,33], which can be regarded as a generalization of the classfication for static tenfoldway TI/SCs.
In a periodically driven, or Floquet, system, the nontrivialty can arise from the nontrivial topology of the unitary time-evolution operator U (t) (with period T ), which can be decomposed into two parts as U (t) = e −iH F t P (t).
Here, the first part describes the stroboscopic evolution at time of multiples of T in terms of a static effective Hamiltonian H F , and the second part is known as the micromotion operator P (t) = P (t + T ) describing the evolution within a given time period [34]. (We will make this decomposition more explicit later). Thus, the nontrivial topology can separately arise from H F as in a Floquet second-order TI/SCs protected by time-glide symmetry/antisymmetry can be mapped to static second-order TI/SCs protected by reflection symmetry/antisymmetry. The dashed line indicates the reflection (time-glide) plane. static topological phase, or from the nontrivial winding of P (t) over one period. Whereas Floquet topological phase in former situation is very similar to a static topological phase as it has a static limit, the latter is purely dynamical and cannot exist if the time-periodic term in the Hamiltonian vanishes. Therefore, systems belong to the latter case are more interesting and are known as the anomalous Floquet topological phases. In a Floquet system, energy is not conserved because of the excplicit time-depdendence of the Hamiltonian. However, one can define quasienergies as eigenvalues of H F = i T ln U (T ), which are only defined modulo the periodic driving frequency ω = 2π/T . This can be intuitively understood due to the existence of energy quanta ω that can be absorbed and emitted. Similar to static topological phases, the quasienergy spectrum can be different with different boundary conditions. Particularly, inside a bulk quasienergy gap (when periodic boundary condition is applied), there may exist topologically protected boundary modes.
In Floquet topological phases protected by nonspatial symmetries (tenfold-way phases), the bulk-boundary correspondence is also expected to hold [32], namely the number of boundary modes inside a particular bulk gap can be fully obtained from the topology of the evolution operator U (t), when periodic boundary condition is applied. Interestingly, when there exists a symmetry relating states at quasienergies and − , then the topological protected boundary modes will appear inside the quasienergy gap at 0 and ω/2, since these are quasienergies that are invariant under the above symmetry operation.
In particular, a bulk micromotion operator with nontrivial topology is able to produce gapless Floquet codimension-one boundary modes at quasienergy ω/2 (which will be made clear later). The natural following question to ask is that how can we create Floquet higher-order topological phases, with protected gapless modes at arbitrary codimensions. In particular, we want to have the topological nontriviality arise from the micromotion operator, otherwise we just need to have H F as a Hamiltonian for a static higher-order topological phase.
Similar to the static situation, when only nonspatial symmetries are involved, the tenfold-way Floquet topo-logical phases are all first-order phases which can only support codimension-one boundary modes. Higher-order phases are yet possible when symmetries relating different spatial points of the system are involved. These symmetries can be static crystalline symmetries, as well as space-time symmetries which relates systems at different times.
Recently, the authors in Refs. [35][36][37][38][39][40] constructed Floquet second-order TI/Scs. Particularly, the authors of Ref. [38] were able to construct Floquet corner modes by exploiting the time-glide symmetry [41], which combines a half-period time translation and a spatial reflection, as illustrated in the left part of Fig. 1.
It turns out that the roles played by such space-time symmetries in Floquet systems cannot be trivially replaced by spatial symmetries. As pointed out in Ref. [38], in protecting anomalous Floquet boundary modes, the space-time symmetries generally have different commutation relations with the nonspatial symmetries, compared to what the corresponding spatial symmetries do.
Since the use of space-time symmetries opens up new possibilities in engineering Floquet topological phases, especially the Floquet HOTI/SCs, it is important to have a thorough topological classification, as well as a general recipe of model construction for such systems.
In this work, we completely classify Floquet HOTI/SCs with an order-two space-time symmetry/antisymmetry realized by an operatorÔ, which can be either unitary or antiunitary. By order-two, it means that the symmetry/antisymmetry operator twice trivially acts on the time-periodic Hamiltonian H(t), namely [Ô 2 , H(t)] = 0,Ô =Û,Â whereÔ can be either unitaryÛ or antiunitaryÂ. We further provide a general recipe of constructing tight-binding Hamiltonians for such Floquet HOTI/SCs in different symmetry classes. Note that the ordertwo static crystalline symmetries/antisymmetries considered in Ref. [11] will be a subset of the symmetries/antisymmetries considered in this work. Our classification and model construction of Floquet HOTI/SCs involve two complementary approaches. The first approach is based on the classification of gapped unitaries [32,41], namely the time-evolution operator U (t) at time t ∈ [0, T ), with U (T ) gapped in its eigenvalues' phases. It turns out that the gapped unitaries can be (up to homotopy equivalence) decomposed as a unitary loop (which is actually the micromotion operator) and a unitary evolution under the static Floquet Hamiltonian H F . Thus, a general gapped unitary is classified by separately considering the unitary loop and the static Hamiltonian H F , where the latter is well known for systems in AZ classes as well as systems with additional crystaline symmetries. The classification of unitary loops on the other hand is less trivial since it is responsible for the existence of anomalous Floquet phases [42], especially when we are considering space-time symmetries. I. Nontrivial space-time symmetry/antisymmetry with subscript T /2 vs. static spatial symmetry/antisymmetry with subscript 0, sharing the same K groups at the same dimension.Û,Â, U and A denote unitary symmetry, antiunitary symmetry, unitary antisymmetry and antiunitary antisymmetry, respectively. The commutation (anticommutation) relations with coexisting nonspatial symmetries are denoted as additional subscripts + (−), while the superscript indicates the square of the operator. In the case of classes BDI, DIII, CII, CII, the first and second ± correspond to time-reversal, and particle-hole symmetries, respectively. We focus on the classification of Floquet unitary loops in this work. In particular, a hermitian map between unitary loops and hermitian matrices is introduced, which is inspired by the dimensional reduction map used in the classification of TI/SCs with scattering matrices [43].
The key observation is that the symmetry constraints on the unitary loops share the same features as the ones on scattering matrices. This hermitian map has advantages over the one used in earlier works [32,41], because it simply maps a unitary loop with a given order-two space-time symmetry/antisymmetry to a static Hamiltonian of a topological crystaline insulator with an ordertwo crystalline symmetry/antisymmetry. This enable us to exploit the full machinary of K theory, to define K groups, as well as the K subgroup series introduced in Ref. [26], for the unitary loops subject to space-time symmetries/antisymmetries.
Based on this approach, we obtain the first important result of the this work, namely, for every order-two nontrivial space-time (anti)unitary symmetry/antisymmetry, which involves a half-period time translation, there always exists a unique order-two static spatial (anti)unitary symmetry/antisymmetry, such that the two symmetries/antisymmetries corresopond to the same K group and thus the same classification. This result is illustrated in Fig. 1 for the case of time-glide vs. reflection symmetries. The explicit relations are summarized in Table I. Because of these relations, all results for the classification [17][18][19][20][21][22][23][24][25] as well as the higher-order bulkboundary correspondence [26] of static HOTI/SCs can be applied directly to the anomalous Floquet HOTI/SCs.
In the second approach, by exploiting the frequencydomain formulation, we obtain the second important result of this work, which is a general recipe of construct-ing harmonically driven Floquet HOTI/SCs from static HOTI/SCs. This recipe realizes the K group isomorphism of systems with a space-time symmetry and systems with a static crystalline symmetry at the microscopic level of Hamiltonians, and therefore provide a very intuitive way of understanding the classification table obtained from the formal K theory.
The rest of the paper is organized as follows. We first introduce the symmetries, both nonspatial symmetries and the order-two space-time symmetries, for Floquet system in Sec. III. Then, in Sec. IV, we introduce a hermitian map which enables us to map the classification of unitary loops to the classification of static Hamiltonians. In Sec. V, by using the hermitian map, we explicitly map the classification of unitary loops in all possible symmetry classes supporting an order-two symmetry, to the classification of static Hamiltonians with an order-two crystalline symmetry. In Sec. VI, we derive the corresponding K groups for unitary loops in all possible symmetry classes and dimensions. In Sec. VII, we introduce the K subgroup series for unitary loops, which enables us to completely classify Floquet HOTI/SCs. In Sec. VIII, the frequency-domain formulation is introduced, which provides a complimentary perspective on the topological classification of Floquet HOTI/SCs. In Sec. IX, we introduce a general recipe of constructing harmonically driven Floquet HOTI/SCs, and provide examples in different situations. Finally, we conclude our work in Sec. X.
Note that it is possible to skip the K-theory classification sections from IV to VII, and understand the main results in terms of the frequency-domain formulation.

