We define a new Z2-valued index to characterize the topological properties of periodically driven two dimensional crystals when the time-reversal symmetry is enforced. This index is associated with a spectral gap of the evolution operator over one period of time. When two such gaps are present, the Kane-Mele index of the eigenstates with eigenvalues between the gaps is recovered as the difference of the gap indices. This leads to an expression for the Kane-Mele invariant in terms of the Wess-Zumino amplitude. We illustrate the relation of the new index to the edge states in finite geometries by numerically solving an explicit model on the square lattice that is periodically driven in a time-reversal invariant way. Introduction. -The recent discovery of the quantum spin Hall effect [1][2][3][4] renewed interest in topological insulating phases which were first encountered in the beginning of the 1980s with the discovery of the quantum Hall effect (QHE) [5][6][7][8]. In his seminal paper [9] of 1981, Laughlin related the quantized Hall conductance to a quantum pump adiabatically driven by the magnetic flux. As shown by Thouless [10], such pumps drive through the insulator an integer number of charges whose origin is topological. In deep analogy, several works interpreted topological insulators and their robust boundary states in terms of quantum adiabatic pumps [11][12][13]. Interestingly, quantum crystals can exhibit original topological features when periodically driven beyond the adiabatic regime. While such modulation was first proposed to trigger a topological phase transition [14][15][16], it can also yield specific topological properties which cannot be understood within the usual framework of topological band theory [17,18]. The search for these so-called Floquet topological states quickly became a very active field and has recently stimulated numerous experimental works. A realization of such phases in condensed matter is quite challenging [19,20]. Several alternative artificial systems have been proposed to simulate and probe analogous phases, such as lattices of photonic resonators periodically driven by electro-optic modulators [21], ring resonator lattices [22], or more recently, photons coupled to excitons in semiconductors [23]. Signatures of topological Floquet states have already been revealed in onedimensional quantum walks with photons [24], as well as in 2D waveguide lattices [25]. Shaken trapped cold atoms were also proposed as a good candidate [26][27][28] and non-trivial topological phases were recently observed there [29].

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Fluids in which both time-reversal and parity are broken can display a dissipationless viscosity that is odd under each of these symmetries. Here, we show how this odd viscosity has a dramatic effect on topological sound waves in fluids, including the number and spatial profile of topological edge modes. Odd viscosity provides a short-distance cutoff that allows us to define a bulk topological invariant on a compact momentum space. As the sign of odd viscosity changes, a topological phase transition occurs without closing the bulk gap. Instead, at the transition point, the topological invariant becomes ill-defined because momentum space cannot be compactified. This mechanism is unique to continuum models and can describe fluids ranging from electronic to chiral active systems.In ordinary fluids, acoustic waves with sufficiently large wavelength have arbitrarily low frequency due to Galilean invariance [1]. When either a global rotation or an external magnetic field is present, Galilean invariance is explicitly broken by either Coriolis or Lorentz forces within the fluid, respectively. Hence, the spectrum of acoustic waves becomes gapped in the bulk. Yet, a peculiar phenomenon can occur at edges or interfaces: chiral edge modes propagate robustly irrespective of interface geometry. This phenomenon analogous to edge states in the quantum Hall effect [2][3][4] was unveiled in the context of equatorial waves [5] and explored in out-of-equilibrium and active fluids [6,7]. Similar phenomena occur in lattices of circulators [8,9], polar active fluids under confinement [10] and coupled mechanical oscillators [11][12][13], including gyroscopes [14,15] and oscillators subject to Coriolis forces [16,17].In addition to Coriolis or Lorentz body forces, fluids in which time-reversal and parity are broken generically exhibit a dissipationless viscosity that is odd under each of these symmetries [18,19]. The viscosity tensor η ijkl relates the strain rate v kl ≡ ∂ k v l to the viscous part of the stress tensor σ ij = η ijkl v kl . Odd viscosity refers to the antisymmetric part of the viscosity tensor η o ijkl = −η o klij [18,19]. In an isotropic two-dimensional fluid, odd viscosity is specified by a single pseudoscalar η o , see Supplementary Information (SI) for details [19]. Odd viscosity changes sign under either time-reversal or parity, and hence must vanish when at least one of these symmetries is present. Conversely, odd viscosity is generically non-vanishing as soon as both time-reversal and parity are broken [20][21][22]. For instance, microscopic Coriolis or Lorentz forces are sufficient to induce a non-zero odd viscosity [23,24], in addition to the corresponding body forces. Odd viscosity has been studied theoretically in various systems (see SI for a partial review) including polyatomic gases [25], magnetized plasmas [24,26], flu-ids of vortices [27][28][29][30], chiral active fluids [31], quantum Hall states and chiral superfluids/superconductors [32][33][34][35][36][37][38][39][40][41][42]. Its presence has been exp...

Periodically driven (Floquet) crystals are described by their quasi-energy spectrum. Their topological properties are characterized by invariants attached to the gaps of this spectrum. In this article, we define such invariants in all space dimensions, both in the case where no symmetry is present and in the case where the unitary chiral symmetry is present. When no symmetry is present, a $\mathbb{Z}$-valued invariant can be defined in each gap in all even space dimensions. This invariant does not capture all the properties of a system where chiral symmetry is present. In even space dimension, chiral symmetry puts constraints on its values in different gaps. In odd space dimension, chiral symmetry also enables to define a $\mathbb{Z}$-valued invariant, only in the chiral gaps $0$ and $\pi$. We relate both gap invariants to the standard invariants characterizing the quasi-energy bands of the system. Examples in one and three dimensions are discussed.Comment: published version, 11 pages, 5 figures; source code available in ancillary file