Recent theoretical work has firmly established that periodic driving can make an ordinary insulator or superconductor topologically nontrivial [12], [13], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38]. Roughly speaking, a proper driving field mixes the bands to fundamentally change their topological characteristics such as the Berry curvature and Chern number [18]. For fast driving, the driven system stroboscopically mimics a static system described by the effective Hamiltonian, which can be computed in a controlled manner by successive approximations. Experimentally, Floquet edge states have been demonstrated in photonic crystals [39] and photonic quantum walks [40]. In particular, time periodically modulated, or shaken, optical lattices [41], [42] have been implemented experimentally to engineer the band structure and generate artificially gauge fields for ultracold atoms [43], [44], [45], [46], [47], [48], [49], [50], [51], [52]. Given the scarcity of available topological superconductors with desirable properties, e.g. Majorana zero modes useful for topological quantum computing, it is desirable to explore to what extent periodic driving can be used to synthesise new topological superconductors using existing materials. This approach may be referred to as “Floquet engineering”.

The topological properties of periodically driven systems are complex, and some of their features are rather unique. Due to the discrete time translational invariance of the Hamiltonian, the quasienergy spectrum of the driven system lives in the Quasienergy Brillouin Zone (QBZ), which is topologically equivalent to a closed circle. For a typical spectrum with *q* bands, there are *q* gaps, one more than the analogous static system. It can therefore support more edge modes, e.g. the so-called *π*-modes inside the gap around quasienergy ±*π*/*T*. Floquet Majorana modes at *π*/*T* in one-dimensional systems were noted by many groups, for example, in [19]. For 2D lattice systems, the *π*-modes have been studied by Rudner et al. [29] and in the context of periodically driven Hofstadter model by us [31], [32] and also in [33], [53], [54]. Kitagawa et al. have shown that the Floquet operator *U*(*T*) can be used to construct the topological invariants for driven lattice systems in one and three dimensions [17]. The topological invariants for 2D driven lattice systems have been constructed by Rudner et al. [29] and Carpentier et al. [55].

In the rest of the section, we review some of the concepts and definitions relevant to our subsequent discussion on driven superconductors. Consider a many-body system described by time-periodic Hamiltonian *H*(*t*)=*H*(*t*+*T*), with *T* the period. The time evolution operator

$$U\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathcal{T}{e}^{-i{\displaystyle {\int}_{0}^{t}}H\mathrm{(}{t}^{\prime}\mathrm{)}\text{d}{t}^{\prime}}\mathrm{,}$$(1)

where 𝒯 denotes the time ordering, *ℏ*=1, and we choose *U*(*t*=0)=1. Following the convention in the literature, we will call *U*(*t*=*T*) the Floquet operator. The eigenvalue problem of *U*(*T*) defines quasienergy *ω*_{ℓ},

$$U\mathrm{(}T\mathrm{)}|{\psi}_{\ell}\u3009\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}^{-i{\omega}_{\ell}T}\mathrm{|}{\psi}_{\ell}\u3009\mathrm{,}$$(2)

where *ℓ* is the band index, *ω*_{ℓ} is equivalent to *ω*_{ℓ}+*m*Ω, with *m* an integer and the fundamental frequency Ω=2*π*/*T*. The first QBZ is usually defined as *ω*∈[–Ω/2, Ω/2]. The effective Hamiltonian ℋ, which is time-independent, is defined from *U*(*T*) through the relation

$$U\mathrm{(}T\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}^{-i\mathscr{H}T}\mathrm{.}$$(3)

In order to understand the topological properties of the driven system, ℋ and *U*(*T*) are, generally speaking, insufficient. One often needs the entire function of *U*(*t*) within a driving cycle, e.g. for *t*∈[0, *T*]. Following Nathan and Rudner [12], we introduce the notion of phase bands. Let **k**=(*k*_{x}, *k*_{y}) be the crystal momentum of a 2D lattice system. For any given time *t*, the eigenvalue problem of *U*(**k**, *t*) yields an instantaneous band structure *ϕ*_{ℓ}(**k**, *t*),

$$U\mathrm{(}k\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}|{\varphi}_{\ell}\mathrm{(}k\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\u3009\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}^{-i{\varphi}_{\ell}\mathrm{(}k\mathrm{,}t\mathrm{)}}\mathrm{|}{\varphi}_{\ell}\mathrm{(}k\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\u3009\mathrm{,}$$(4)