In a Floquet system, the Hamiltonian
is periodic in time with period T = 2π/ω, where ω is the angular frequency. In a d-dimensional system with translational symmetry and periodic boundary condition, we have well defined Bloch wave vector k in the d dimensional Brillouin zone T d (torus). The system can thus be characterized by a time-periodic Bloch Hamiltonian H(k, t).
In the presence of a d def −dimensional topological defect, the wave vector k is no longer a good quantum number due to the broken translational symmetry. However, the topological properties of the defect can be obtained by considering a large D = (d − d def − 1)-dimensional surface, on which the translational symmetry is asymptotically restored so that k can be defined, surrounding the defect. We will denote r as the real space coordinate on this surrounding surface, or a D-sphere S D , which will determine the topological classification. Thus, we have a time-periodic (t ∈ S 1 ) Bloch Hamiltonian H(k, r, t) defined on T d × S D+1 . In the following, we will denote the dimension of such a system with a topological defect as a pair (d, D).
The topological properties for a given Hamiltonian H(k, r, t), can be derived from its time-evolution operator whereT denotes the time-ordering operator. The Floquet effective Hamiltonian H F (k, r) is defined as Note that different H F s defined at different t 0 s are related by unitary transformations, and thus the eigenvalues of the Floquet effective Hamiltonian are uniquely defined independent of t 0 . which is independent of t 0 . We also introduce n (k, r) ∈ [−π/T, π/T ] to denote the nth eigenvalue of H F (k, r), and call it the nth quasienergy band. Although H F captures the stroboscopic evolution of the system, it does not produce a complete topological classification of the Floquet phases. It is known that one can have the so called anomalous Floquet phases even when H F is a trivial Hamiltonian.
To fully classify the Floquet phases, we need information of the evolution operator at each t within the period. In order to have a well defined phase, we will only consider gapped unitary evolution operators, whose quasienergy bands are gapped at a particular quasienergy gap . Thus, given a set of symmetries the system respects, one needs to classify these gapped unitaries defined from each gapped quasienergies gap . The most common considered gapped energies in a system with particle-hole or chiral symmetry are 0 and ω/2, since such energies respect the symmetry. Note that the gap = ω/2 case is more interesting since they correspond to anomalous Floquet phases [42], which has no static analog. When neither of the two above mentioned symmetries exists, the gapped energy can take any value, but one can always deform the Hamiltonian such that the gapped energy appears at ω/2 without changing the topological classification. Hence, in the following we will fix gap = ω/2.
It is evident that the initial time t 0 in the evolution operator does not affect the classification, since it corresponds to different ways of defining the origin of time. Thus, from now on, we will set t 0 = 0 and denote A less obvious fact, is that one can define the symmetrized time-evolution operator [32] centered around time τ as which will also give rise to the same topological classification. This statement is proved in Appendix A. In fact, U τ (k, r, T ) leads to the same quasienergy band structure independent of the choice of τ . This is because (the explicit k, r dependence is omitted) with unitary matrix W = U (τ + T /2)U † (T /2). Thus, U τ (T )s at different τ s are related by unitary transformation, and we will in the following use U τ (k, r, t) to classify Floquet topological phases. For classification purpose, we need to setup the notion of homotopy equivalence between unitary evolutions. Let us consider evolution operators gapped at a given quasienergy. Following the definition in Ref. [32], we say two evolution operators U 1 and U 2 are homotopic, denoted as U 1 ≈ U 2 , if and only if there exists a continuous unitary-matrix-valued function f (s), with s ∈ [0, 1], such that where f (s) is a gapped evolution operator for all intermediat s. It is worth mentioning that when dealing with symmetrized evolution operators instead of ordinary evolution operators, the definition of homotopy equivalence is similar except one needs to impose that the interpolation function f (s) for all s is also a gapped symmetrized evolution operator. When comparing evolution operators with different number of bands, the equivalence relation of stable homotopy can be further introduced. Such a equivalence relation is denoted as U 1 ∼ U 2 if there exist two trivial unitaries U 0 n1 and U 0 n2 , with n 1 and n 2 bands respectively, such that where ⊕ denotes the direct sum of matrices.
We will now define how to make compositions between two symmetrized evolution operators. Using the notation in Ref. [32], we write the evolution due to U τ,1 followed by U τ,2 as U τ,1 * U τ,2 , which is given by the symmetrized evolution under Hamiltonian H(t) given by where H 1 (t) and H 2 (t) are the corresponding Hamiltonians used for the evolution operators U τ,1 and U τ,2 , respectively.
As proved in Ref. [32], with such definitions of homotopy and compositions of evolution operators, one can obtain the following two important theorems. First, every gapped symmetrized evolution operator U τ is homotopic to a composition of a unitary loop L τ , followed by a constant Hamiltonian evolution C τ , unique up to homotopy. Here the unitary loop is a special time evolution operator such that it becomes an identity operator after a full period evolution. Second, L τ,1 * C τ,1 ≈ L τ,2 * C τ,2 if and only if L τ,1 ≈ L τ,2 and C τ,1 ≈ C τ,2 , L τ,1 , L τ,2 are unitary loops, and C τ,1 , C τ,2 are constant Hamiltonian evolutions. For completeness, we put the proof of the two theorems in Appendix B.
Because of these two theorems, classifying generic time-evolution operators reduces to classifying separately the unitary loops and the constant Hamiltonian evolutions. Since the latter is exactly the same as classifying static Hamiltonians, we will in this work only focus on the classification of unitary loops. In the following, all the following time-evolution operators are unitary loops, which additionally satisfy U τ (k, r, t) = U τ (k, r, t + T ).

III. SYMMETRIES IN FLOQUET SYSTEMS
In this section, we will summarize the transformation properties of the time evolution opeerator under various of symmetry operators.

A. Nonspatial symmetries
Let us first look at the nonspatial symmetries and consider systems belong to one of the ten AZ classes (see Table II), determmined by the presence or absence of time-reversal, particle-hole and chiral symmetries, which are defined by the operatorsT = U TK ,Ĉ = U CK and S = U S =TĈ respectively, such that whereT = U TK ,Ĉ = U CK are antiunitary operators with unitary matrices U T , U C and complex conjugation operatorK. Here r is invariant in the above equations, because of the nonspatial nature of the symmetries. For a Floquet system, the action of symmetry oper-ationsT ,Ĉ, andŜ on the symmetrized unitary loops U τ (k, r, t) can be summarized aŝ which follow directly from Eqs. (11). For later convenience, we further introduce notations B. Order-two space-time symmetry In addition to the nonspatial symmetries, let us assume the system supports an order-two space-time symmetry realized byÔ, as defined in Eq. (1). Moreover, we assumê O commute or anticommute with the operators for the nonspatial symmetries of the system.
While the nonspatial symmetries leave the spatial coordinate r invariant, the order-two space-time symmetry transforms r nontrivially. To determine the transformation law, we follow Ref. [11] and consider a D-dimensional sphere S D surrounding the topological defect, whose coordinates in Euclidean space are determined by n 2 = a 2 , n = (n 1 , n 2 , . . . , n D+1 ), with radius a > 0. SinceÔ maps S D into itself, we have with n = (n 1 , n 2 , . . . , n D ), and n ⊥ = (n D +1 , n D +2 , . . . , n D+1 ) in a diagonal basis ofÔ. When D ≤ D, we can introduce the coordinate r ∈ S D by which leads to Here, r = (r 1 , r 2 , . . . , r D ) and r ⊥ = (r D +1 , r D +2 , . . . , r D ).
Thus, we need to introduce (d, d , D, D ) to characterize the dimension of the system according to the transformation properties of the coordinates, where d and D are defined the same as defined previously, while d and D denote the dimensions of the flipping momenta and the defect surrounding coordinates, respectively. For example, a unitary symmetry with (d, d , D, D ) = (2, 1, 1, 1) correspond to the reflection in 2D with a point defect on the reflection line, while a unitary symmetry with (d, d , D, D ) = (3, 2, 2, 2) is a two-fold rotation in 3D with a point defect on the rotation axis.
Next, let us consider the action of the order-two spacetime symmetry on the time arguement. For unitary symmetries, an action on t can generically have the form t → t + s. Due to the periodicity in t and the order-two nature of the symmetry, s can either be 0 or T /2.
For antiunitary symmetries, we have t → −t+s. When the system does not support time-reversal or chiral symmetry, as in classes A, C, and D, the constraints due to time-periodicity and the order-two nature do not restrict the value s takes. Hence, s is an arbitrary real number in this situation.
However, when the system has at least one of the timereversal and chiral symmetries, denoted asP, s will be restricted to take a few values as shown in the following. The composite operationPÔ shift the time as t → −s + t. On the other hand, sincePÔ is another order-two symmetry, s can be either 0 or T /2 (note that s is defined modulo T ).
To summarize, for a Hamiltonian H(k, r, t) living in dimension (d, d , D, D ), under the action ofÔ, it transforms aŝ in the diagonal basis ofÔ. for unitary and antiunitary symmetries. Let us supposeÔ 2 = O = ±1, andÔ commutes or anticommutes with coexisting nonspatial symmetries according tô where η T = ±1, η C = ±1, and η S = ±1. Note that whenÔ =Û, we can always set O = 1 with the help of multiplyingÔ by imaginary unit i, but this changes the (anti)commutation relation withT and/orĈ at the same time.
One can also consider an order-two antisymmetry O defined by (23) where O can be either unitary U or antiunitary A. Such an antisymmetry can be realized by combining any of order-two symmetries with chiral or particle-hole symmetry. Similar toÔ, we define In the following, we will discuss each symmetry/antisymmetry operator separately, and choose a particular value of τ for each case, since we know the classification would not depend on what the value τ takes. ForÛ s and A s s = 0, T /2, and we take τ = T /2. By using and omitting the subscript τ from U τ (k, r, t) from now on for simplicity, we get When considering U s andÂ s in classes A, C and D, we can choose τ = s/2, which giveŝ This implies that the value s here actually does not play a role in determining topological classification.
In the remaining classes, we have s = 0, T /2, and we will choose τ = T /2. This leads tô