where the phase *ϕ*_{ℓ}∈[–*π*, *π*]. In particular, the phase bands at *t*=*T* are nothing but the quasienergy bands, *ϕ*_{ℓ}(**k**, *t*=*T*)=*ω*_{ℓ}(**k**)*T*. One can visualise the phase bands as a set of “membranes” hovering above the 2D Brillouin Zone (BZ). As time goes on, these membranes change their shapes smoothly, and quite often during the process, they intersect with or touch each other. These locations are where, to use a loose analogy, the “knots are tied”. It is illuminating to compare the actual evolution of the phase bands to that of a hypothetical static system described by the effective Hamiltonian ℋ above,

$$U\mathrm{(}k\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}^{-iH\mathrm{(}k\mathrm{)}t}\mathrm{,}$$(5)

whose phase bands are simply given by *ω*_{ℓ}(**k**)*t*, i.e. linearly dispersing with time. As shown by Nathan and Rudner [12], the scrambling of the phase bands of *U* during the time evolution may render it topologically distinct from the linear evolution according to 𝒰. Namely, it may be impossible to smoothly deform one to the other,

$$U\mathrm{(}k\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}\overline{)\iff}\text{\hspace{0.17em}}U\mathrm{(}k\mathrm{,}\text{\hspace{0.17em}}t\mathrm{}\mathrm{)}\mathrm{.}$$(6)

It then follows that the degeneracies in the phase bands, i.e. when and where the phase bands touch each other, hold the key to understand the topology of *U*(**k**, *t*) and the corresponding edge states. From this argument, it is also clear that the effective Hamiltonian ℋ by itself *cannot* provide a complete description of the topological properties of the driven system in all cases [12]. Otherwise, it would have implied that the time evolution is always topologically equivalent to 𝒰 and in turn equivalent to a static system. The presence of degeneracies in the phase band obstructs the deformation from *U* to 𝒰 and gives rise to edge modes unique to periodically driven (Floquet) systems.

The knottiness of *U*(**k**, *t*) can be captured by constructing its topological invariants. One example is the winding number introduced by Rudner et al. [29] for 2D lattice systems. For the *ℓ*-th gap of the quasienergy spectrum,

$${w}_{\ell}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \int \frac{\text{d}{k}_{x}\text{d}{k}_{y}\text{d}t}{24{\pi}^{2}}{\u03f5}^{\mu \nu \rho}\text{Tr[}\mathrm{(}{u}^{-1}{\partial}_{\mu}u\mathrm{)}\mathrm{(}{u}^{-1}{\partial}_{\nu}u\mathrm{)}\mathrm{(}{u}^{-1}{\partial}_{\rho}u\mathrm{)}\text{]}}\mathrm{.}$$(7)

Here the Greek indices loop through *k*_{x}, *k*_{y}, *t* and are summed over. Note that *u*(**k**, *t*) is an extrapolation of *U*(**k**, *t*), for example [29],

$$u\mathrm{(}k\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}U\mathrm{(}k\mathrm{,}\text{\hspace{0.17em}}2t\mathrm{)}\Theta \mathrm{(}\frac{T}{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{e}^{-i2{H}_{\ell}\mathrm{(}k\mathrm{)}\mathrm{(}T\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\mathrm{)}}\Theta \mathrm{(}t\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{T}{2}\mathrm{)}\mathrm{,}$$(8)

where Θ is the step function. The dependence of *u*(**k**, *t*) on *ℓ* is through the definition of ℋ_{ℓ}=(*i*/*T*)log*U*(*T*) where the branch cut of the logarithm is chosen to lie within the *ℓ*-th gap [29]. The second term was added to unwind the evolution due to ℋ and ensure *u*(*T*)=1. Via the bulk-boundary correspondence, the authors of [29] showed that *w*_{ℓ} is equal to the net chirality *ν* of the edge modes within the *ℓ*-th gap, i.e. the number of chiral edge modes with positive group velocity minus the number of chiral edge states with negative group velocity.