IV. HERMITIAN MAP
One observation that can be made from Eqs. (12)(13)(14) is that at fixed r and t, the transformation properties for the unitary loops U (k, r, t) under the actions ofT , C, andŜ are exactly the same as the ones for unitary boundary reflection matrices introduced in, for example, Refs. [18,43]. In these works, an effective hermitian matrix can be constructed from a given reflection matrix, which maps the classification of reflection matrices into the classification of hermitian matrices.
Here, we can borrow the same hermitian mapping defined as if U (k, r, t) has a chiral symmetry, and if U (k, r, t) does not have a chiral symmetry. In the latter case, H(k, r, t) aquires a new chiral symmetry where we have introduced a set of Pauli matrices ρ x,y,z in the enlarged space. Note that when the unitary loop U(k, r, t) does not have a chiral symmetry, our hermitian map is the same as the one used in Refs. [32,41]. When the unitary loop does have a chiral symmetry, however, we chose a new map which maps the unitary loop into a hermitian matrix without unitary symmetry.
The advantage of the hermitian map defined here over the one in the previous works will become clear soon. Note that the hermitian matrix H(k, r, t) can be regarded as a static spatially modulated Hamiltonian in (d, D + 1) dimension, because the time arguement transforms like a spatial coordinate similar to r. The classification of unitary loops in (d, D) dimension in a given symmetry class, is then the same as the classification of static Hamiltonians in (d, D + 1) dimension in the symmetry class shifted upward by one (s → s − 1) (mod 2 or 8 depending for complex or real symmetry classes), where s is used to order the symmetry classes according to Table II. Thus, one can directly apply the classification scheme of the static Hamiltonians H(k, r) using K theory, as was done in Ref. [11]. This is provided by a homotopy classification of maps from the base space (k, r) ∈ S d+D to the classifying space of Hamiltonians H(k, r) subject to the given symmetries, which we denoted as C s or R s as shown in the table.
Because of the Bott periodicity in the periodic table of static TI/SCs [4][5][6][7][8], the classification is unchanged when simultaneously shifting the dimension D → D + 1 and the symmetry class upward by one s → s − 1 (mod 2 or 8 for complex or real symmetry classes). It turns out that the classification of unitary loops is the same as the classification of the static Hamiltonian in the same symmetry class and with the same dimension (d, D). In the following, we will explicitly derive the action of the hermitian map on each symmetry classes.

A. Classes A and AIII
We first consider the two complex classes. Under the hermitian map defined above, classifying unitary loops in (d, D) dimension in class A is the same as classifying hermitian matrices in (d, D + 1) dimension in class AIII. On the other hand, classifying unitary loops in (d, D) dimension in class AIII is the same as classifying hermitian matrices in (d, D + 1) dimension in class A.

B. Classes AI and AII
Now we turn to real symmetry classes. Since classes AI and AII have only time-reversal symmetry, we need to apply the hermitian map defined in Eq. (33). By using Eq. (12) with τ = T /2, or we have effective time-reversal symmetry with U T = ρ x ⊗ U T , and effective particle-hole symmetry with U C = iρ y ⊗ U T . Note that the effective time-reversal and particle-hole symmetries combines into the chiral symmetry as expected. The types of the effective time-reversal and particle-hole symmetries of H(k, t) are determined from where ρ 0 is the two-by-two identity matrix in the extended space. Under the hermitian map, classifying unitary loops in (d, D) dimension in classes AI and AII, are the same as classifying hermitian matrices in (d, D + 1) dimension in classes CI and DIII.

C. Classes C and D
Let us consider classes C and D with only particle-hole symmetry. We need to apply the hermitian map defined in Eq. (33). By using Eq.(13), one can define effective time-reversal symmetry with U T = ρ 0 ⊗U C , and particlehole symmetry with U C = ρ z ⊗ U C , such that Note that U T and U C combines into the chiral symmetry as expected. The types of these effective symmetries are determined by Under the hermitian map, classifying unitary loops in (d, D) dimension in classes C and D, are the same as classifying hermitian matrices in (d, D + 1) dimension in classes CII and BDI.

D. classes CI, CII, DIII, and BDI
Here we consider symmetry classes where timereversal, particle-hole, and chiral symmetries are all present In this case, This can be used to show that Notice that U C U * C = ±1 and U T U * T = ±1 are just numbers.
The effective Hamiltonian H(k, t) defined in Eq.(32) has the property This gives rise to time-reversal or particle-hole symmetry depending on (U C U * C )(U T U * T ) = 1 or −1, respectively. Therefore, under the hermitian map, the unitary loops in (d, D) dimension in classes CI, CII, DIII, and BDI, map to hermitian matrices in (d, D + 1) dimension in classes C, AII, D, and AI, respectively.

V. CLASSFICATION WITH ADDITIONAL ORDER-TWO SPACE-TIME SYMMETRY
After introducing the hermitian map which reduces the classification of unitary loops to the classification of static hermitian matrices, or Hamiltonians, in the AZ symmetry classes, let us now assume the system supports an additional order-two space-time symmetry/antisymmetry, which is either unitary or antiunitary, as defined in Sec. III B. In the following, we will focus on each class separately.

A. Complex symmetry classes
The complex classes A and AIII are characterized by the absence of time-reversal and particle-hole symmetries.

Class A
Let us start with Class A, with additional symmetry realized byÔ or O, whose properties are summarized as (A,Ô O ) or (A, O O ). For unitary symmetry realized bŷ U and U, one can fix U = 1 or U = 1.
a.Ô =Û 0 We havê whereÛ 0 = ρ 0 ⊗Û 0 behaves as an order-two crystalline symmetry if one regards t ∈ S 1 as an additional defect surrounding parameter. Recall that H(k, r, t) has chiral symmetry realized by operatorŜ = U S = ρ z ⊗I, we have This means under the hermitian map, unitary loops with symmetry (A,Û + 0 ) in dimension (d, d , D, D ) are mapped to static Hamiltonians with symmetry (AIII,Û + + ) in dimension (d, d , D + 1, D ). Here, we use the notation (AIII,Ô O ηΓ ) to denote class AIII with an additional symmetry realized byÔ, which squares to O and commutes (η S = 1) or anticommutes (η S = −1) with the chiral symmetry operatorŜ . One can also replaceÔ by O to define class AIII with an additional antisymmetry in the similar way.
c. O = U s The unitary antisymmetry U s leads to an order-two symmetry on H(k, r, t) with where U s = ρ x ⊗ U s . Moreover, we have U

Class AIII
In class AIII, we have a chiral symmetry realized byŜ. We assume an additional order-two symmetryÛ U η S or antisymmetry U U η S . Moreover, we can fix U = 1 and U = 1 for unitary symmetries and antisymmetries realized bŷ U and U respectively. For unitary (anti)symmetries, note that U η S in class AIII is essentially the same aŝ U η S , because they can be converted to each other by U ηs =ŜÛ η S . Similarly, for antiunitary (anti)symmetries, Hence in the following, we only discuss unitary and antiunitary symmetries.

B. Real symmetry classes
Now let us consider real symmetry classes, where at least one antiunitary symmetry is present.
In classes AI and AII, only time reversal symmetry is present.

Classes AI and AII
a.Ô =Û 0 The new hermitian matrix H(k, r, t) under the hermitian map defined by Eq.(33) aquires new time-reversal and particle-hole symmetries, realized bŷ T = ρ x ⊗T andĈ = iρ y ⊗T , respectively. Due to the order-two symmetry realized byÛ 0 , we havê Under the hermitian map, unitary loops with symmetry (AI,Û U 0,η T ) and (AII, andÛ 2 T /2 = U . Under the hermitian map, unitary loops with symmetry (AI,Û U T /2,η T ) and (AII,Û U T /2,η T ) in dimension (d, d , D, D ) are mapped to static Hamiltonians with symmetry (CI,Û U η T ,−η T ) and (DIII, c. O = U 0 Due to the order-two antisymmetry realized by U 0 , we have and U d. O = U T /2 Due to the order-two antisymmetry realized by U T /2 , we haves and U 2 T /2 = U . Under the hermitian map, unitary loops with symmetry (AI, U U T /2,η T ) and (AII,

Classes C and D
a.Ô =Û 0 The new hermitian matrix H(k, r, t) under the hermitian map defined by Eq.(33) aquires new time-reversal and particle-hole symmetries, realized bŷ T = ρ 0 ⊗Ĉ andĈ = ρ z ⊗T , respectively. Due to the order-two symmetry realized byÛ 0 , we havê Under the hermitian map, unitary loops with symmetry (C,Û U 0,η C ) and (D,Û U 0,η C ) in dimension (d, d , D, D ) are mapped to static Hamiltonians with symmetry (CII,Û U η C ,η C ) and (BDI, b.Ô =Û T /2 Due to the order-two symmetry realized byÛ T /2 , , we havê c. O = U s Due to the order-two antisymmetry realized by U s , we have with U s = ρ x ⊗ U s , which satisfieŝ and U 2 s = U . Hence, under the hermitian map, unitary loops with symmetry (C,

Classes CI, CII, DIII, and BDI
In these classes, the time-reversal, particle-hole and chiral symmetries are all present. Without loss of generality, we assumeŜ =TĈ andŜ 2 = 1. The hermitian matrix H(k, r, t) defined according to Eq.(32) has either time-reversal or particle-hole symmetry realized by depending on whether C T is 1 or −1. a.Ô =Û 0 Due to the order-two symmetry realized byÛ 0 , we havê Under the hermitian map, unitary loops in dimension (d, d , D, D ) with a given symmetry are mapped to static Hamiltonians in dimension (d, d , D + 1, D ) with another symmetry according to with X = CI, CII, DIII, BDI, and Y = C, AII, D, AI respectively. b.Ô =Û T /2 Due to the order-two symmetry realized byÛ T ?2 , we have Moreover, we have (ŜÛ T /2 )Ĉ = η C C TĈ (ŜÛ T /2 ), (ŜÛ T /2 ) 2 = η T η C U . Under the hermitian map, unitary loops in dimension (d, d , D, D ) with a given symmetry are mapped to static Hamiltonians in dimension (d, d , D + 1, D + 1) with another symmetries according to with X = CI, CII, DIII, BDI, and Y = C, AII, D, AI respectively.

VI. K GROUPS IN THE PRESENCE OF ORDER TWO SYMMETRY
Using the hermitian map introduced in the previous sections, the unitary loops with an order-two spacetime symmetry/antisymmetry are successfully mapped into static Hamiltonians with an order-two crystalline symmatry/antisymmetry, whose classfication has already been worked out in Ref. [11]. Thus, the latter result can be directly applied to the classification of unitary loops.
We first summarize the K-theory-based method used for classifying static Hamiltonians, and then finish the classification of unitary loops. Let us consider static Hamiltonians defined on a base space of momentum k ∈ T d and real space coordinate r ∈ S D . For the classification of strong topological phases, one can instead simply use S d+D as the base space [5,7]. To classify these Hamiltonians, we will use notion of stable homotopy equivalence as we defined for unitaries in Sec. II, by identifying Hamiltonians which are continuously deformable into each other up to adding extra trivial bands, while preserving an energy gap at the chemical potential. These equivalence classes can be formally added and they form an abelian group.
For a given AZ symmetry class s, the classification of static Hamiltonians is given by the set of stable equivalence classes of maps H(k, r), from the base space (k, r) ∈ S d+D to the classifying space, denoted as C s or R s , for complex and real symmetry classes, as listed in Table II. The abelian group structure inherited from the equivalence classes leads to the group structure in this set of maps, which is called the K group, or classification group.
For static topological insulators and superconductors of dimension (d, D) in an AZ class s without additional spatial symmetries, the K groups are denoted as K C (s; d, D) and K R (s; d, D), for complex and real symmetry classes, respectively. Note that for complex symmetry classes, we have s = 0, 1 mod 2, whereas for real symmetry classes, s = 0, 1, . . . , 7 mod 8.
These K groups have the following properties known as the Bott periodicity, where π 0 denotes the zeroth homotopy group which counts the number of path connected components in a given space. In the following, we will introduce the K groups for Hamiltonians supporting an additional order-two spatial symmetry/antisymmetry following Ref. [11]. Because of the hermitian map, these K groups can also be associtated with the unitary loops, whose classification is then obtained.
A. Complex symmetry classes with an additional order-two unitary symmetry/antisymmetry When a spatial or space-time symmetry/antisymmetry is considered, one needs to include the number of "flipped" coordinates for both k and r, into the dimensions.
For a static Hamiltonian of dimension (d, d , D, D ) in complex AZ classes with an additional order-two unitary symmetry/antisymmetry, the K group is denoted as K U C (s, t; d, d , D, D ), where the additional parameter t = 0, 1 mod 2, specifies the coexisting ordertwo unitary symmetry/antisymmetry. These K groups satisfy the following relation Thus, for classification purpose, one can use the pair (δ, δ ) instead of (d, d , D, D ) to denote the dimensions of the base space, on which the static Hamiltonian is defined.
To define K groups for unitary loops, we use the fact that the K group for certain unitary loops should be the same as the one for the corresponding static Hamiltonians under the hermitian map. The K groups for unitary loops are explicitly defined in Table III, where the two arguements s, t label the AZ class and the coexisting order-two space-time symmetry/antisymmetry.

B. Complex symmetry classes with an additional order-two antiunitary symmetry/antisymmetry
We now consider static Hamiltonians of dimension (d, d , D, D ), in complex AZ classes, with an order-two antiunitary symmetry/antisymmetry, realized byÂ or A. It turns out that complex AZ classes acquire real structures because of the antiunitary symmetry [11]. Indeed, effective time-reversal or particle-hole symmetry realized byÂ or A emerges, if we regard (k ⊥ , r ) as "momenta", and (k , r ⊥ ) as "spatial coordinates". Thus, a system in complex AZ classes with an antiunitary symmetry can be mapped into a real AZ class without additional spatial symmetries.
The K groups for these Hamiltonians are denoted as Similar to the previous case, the unitary loops with an antiunitary space-time symmetry/antisymmetry can also be associated with these K groups.
If we group these antiunitary symmetries and antisymmetries in terms of the index s = 0, . . . , 7 mod 8, according to Table IV, then K A C (s) can further be reduced to K R (s) ≡ K R (s; 0, 0).

C. Real symmetry classes with an additional order-two symmetry
In real symmetry classes, there are equivalence relations between order-two unitary and antiunitary symmetries/antisymmetries, as discussed previously. Thus, one can focus on unitary symmetries/antisymmetries only. The existence of an additional order-two unitary symmetry divide each class into four families (t = 0, . . . , 3 mod 4), as summarized in Table V, where we have used the equivalence of K groups for static Hamiltonians and unitary loops in terms of the hermitian map.

D. Nontrivial space-time vs static spatial symmetries/antisymmetries
The classification of unitary loops with an ordertwo space-time symmetry/antisymmetry is given by As can be seen in Tables III-V, for every order-two spacetime (anti)unitary symmetry/antisymmetry that is nontrivial, namely the half-period time translation is involved, there always exists a unique static spatial (anti)unitary symmetry/antisymmetry, such that both symmetries/antisymmetries give rise to the same K group. It is worth mentioning that when looking at the static symmetries/antisymmetries alone, the corresponding K groups for unitary loops are defined in the same way as the ones for Hamiltonians introduced in Ref. [11], as expected.
The explicit relations between the two types of symmetries/antisymmetries (nontrivial space-time vs static) with the same K group can be summarized as follows. Recall that we use η S (η S ), η T (η T ) and η C (η C ) to characterize the commutation relations between the ordertwo symmetry (antisymmetry) operator and the nonspatial symmetry operators. For two unitary order-two symmetries giving rise to the same K group, the η S s and η C s for the two symmetries take opposite signs, whereas η T s are the same. For two antiunitary order-two symmetries, we have η S s take opposite signs. For two unitary antisymmetries, the η T s have opposite signs. Finally, for class A, the antiunitary space-time antisymmetry operator A ± T /2 have the same K group as the one for A ∓ 0 . These relations are summarized in Table I, and can be better understood after we introduce the frequency-domain formulation of the Floquet problem in Sec. VIII B.

E. Periodic table
From the K groups introduced previously, we see that in addition to the mod 2 or mod 8 Bott periodicity in δ, there also exists a periodic structure in flipped di-mensions δ , because of the twofold or fourfold periodicity in t, which accounts for the additional ordertwo symmetry/antisymmetry. In particular, for complex symmetry classes with an order-two unitary symmetry/antisymmetry, the classification has a twofold periodicity in δ , whereas for complex symmetry classes with an order-two antiunitary symmetry/antisymmetry, and for real symmetry classes with an order-two unitary/antisymmetry, the periodicity in δ is fourfold. These periodic features are the same as the ones obtained in Ref. [11] for static Hamiltonians with an order-two crystalline symmetry/antisymmetry. We summarize the periodic tables for the four (δ = 0, . . . , 3 mod 4) different families below in the supplemental material [44].
Note that in obtaining the classification Tables, we made use of the K groups in their zero dimensional forms defined in Eqs. (91), (92) and (93), as well as the following relations where C s (s = 0, 1 mod 2) and R s (s = 0, . . . , 7 mod 8) represent the classifying space of complex and real AZ classes, see Table II.