To identify the features of the phase bands that give rise to a nonzero winding number, we review the argument given by Nathan and Rudner [12]. Higher dimensional degeneracy manifolds of the phase band (such as surfaces and lines) may be shrunk to isolated points or entirely eliminated by introducing additional perturbations to lift the degeneracy. For 2D lattice systems, the only topologically stable degeneracies appear to be isolated points in the space of (*k*_{x}, *k*_{y}, *t*). These band touching points are known as Weyl points in the study of semimetals [56] and nodal superconductors [57], [58] or diabolical points in a more general context. As well known, a Weyl point can be viewed as a magnetic monopole [6] with topological charge *q*=1 or –1. Now imagine that the phase bands are smoothly deformed to become flat at *ϕ*=0 except for the neighbourhoods of these point degeneracies and a small time interval of linear evolution, which does not contribute to *w*_{ℓ} [12]. Consider the quasienergy gap at the boundary of QBZ, and suppose the phase bands above and below (noting that QBZ is periodic) touch each other during the time evolution at a few isolated degeneracy (Weyl) points. Let *q*_{i} be the topological charge of the *i*-th degeneracy point. Their total charge $Q\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle {\sum}_{i}}{q}_{i}$ is a topological invariant. Hereafter, we will follow [12] and refer to these degeneracy points as “zone edge singularities.” The winding number of the *ℓ*-th quasienergy gap is then given by [12]

$${w}_{\ell}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{n\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}\ell}}{C}_{n}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}Q\mathrm{,}$$(9)

where *C*_{n} is the Chern number of the *n*-th quasienergy band. In particular, for the gap at the zone edge, the Chern number sum over all the bands will vanish, and *w*_{ℓ} is nothing but –*Q*. Thus, finding the winding number or the net chirality for the zone-edge gap is reduced to counting the total charge of the corresponding point (Weyl) singularities. This is a very simple but useful result. One of the goals of our paper is to provide concrete examples to illustrate the topological singularities in the phase bands, which make the driven system interesting and distinct from static systems.

Why do we care about the topological singularities in the phase band if we already know how to evaluate the topological invariant *w*_{ℓ} from *U*(*t*)? The first reason is that *w*_{ℓ} contains only the total charge *Q*. In contrast, the complete list of {*q*_{i}} and their locations, the “charge distribution map” (see Fig. 1 below for example), obtained from the phase band analysis contain much more information. Imagine a scenario that, due to additional symmetries, the degeneracy points always come in pairs of opposite charge. Then *Q* will be identically zero. Vanishing *Q* or *w*_{ℓ}, however, does not mean there are no robust edge states. We have previously shown that this occurs in the periodically driven Hofstadter model [31], [32]. For the gap at quasienergy *π*/*T*, the winding number is zero but there can be pairs of counter-propagating edge modes. This model will be analysed briefly in Section 3.

Figure 1: Topological singularities in the phase bands of a periodically driven Hofstadter model at fixed flux 1/3. *k*_{x}=0, *θ*_{x}=*π*/3. The panel on the right shows the locations of the zone edge singularity with monopole charge +1 (empty circle) or –1 (solid dot) for *t*=*T*_{1}+2*π*/3*J*_{y}.

The second reason why topological singularity is such a useful concept has to do with Floquet systems with gapless quasienergy spectra. Previous theoretical works have largely focused on fully gapped Floquet topological insulators/superconductors. But gapless Floquet systems can also be topologically nontrivial with interesting edge states. The band flattening procedure and the winding number mentioned above become ill defined when the quasienergy gap closes, say, at isolated points in **k** space. We will provide an example in Section 6 to show that even in these cases, understanding the degeneracies in the phase band helps identify the topological invariants and the corresponding edge states.

Despite being very useful, the phase band analysis may be a messy business. First of all, degeneracies are ubiquitous in the phase bands of periodically driven systems even for topological trivial cases. Secondly, as mentioned above, the manifolds of degeneracy may be of finite dimensions in the form of lines or surfaces. It is theoretically plausible that the continuous degeneracy can be either lifted or reduced to isolated points by perturbations. In practice, however, it remains a nontrivial task to perform such topological surgeries numerically for arbitrary phase bands. Thirdly, the degeneracies of all phase bands for the entire driving cycle *t*∈[0, *T*] have to be exhausted and classified. And finally, in some cases (see example in Section 5), direct application of (9) is impossible either because the quasienergy bands are overlapping (so that Chern numbers are ill-defined unless additional perturbations are introduced) or *Q* itself is ill-defined, e.g. when the spectrum is gapless at the zone edge. In Section 5, we will show an example how such difficulty can be overcome by generalizing (9). We hope that our case studies presented here can stimulate further application of the phase band analysis to other Floquet systems.

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