VII. FLOQUET HIGHER-ORDER TOPOLOGICAL INSULATORS AND SUPERCONDUCTORS
In the previous sections, we obtained a complete classification of the anomalous Floquet TI/SCs using K theory, where the K groups for the unitary loops were de-fined as the same ones for the static Hamiltonians, according to the hermitian map.
Noticeably, the classification obtained in this way is a bulk classification, since the only the bulk unitary evolution operators were considered. These bulk K groups include the information of topological classification at any order. For static tenfold-way TI/SCs, in which the topological property is determined from the nonspatial symmetries, there is a bulk-boundary correspondence which essentially says that the nontrivial topological bulk indicates protected gapless boundary modes living in one dimension lower. This boundary modes is irrespective of boundary orientation and lattice termination. The same is true for tenfold-way Floquet TI/SCs with only nonspatial symmetries. In this situation, since only firstorder topological phases are allowed, this bulk K group is enough to understand the existence of gapless boundary modes.
However, when an additional crystalline symmetry/antisymmetry is taking into account, the existence of gapless boundary modes due to nontrivial topological bulk is not guaranteed unless the boundary is invariant under the nonlocal transformation of the symmetry/antisymmetry [9,12].
A more intriguing fact regarding crystalline symmetries/antisymmetries is that they can give rise to boundary modes with codimension higher than one, such as corners of 2D or 3D systems, as well as hinges of 3D systems [17][18][19][20][21][22][23][24][25]. Such systems are known as HOTI/SCs, in which the existence of the high codimension gapless boundary modes is guarenteed when the boundaries are compatible with the crystalline symmetry/antisymmetry, i.e. a group of boundaries with different orientations are mapped onto each other under the nonlocal transformation of a particular crystalline symmetry/antisymmetry. For example, to have a HOTI/SC protected by inversion, one needs to create boundaries in pairs related by inversion [24,25].
An additional requirement for these corner or hinge modes is that they should be intrinsic, namely their existence should not depend on lattice termination, otherwise such high codimension boundary modes can be thought as a (codimension one) boundary modes in the low dimensional system, which is then glued to the original boundary. In other words, an nth order TI/SCs has codimension-n boundary modes which cannot be destroyed through modifications of lattice terminations at the boundaries while preserving the bulk gap and the symmetries. According to this definition, the tenfoldway TI/SCs are indeed intrinsic first-order TI/SCs. In Ref. [26], a complete classification of these intrinsic corner or hinge modes was derived and a higher-order bulk-boundary correspondence between these high codimension boundary modes and the topological bulk was obtained. These were accomplished by considering a K subgroup series for a d-dimensional crystal, where K ≡ K (0) is the K group which classifies the bulk band structure of Hamiltonians with coexisting order-two symmetry/antisymmetry, defined in the previous section. K (n) ⊆ K is a subgroup excluding topological phases of order n or lower, for any crystalline-symmetry compatible boundaries. For example, K classifies the "purely crystalline phases" [23,26], which exclude the tenfoldway topological phases, which are first-order topological phases protected by nonspatial symmetries alone and have gapless modes at any codimension-one boundaries. This purely crystalline phases can have gapless modes only when the boundary preserves the crystalline symmetry, and the gapless modes will be gapped when the crystalline symmetry is broken. From a boundary perspective, one can define the boundary K group K , which classifies the tenfold-way topological phases with gapless codimension-one boundary modes irrespective of boundary orientations, as long as the crystal shape and lattice termination are compatible crystalline symmetries. According to the above definitions, K can be identified as the quotient group Generalizing this idea, a series of boundary K groups denoted as K (n) can be defined, which classify the intrinsic n-th order TI/SCs with intrinsic gapless codimensionn boundary modes, when the crystal has crystallinesymmetry-compatible shape and lattice termination. In Ref. [26], the authors proved the following relation, known as the higher-order bulk-boundary correspondence: an intrinsic higher-order topological phase is uniquely associated with a topologically nontrivial bulk. Moreover, the above equation provides a systematic way of obtaining the complete classification of intrinsic HOTI/SCs from K subgroup series, which were computed for crystals up to three dimensions with order-two crystalline symmetries/antisymmetries. We can generalize these results to anomalous Floquet HOTI/SCs, by considering unitary loops U (k, t) in d dimension without topological defect. To define a K subgroup series for unitary loops with an ordertwo space-time symmetry/antisymmetry, one can exploit the hermitian map and introduce the K groups according to their corresponding Hamiltonians with an ordertwo crystalline symmetry/antisymmetry. One obtains that the K subgroup series for each nontrivial spacetime symmetry/antisymmetry are the same as the ones for a corresponding static order-two crystalline symmetry/antisymmetry, according to the substitution rules summarized in Sec. VI D and Table I. On the other hand, the K groups are the same for unitary loops and Hamiltonians when static order-two symmetries/antisymmetries are considered. Using the results from Ref. [26], we present the K subgroup series for unitary loops with an order-two space-time symmetry/antisymmetry in Tables VI-XI, for systems up to three dimensions. In these TABLE VI. Subgroup series K (d) ⊆ · · · ⊆ K ⊆ K for zero-(d = 0), one-(d = 1), and two-dimensional (d = 2) anomalous Floquet HOTI/SCs with a unitary order-two space-time symmetry/antisymmetry in complex classes. The number of flipped dimensions for the symmetry/antisymmetry is denoted as d .
tables, we use the notation G 2 to denote G ⊕ G, with G = Z, 2Z, Z 2 . One also notices that the largest K group K (0) in the series is actually the ones shown in Tables of the supplemental material [44]. The classification of intrinsic codimension-n anomalous Floquet boundary modes is then given by the quotient K (n) = K (n−1) /K (n) .

VIII. FLOQUET HOTI/SCS IN FREQUENCY DOMAIN
In this section, we take an alternative route to connect a Floquet HOTI/SC with a nontrivial space-time symmetry/antisymmetry, to a static HOTI/SC with a corresponding crystalline symmetry/antisymmetry. This connection is based on the frequency-domain formulation of the Floquet problem [42], which provides a more intuitive perspective to the results obtained by K theory.

A. Frequency-domain formulation
In the frequency-domain formulation of the Floquet problem, the quasienergies are obtained by diagonalizing the enlarged Hamiltonian where the matrix blocks are given by Here, the appearance of the infinite dimensional matrix H can be subtle, and should be defined more carefully. Since later we would like to discuss the gap at gap = ω/2, we will assume that the infinite dimensional matrix H should be obtained as taking the limit n → ∞ of a finite dimensional matrix whose diagonal blocks are given from h 0 + nω to h 0 − (n − 1)ω, with n a positive integer. With this definition, ω/2 will be the particlehole/chiral symmetric energy whenever the system has particle-hole/chiral symmetries.
As a static Hamiltonian, H (k, r) has the same nonspatial symmetries as the original H(k, r, t) does. Indeed, one can define the effective time-reversal T , particle-hole C and chiral S symmetries for the enlarged TABLE VIII. Subgroup series K (d) ⊆ · · · ⊆ K ⊆ K for zero-(d = 0), one-(d = 1), and two-dimensional (d = 2) anomalous Floquet HOTI/SCs with a unitary order-two space-time symmetry/antisymmetry in real classes. The number of flipped dimensions for the symmetry/antisymmetry is denoted as d .
TABLE IX. Subgroup series K (d) ⊆ · · · ⊆ K ⊆ K for three-dimensional (d = 3) anomalous Floquet HOTI/SCs with a unitary order-two space-time symmetry/antisymmetry in complex classes. The number of flipped dimensions for the symmetry/antisymmetry is denoted as d .

Symmetry
Class TABLE XI. Subgroup series K (d) ⊆ · · · ⊆ K ⊆ K for three-dimensional (d = 3) anomalous Floquet HOTI/SCs with a unitary order-two space-time symmetry/antisymmetry in real classes. The number of flipped dimensions for the symmetry/antisymmetry is denoted as d .

Symmetry
Class Hamiltonian H (k, r) as On the other hand, when the original H(k, r, t) has a nontrivial space-time symmetry/antisymmetry, the enlarged Hamiltonian H (k, r) will acquire the spatial (crystalline) symmetry/antisymmetry inherited from the spatial part of the space-time symmetry/antisymmetry. Let us first considerÛ T /2 defined in Eq. (20) for s = T /2, which is an unitary operation together with a halfperiod time translation. Sincê U T /2 h n (k, r)Û −1 T /2 = (−1) n h n (−k , k ⊥ , −r , r ⊥ ), (103) the enlarged Hamiltonian thus respects a unitary spatial symmetry defined by where the unitary operator is inherited fromÛ T /2 . Next, we consider A T /2 . Since We now consider symmetry operatorsÂ T /2 and U T /2 , for symmetry classes other than A, C, and D. ForÂ T /2 , we havê (109) Thus, the enlarged Hamiltonian H (k, r) also has an antiunitary spatial symmetry inherited fromÂ T /2 , given by where the antiunitary operator Finally, for U T /2 , it satisfies the enlarged Hamiltonian will satisfy

B. Harmonically driven systems
To simplify the discussion, it is helpful to restrict ourselves to a specific class of periodically driven systems, the harmonically driven ones, whose Hamiltonians have the following form To discuss the band topology around at gap = ω/2, one can further truncate the enlarged Hamiltonian H to the 2×2 block, containing two Floquet zones with energy difference ω, namely where ρ 0 is the identity in the two Floquet-zone basis.
For later convenience, we use ρ x,y,z to denote the Pauli matrices this basis. Since the last term in Eq. (116) is a shift in energy by ω/2, we have a Floquet HOTI/SC at gap = ω/2 if and only if the first term in Eq. (116) is a static HOTI/SC. When restricted to the two Floquet-zone basis, the nonspatial symmetries can be conveniently writtens The spatial symmetries/antisymmetries for H , which are inherited from the space-time symmetries/antisymmetries, can also be written simply as From these relations, one arrives at the same results as the ones from K theory in the previous sections. When a spatial symmetry O, with O = U , A , coexists with the particle-hole or/and chiral symmetry operators C , S , O will commute or anticommute with C or/and S . Let us write with χ T = ±1, then we would have because ρ y is imaginary. Because of this, we can also obtain

IX. MODEL HAMILTONIANS FOR FLOQUET HOTI/SCS
In this section, we introduce model Hamiltonians, which are simple but still sufficiently general, for Floquet HOTI/SCs in all symmetry classes. Particularly, we consider harmonically driven Floquet HOTI/SCs Hamiltonians with a given nontrivial space-time symmetry/antisymmetry, realized byÛ T /2 , A T /2 ,Â T /2 , or U T /2 . One should notice that the latter two symmetries/antisymmetries are only available when the system is not in classes A,C or D, because in these classes, the symmetries with s = 0 and T /2 are the same up to redefining the origin of time coordinate.

A. Hamiltonians
The harmonically driven Floquet HOTI/SCs in ddimension to be constructed have Bloch Hamiltonians of the following general form and {Γ i , Γ j } = 2δ ij I, with I the identity matrix. Here ". . . " represents k-independent symmetry allowed perturbations that will in general gap out unprotected gapless modes.
One can further choose a representation of these Γ j s such that for j = 1, . . . , d. By the transformation properties of the symmetry/antisymmetry operators, we have, in this representation,T ,Û T /2 andÂ T /2 are block diagonal, namely they act independently on the two subspaces with τ z = ±1, whereas the operatorsĈ,Ŝ, U T /2 and A T /2 are block off-diagonal, which couple the two subspaces. In this representation, the enlarged Hamiltonian H (k) truncated to two Floquet zones, up to the constant shift ω/2, can be decoupled into two sectors with ρ z τ z = ±1. Hence, one can write it as a direct sum with Here the matricesΓ j s have a two-by-two block structure when restricting to the ρ z τ z = ±1 sectors of H (k). If we abuse the notation by still using τ x,y,z for this twoby-two degree of freedom, we can identifyΓ j = Γ j , for j = 0, . . . , d.
It is straightforward to verify that the static Hamiltonian h(k, m) respects the same nonspatial symmetries as the harmonically driven Hamiltonian H(k, t, m) does, with the same symmetry operators. Moreover, if H(k, t, m) respects a nontrivial space-time symmetry, realized byÛ T /2 orÂ T /2 , then h(k, m) will respect a spatial symmetry, realized by Γ 0ÛT /2 or Γ 0ÂT /2 , respectively. However, if H(k, t, m) respects a nontrivial space-time antisymmetry, realized by U T /2 or A T /2 , then h(k, m) will respect a spatial antisymmetry, realized by −iΓ 0ÛT /2 or −iΓ 0ÂT /2 , respectively. These relations can be worked out by using the block diagonal or off-diagonal properties of the operators of space-time symmetries/antisymmetries, as well as the relations in Eq. (118).
Thus, we have established a mapping between harmonically driven Hamiltonians H(k, t, m) and static Hamiltonians h(k, m), as well as their transformation properties under symmetry/antisymmetry operators. On the other hand, h(k, m) given in Eq. (128) are well studied models for static HOTI/SCs [23,26]. It is known that for −2 < m < 0, the Hamiltonian h(k, m) is in the topological phases (if the classification is nontrivial), whereas for m > 0 the Hamiltonian is in a trivial phase. A topological phases transition occurs at m = 0 with the band gap closing at k = 0.
Since the enlarged Hamiltonian H (k), up to a constant ω/2 shift, can be written as a direct sum of h(k, m ± ω/2), the static Hamiltonian H (k) will be in the topological phase (with chemical potential inside the gap at ω/2) if −2 < m − ω/2 < 0 and m + ω/2 > 0. This is also the condition when H(k, t, m) is in a Floquet topological phase at gap = ω/2.

B. Symmetry/antisymmetry-breaking mass terms
Let us consider −2 < m − ω/2 < 0 and m + ω/2 > 0. In this parameter regime, h(k, m + ω/2) is always in a trivial insulating phase, whereas h(k, m − ω/2) is in a nontrivial topological phase, if there exists no mass term M that respect the nonspatial symmetries, as well as the spatial symmetry/antisymmetry inherited from the space-time symmetry/antisymmetry of H(k, t, m). Here, the mass term in addition satisfies M 2 = 1, M = M † and {M, h(k, m)} = 0. Such a mass term will gap out any gapless states that may appear in a finite-size system whose bulk is given by h(k, m − ω/2). When M exist, one can define a term M cos(ωt) respecting all nonspatial symmetries and the space-time symmetry/antisymmetry of H(k, m, t), and it will gap out any gapless Floquet boundary modes at quasienergy gap = ω/2.
If no mass term M , which satisfies only the nonspatial symmetries irrespective of the spatial symmetry/antisymmetry, exists, then h(k, m−ω/2) (H(k, m, t)) is in the static (Floquet) tenfold-way topologogical phases, as it remains nontrivial even when the spatial (space-time) symmetry/antisymmetry is broken. Thus, the tenfold-way phases are always first-order topological phases. However, if such a M exists, h(k, m − ω/2) (H(k, m, t)) describes a static (Floquet) "purely crystalline" topological phase, which can be higher-order topological phases, and the topological protection relies on the spatial (space-time) symmetry/antisymmetry.
As pointed out in Ref. [26], several mutually anticommuting spatial-symmetry/antisymmetry-breaking mass terms M l can exist for h(k, m − ω/2), where M l also anticommutes with h. Furthermore, if h has the minimum possible dimension for a given "purely crystalline" topological phase, then the mass terms M l all anticommute (commute) with the spatial symmetry (antisymmetry) operator of h(k, m − ω/2). In this case, one can relate the number of these mass terms M l and the order of the topological phase [26]: When n mass terms M l exist, with l = 1, . . . , n, boundaries of codimension up to min(n, d ) are gapped, and one has a topological phase of order min(n+1, d +1) if min(n+1, d +1) ≤ d. However, if min(n + 1, d + 1) > d, the system does not support any protected boundary modes at any codimension. See Ref. [26], or Appendix C for the proof of this statement.
Hence, the order of the Floquet topological phase described by H(k, t, m) is reflected in the number of symmetry/antisymmetry-breaking mass terms M l , due to the mapping between H(k, t, m) and h(k, m − ω/2). In the following, we explicitly construct model Hamiltonians for Floquet HOTI/SCs with a given space-time symmetry/antisymmetry.

C. First-order phase in d = 0 family
When d = 0, the symmetries/antisymmetries are onsite. From Tables VI-XI, we see that the onsite symmetries/antisymmetries only give rise to first-order TI/SCs, since only the K (0) in the subgroup series can be nonzero. This can also be understood from the fact that min(n + 1, d + 1) = 1 in this case. We will in the following provide two examples in which we have anomalous Floquet boundary modes of codimension one which are protected by the unitary onsite space-time symmetry.

T /2,−
The simplest static topological insulator protected by unitary onsite symmetry is the quantum spin Hall insulator with additional two-fold spin rotation symmetry around the z axis [11]. This system is in class AII with time-reversal symmetryT 2 = −1. It is known that either a static or a Floquet system of class AII in 2D will have a Z 2 topological invariant [8,32]. However, with a static unitary d = 0 symmetry (such as a two-fold spin rotation symmetry), realized by the operatorÛ + 0,− that squares to one and anticommutes with the timereversal symmetry operator, a K (0) = Z topological invariant known as the spin Chern number can be defined. In fact, such a Z topological invariant (see Table VIII) can also appear due to the existence of space-time symmetry realized byÛ + T /2,− at quasienergy gap gap = ω/2. A lattice model that realizes a spin Chern insulator can be defined using the following Bloch Hamiltonian h(k, m) = (m + 2 − cos k x − cos k y )τ z + (sin k x τ x s z + sin k y τ y ), where s x,y,z and τ x,y,z are two sets of Pauli matrices for spins and orbitals. This Hamiltonian has time-reversal symmetry realized byT = −is yK as well as the unitary symmetry realized by operatorÛ + 0,− = s z . When we choose an open boundary condition along x while keep the y direction with a periodic boundary condition, there will be gapless helical edge states inside the bulk gap propagating along the x edge at k y = 0 for −2 < m < 0.
The corresponding harmonically driven Hamiltonian can be written as where the time-reversal and the half-period time translation onsite symmetry operators are defined asT = −is yK andÛ + T /2,− = s z τ z respectively. When −2 < m−ω/2 < 0 and m+ω/2 > 0 are satisfied, this model supports gapless helical edge states at k y = 0 inside the bulk quasienergy gap gap = ω/2 when the x direction has an open boundary condition. Furthermore, such gapless Floquet edge modes persist as one introduces more perturbations that preserve the time-reversal and theÛ + T /2,− symmetry.

T /2,−
For 2D, either static or Floquet, superconductors in class D with no additional symmetries, the topological invariant is Z given by the Chern number of the Bogoliubov-de Gennes (BdG) bands. When there exists a static unitary d = 0 symmetry, realized byÛ + 0,+ which commutes with the particle-hole symmetry operator, the topological invariant instead becomes to K (0) = Z ⊕ Z, see Table VIII. The same topological invariant can also be obtained from a space-time unitary symmetry realized bŷ U + T /2,− , which anticommutes with the particle-hole symmetry operator. In the following, we construct a model Hamiltonian for such a Floquet system. Let us start from the static 2D Hamiltonian in class D given by with particle-hole symmetry and the unitary onsite symmetries realized byĈ = τ xK andÛ + 0,+ = s z , where τ x,y,z are the Pauli matrices for the Nambu space. Here, the unitary symmetry can be thought as the mirror reflection with respect to the xy plane, and bs z is the Zeeman term which breaks the time-reversal symmetry.
The Z ⊕ Z structure is coming from the fact thatÛ + 0,+ , C and h(k, m) can be simultaneously block diagonalized, according to the ±1 eigenvalues ofÛ + 0,+ . Each block is a class D system with no additional symmetries, and thus has a Z topological invariant. Since the two blocks are independent, we have the topological invariant of the system should be a direct sum of the topological invariant for each block, leading to Z ⊕ Z.
The harmonically driven Hamiltonian with a unitary space-time onsite symmetry realized byÛ + T /2,− = s z τ z can be written as H(k, t, m) = (m + 2 − cos k x − cos k y + bs z )τ z + (sin k x s z τ x − sin k y τ y ) cos(ωt).
The particle-hole symmetry operator for this Hamiltonian isĈ = τ xK .
D. Second-order phase in d = 1 family When a d = 1 space-time symmetry/antisymmetry is present, the system can be at most a second-order topological phase, since the order is given by min(n + 1, d + 1) ≤ 2. Note that the unitary symmetry in this case is the so-called time-glide symmetry, which has been already discussed thoroughly in Refs. [38,41], we will in the following construct models for second-order topological phases with antiunitary symmetries, as well as models with unitary antisymmetries.

T /2,−
For 2D systems in class AIII without any additional symmetries, the topological classification is trivial, since the chiral symmetry will set the Chern number of the occupied bands to zero. However, in Table VII, we see that when the 2D system has an antiunitary symmetry realized by eitherÂ + 0,+ orÂ + T /2,− , the K subgroup series is 0 ⊆ Z 2 ⊆ Z 2 .
Let us first understand the K (0) = Z 2 classification in the case ofÂ + 0,+ in a static system with Hamiltonian h(k x , k y ). Let us assume thatÂ + 0,+ corresponds to the antiunitary reflection about the x axis, then we havê On the other hand, the chiral symmetry imposes the following condition Thus, if we regard k x ∈ S 1 as a cyclic parameter, then at every k x , h(k x , k y ) as a function of the Bloch momentum k y is actually a 1D system in class BDI. Thus, the topological classification in this case is the same as the one for a topological pumping for a 1D system in class BDI described by a Hamiltonian h (k, t), with momentum k and periodic time t. This gives rise to a Z 2 topological invariant, corresponding to either the fermion parity has changed or not after an adiabatic cycle [7], when the 1D system has an open boundary condition. Since the bulk is gapped at any t, such a fermion pairty switch is allowed only when the boundary becomes gapless at some intermediate time t. Since our original Hamiltonian h(k x , k y ) is related to h (k, t) by replacing k ↔ k y and t ↔ k x , a nontrivial phase for h(k x , k y ) implies the existance of a counter propagating edge modes on the x edge when we choose an open boundary condition along y.
Let us understand the pure crystalline classification K = Z 2 . One can consider the edge Hamiltonian for a pair of counter propagating gapless mode on the edge parallel to x as H edge = k x σ z , withŜ = σ x andÂ + 0,+ = K. This pair of gapless mode cannot be gapped by any mass term. However, if there exist two pairs of gapless modes, whose Hamiltonian can be written as H edge = k x τ 0 σ z , one can then add a mass term mτ y σ y to H edge to gap it out. On the other hand, if the edge does not preserve the antiunitary symmetry given byÂ + 0,+ , then a mass term mσ y can be added to gap out a single pair of gapless mode, which implies that there is no intrinsic codimension-one boundary modes. Thus, K = Z 2 , and K = 0.
Instead of intrinsic codimension-one boundary modes, the system supports intrinsic codimension-two boundary modes, implying it as a second-order TI. If one creates a corner that is invariant under the reflection x → −x, this corner will support a codimension-two zero mode, with a K = K /K = Z 2 classification.
An explicit Hamiltonian that realizes this phases can have the following form where τ x,y,z and σ x,y,z are two sets of Pauli matrices, and the parameter b, which gaps out the y edge, is numerically small. One can show that this Hamiltonian has desired chiral and antiunitary reflection symmetries given byŜ = τ x σ z andÂ + 0,+ =K, respectively. When −2 < m < −0, there are counter propagating edge modes on each x edge at momentum k x = 0. On the other hand, a corner, which is invariant under reflection x → −x, will bound a zero mode. These two different boundary conditions are illustrated in Fig. 2.

2D system in class AI with
For 2D systems in class AI, with only spinless timereversal symmetryT 2 = 1, the topological classification is trivial. However, with a unitary, either static or spacetime, d = 1 antisymmetry realized by U + 0,+ or U + T /2,− , the K group subseries is 0 ⊆ Z ⊆ Z, as given in Table VIII.
Let us start by considering a Hamiltonian h(k x , k y ) with a static d = 1 antisymmetry, given by in addition to the spinless time-reversal symmetry. At the reflection symmetric momenta k x = 0, π, the Hamiltonian as a function of k y reduces to a 1D Hamiltonian in class BDI, which has a Z winding number topological invariant. One can also undertand the topological classification from the edge perspective. At reflection invariant edge, the x edge in this case, multiple pairs of counter propagating edge modes can exist. One can write the edge Hamiltonian as H edge = k x Γ x + mΓ m , with a possible mass term of magnitude m. Here the matrices Γ x and Γ m anticommute with each other and squares to identity. Since the edge is reflection invariant, we have [Γ x , U When the edge is deformed away symmetrically around a corner at x = 0, mass terms m 1 (x)σ x + m 2 (x)σ z , with m i (x) = −m i (−x), i = 1, 2, can be generated. This gives rise to (n + −n − ) zero energy corner modes, corresponding to K = K /K = Z.
An explicit Hamiltonian for h(k x , k y ) can have the following form h(k, m) = (m + 2 − cos k x − cos k y )τ z + sin k x τ x σ y + sin k y τ y + bτ z σ z withT =K and U + 0,+ = τ x , and numerically small b. When −2 < m < 0, there exist counter propagating gapless modes on the x edges when the system has an open boundary condition in the y direction.
The corresponding harmonically driven Hamiltonian with a unitary space-time antisymmetry has the following form where the time-reversal symmetry and the unitary spacetime antisymmetry are realized byT =K and U + T /2,− = τ y , respectively. Gapless Floquet edge modes, or Floquet corner modes at gap = ω/2, can be created, with appropriately chosen boundary conditions, when both −2 < (m − ω/2) < 0 and (m + ω/2) > 0 are satisfied.
E. Third-order phase in d = 2 family When a Floquet system respects a d = 2 space-time symmetry/antisymmetry, it can be at most a third-order topological phase, because min(n + 1, d + 1) ≤ 3. In the following, we construct a model Hamiltonian for a third-order TI representing such systems.
3D system in class AIII withÂ + T /2,− It is known that for 3D system in class AIII without any additional spatial symmetries, the topological classification is Z [8], which counts the number of surface Dirac cones at the boundary of the 3D insulating bulk. When there exists an antiunitary two-fold rotation symmetry, eitherÂ + 0,+ orÂ + T /2,− , the topological invariants are given by the K subgroup series 0 ⊆ Z 2 ⊆ Z 2 ⊆ Z 2 in Table X.
Indeed, because of the additional symmetry realized bŷ A + 0,+ orÂ + T /2,− , the symmetry invariant boundary surface is able to support gapless Dirac cone pairs. As will be shown in the following, it turns out that the number of such pairs is maximum to be one, which gives rise to the K (0) = Z 2 topological invariant.
Let us first look at the static antiunitary two-fold rotation symmetry, realized byÂ + 0,+ , which transforms a (b) (a) static Bloch Hamiltonian aŝ With an appropriate basis, one can writeŜ = τ z , and A + 0,+ =K. At the symmetry invariant boundary surface perpendicular to z, while keeping the periodic boundary condition in both x and y directions, a single Dirac cone pair with a dispersion h surf = τ x (σ x k x + σ z k y ) can exist. This Dirac cone pair cannot be gapped by an additional mass term preserving theÂ + 0,+ symmetry, which requires the mass term to be real. However, when there are two pairs of Dirac cones, described by the surface Hamiltonian h surf = µ 0 τ x (σ x k x + σ z k y ), with µ 0 a twoby-two identity matrix for another spinor degree of freedom, for which we also introduce a new set of Pauli matrices µ x,y,z . Noticeably, a mass term which couple the two pairs of Dirac cones and gap them out can be chosen as µ y σ x τ y , which preserves the antiunitary two-fold symmetry. Hence, we have a K (0) = Z 2 .
When the surface is tilted away from the rotation invariant direction, two mutually anticommuting rotationsymmetry-breaking mass terms exist, and can be written as m 1 τ y σ 0 + m 2 τ x σ y , in which m 1,2 must change signs under two-fold rotation. Hence, boundaries of codimension up to min(n, d ) = 2 are gapped. This leads to K = K = 0, which implies K = K = K (0) = Z 2 . Moreover, at the symmetry invariant corner, this mass must vanish, and thus the system can host zero-energy corner mode.
One can write down the following concrete model Hamiltonian with eight bands, where the parameters b 1 and b 2 are numerically small. Here, the chiral and the antiunitary two-fold rotation symmetries are realized byŜ = µ y τ x σ y , andÂ + 0,+ =K, respectively. This Hamiltonian supports a single pair of Dirac cones on the boundary surfaces perpendicular to the z axis, at k x = k y = 0 for −2 < m < 0, as illustrated in Fig. 3(a). When the surface perpendicular to the rotation axis gets deformed from Fig. 3(a) to (b), the rotation invariant corner then bounds a codimension-two boundary mode.
The corresponding harmonically driven model has the following Hamiltonian Here, the chiral symmetry is realized byŜ = µ y τ x σ y , while the the antiunitary two-fold time-screw symmetry is realized byÂ + T /2,− = τ zK . This Hamiltonian is able to support a pair of Dirac cones on the boundary surface perpendicular to z direction inside the bulk quasienergy gap around gap = ω/2 ( Fig. 3(a)), as well as codimension-two mode with quasienergy ω/2 localized at the rotation invariant corner of the system (Fig. 3(b)).

F. Higher-order topological phases in d = 3 family
Unlike the symmetries discussed previously, the d = 3 symmetry (antisymmetry) operatorP (P) does not leave any point invariant in our three dimensional world. In particular, since the surface of a 3D system naturally breaks the inversion symmetry, the topological classification of the gapless surface modes (if exist), should be the same as the 3D tenfold classification disregarding the crystalline symmetry, in the same symmetry class. Hence, we have the boundary K group where K TF is the corresponding K group for the tenfoldway topological phase, with only nonspatial symmetries considered. However, inversion related pairs of boundaries with codimension larger than one are able to host gapless modes, which can not be gapped out without breaking the symmetry (antisymmetry) realized byP (P). This can be understood by simply considering the surface Hamiltonian h(p ,n) withn ∈ S 2 . Here p is the momenta perpendicular ton. Let us assume there are n spatially dependent mass terms m l (n)M l , with l = 1, . . . , n, that can gap out the surface Hamiltonian h(p ,n). The inversion symmetry/antisymmetry restricts m l (n) = −m l (−n) (see Appendix C for details), which implies that there must exist a 1D inversion symmetric loop S 1 ⊆ S 2 , such that m l (n) = 0, forn ∈ S 1 . This 1D loop for different ls can be different, but they all preserve the inversion symmetry, and cannot be removed. Hence, for n = 1, we have a 1D massless great circle, whereas for n = 2 we have a pair of antipodal massless points. The 1D or 0D massless region are irremovable topological defects which are able to host gapless modes.
Since inversion operation maps one point to another point, the stability of the gapless modes on the massless 1D or 0D region must be protected by the nonspatial symmetries alone [24]. Hence, the codimension-k gapless modes are stable only when the (4 − k)-dimensional system has a nontrivial tenfold classification, namely K TF = 0.
Moreover, the number of these gapless modes is at most one [24]. Indeed, a system consisting of a pair of inversion symmetric systems with protected gapless modes can be deformed into a system with completely gapped boundaries without breaking the inversion symmetry. This statement can be understood by considering a pair of inversion symmetric surface Hamiltonians where the + (−) sign is taken when we have a inversion symmetry (antisymmetry). In this situation, the h (p ,n) has a inversion symmetry or antisymmetry realized bŷ Now one can introduce mass terms In this case m l (n) can be nonzero for alln ∈ S 2 , and therefore h (p ,n) can always be gapped. Hence, we obtain the boundary K groups K (k) which classifies boundary modes of codimension k = 2 and 3 as (148) Having understood the general structure of K subgroup series, let us in the following construct model Hamiltonians for Floquet HOTI/SCs in class DIII with a unitary space-time symmetry realized byÛ + T /2,++ (d = 3), as an example.
From Table XI, we see that the K subgroup series is 4Z ⊆ 2Z ⊆ Z ⊆ Z 2 , which implies we can have first-order phase classified by K = Z 2 /Z = Z, second-order phase classified by K = Z/2Z = Z 2 , and third-order phase classified by K (3) = 2Z/4Z = Z 2 .

First-order topological phase
Under the operatorÛ + T /2,++ , no points on the surface of a 3D bulk are left invariant. Hence, the existence of codimension-one boundary modes is due to the protection from the nonspatial symmetries alone. A tightbinding model realizing such a phase can be constructed from its static counter part, namely, a model in class DIII with a static inversion symmetry realized byÛ + 0,+− . The static model can have the following Hamiltonian where the time-reversal, particle-hole, chiral and the inversion symmetres are realized byT = −iσ yK ,Ĉ = σ y τ yK ,Ŝ = τ y , andÛ + 0,+− = τ z , respectively. When −2 < m < 0, this model hosts hosts a gapless Dirac cone with chirality ±1 on any surfaces of the 3D bulk.
Hence, the Hamiltonian for the corresponding Floquet first-order topological phase with a space-time symmetry can be written as where the space-time symmetry is realized byÛ + T /2,++ = I, and the nonspatial symmetry operators are the same as in the static model. When −2 < (m − ω/2) < 0 and (m + ω/2) > 0 are satisfied, H ± (k, t, m) will host a gapless Dirac cone at quasienergy ω/2 with chirality ±1.

Second-order topological phase
Similar to the construction of the first-order phase, let us start from the corresponding static model. A static second-order phase can be obtained by couple h + (k, m 1 ) and h − (k, m 2 ). When both m 1 and m 2 are within the interval (−2, 0), the topological invariant for the codimension-one boundary modes vanishes and their exists a mass term on the surface which gaps out all boundary modes of codimenion one.
Explicitly, one can define the following Hamiltonian and introduce a set of Pauli matrices µ x,y,z for this newly introduced spinor degrees of freedom. There is only one mass term M l = τ x µ x , which satisfies that preserves all symmetries, to h(k, m 1 , m 2 ). This perturbation gaps out all codimension-one surfaces and left a codimension-two inversion invariant loop gapless, giving rise to a second-order topological phase.
The Floquet second-order topological phase can therefore be constructed by addition the perturbation V to the following Hamiltonian (153) In Fig. 4(a), we show the spectral weight of the codimension-one Floquet boundary mode at ω/2, when the system is cutted to an approximate sphere geometry. This boundary mode is localized on an inversion invariant loop.

Third-order topological phase
To construct a model for the third-order topological phase, one needs to find two anticommuting masses M 1 , M 2 , which satisfy the same conditions discussed previously. This can be realized by introducing another spinor degrees of freedom, as one couples two copies of h(k, m 1 , m 2 ). Explicitly, one can take the following Hamiltoniañ as well as the corresponding Pauli matricesμ x,y,z for the spinor degrees of freedom. Thus, two anticommuting mass terms M 1 = τ x µ x and M 2 = τ x µ yμy can be found. Therefore, one can introduce the symmetry preserving perturbatioñ which in general gaps out all boundary modes except at two antipodal points, at which codimension-three modes can exist.
The Floquet version of such a third-order topological phase is constructed by adding the perturbationṼ to the following periodically driven Hamiltoniañ In Fig. 4(b), the spectral weight of the zerodimensional (codimension-three) Floquet modes at quasienergy ω/2 is shown in a system with an approximate sphere geometry. The other zero-dimensional mode is located at the antipodal point.

X. CONCLUSIONS
In this work, we have completed the classification of the Floquet HOTI/SCs with an order-two space-time symmetry/antisymmetry. By introducing a hermitian map, we are able to map the unitary loops into hermitian matrices, and thus define bulk K groups as well as K subgroup series for unitary loops. In particular, we show that for every order-two nontrivial space-time (anti)unitary symmetry/antisymmetry involving a half-period time translation, there always exists a unique order-two static spatial (anti)unitary symmetry/antisymmetry, such that the two symmetries/antisymmetries share the same K group, as well as the subgroup series, and thus have the same topological classification.
Further, by exploiting the frequency-domain formulation, we introduce a general recipe of constructing tightbinding model Hamiltonians for Floquet HOTI/SCs, which provides a more intuitive way of understanding the topological classification table.
It is also worth mentioning that although in this work we only classify the Floquet HOTI/SCs with an ordertwo space-time symmetry/antisymmetry, the hermitian map introduced here can also be used to map the classification of unitary loops involving more complicated spacetime symmetry, to the classification of Hamiltonians with other point group symmetries. Similarly, the frequencydomain formulation and the recipe of constructing Floquet HOTI/SCs should also work with some modifications. In this sense, our approach can be more general than what we have shown in this work.
Finally, we comment on one possible experimental realization of Floquet HOTI/SCs. As lattice vibrations naturally break some spatial symmetries instantaneously, while preserving the certain space-time symmetries, one way to engineer a Floquet HOTI/SC may involve exciting a particular phonon mode with a desired space-time symmetry, which is investigated in Ref. [45]. deformation f (s) = [U τ * C − (s)] * C + (s). (B1) We have f (0) = U , and f (1) = L * C + (1), which is a composition of a unitary loop followed by a constant Hamiltonian evolution.
Appendix C: Order of HOTI/SCs and symmetry-breaking mass terms Consider a static HOTI/SCs in d-dimension described by the Hamiltonian h(k, m) given in Eq. (128). Let us denote the spatial symmetry (antisymmetry) operator asP (P), and assume there are n mutually anticommuting M l , with l = 1, . . . , n, {M l , h(k, m)} = 0, and {M l ,P} = 0 ([M l , P] = 0). We further consider a slowly position-dependent parameter m = m(r), which produces a position-dependent Hamiltonian h(k, m(r)). If there is a region with m(r) < 0 and m(r) > 0 outside this region, such that the boundary defined by m(r) = 0 is topologically the same as S d−1 , then there may exist gapless modes localized at the boundary. One can try to gap out the possible gapless modes, while preserving the spatial symmetry of h(k, m(r)), by introducing a perturbation Let us focus on a point on the boundary defined by its normal unit vectorn (pointing toward the m > 0 region), one can then define p ⊥ = k ·n, p ·n = 0, and x ⊥ = r ·n. Thus, the low energy Hamiltonian near this point at the boundary can be written as The wave function for a bound state of h boundary can be written as The gapless mode corresponds to the solution (Γ 0 + in · Γ)ψ(p ) = 0. According to this, one can define the projector into this gapless sector as Hence, we have the Hamiltonian with the additional perturbation V projected into the boundary low-energy sector wheren j is the jth component ofn. Note that the second term gaps out the boundary, and we can have gapless boundary modes only at locations satisfying i . Since ker B is a linear subspace of R d of dimension d − min(n, d ), we find the gapless set is given by This means one can have gapless boundary modes of codimension min(n + 1, d + 1) if min(n + 1, d + 1) ≤ d, otherwise the boundary is completely gapped.
In this supplement, the explicit form of the K groups for unitary loops with an order-two space-time symmetry/antisymmetry, in different dimensions are listed.
a. δ = 0 family In this family, the additional symmetry includes nonspatial symmetry, such as spin rotations with and without a simultaneous half-period time translation. We summarize the classification table for δ = 0 mod 2 in complex symmetry classes with an order-two unitary symmetry in Table XII. In Table XIII and XIV, we give the classification for δ = 0 mod 4 in complex symmetry classes with an order-two antiunitary symmetry, and in real symmetry classes with an order-two unitary symmetry, respectively. b. δ = 1 family This family includes Floquet topological phases protected by reflection symmetry and time-glide symmetry, where only one direction of the momenta is flipped. We summarize the classification table for δ = 1 mod 2 in complex symmetry classes with an order-two unitary symmetry in Table XV. In Table XVI and XVII, we give the classification for δ = 0 mod 4 in complex symmetry classes with an order-two antiunitary symmetry, and in real symmetry classes with an order-two unitary symmetry, respectively.
The additional symmetry includes twofold spatial rotation and twofold time-screw rotation, in which the momenta along two directions are flipped. For δ = 2, whose classification is the same as δ = 0 mod 2 in complex symmetry classes with an order-two unitary symmetry, as shown in Table XII. We summarize the classification for δ = 2 mod 4 in complex symmetry classes with an order-two antiunitary symmetry, and in real symmetry classes with an order-two unitary symmetry in Table XVIII and XIX, respectively.
The order-two symmetry in this family includes inversion with and without a simultaneous half-period time translation. For δ = 3, whose classification is the same as δ = 1 mod 2 in complex symmetry classes with an order-two unitary symmetry, as shown in Table XV. We summarize the classification for δ = 3 mod 4 in complex symmetry classes with an order-two antiunitary symmetry, and in real symmetry classes with an order-two unitary symmetry in Table XX and XXI, respectively.

Symmetry
Class Classifying space