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Topological classification of gapless Floquet states
When a quantum system is defined on lattice, the discrete spatial translational symmetry leads to a periodic structure in momentum space, i.e. the Brillouin zone. The topology of the Brillouin zone imposes various constraints on realizable band structures. For example, the Nielsen-Ninomiya theorem [1,2] shows that a single chiral fermion cannot be realized in lattice systems. On the other hand, in periodically driven quantum systems (called Floquet systems), the time translational symmetry is also discrete, and therefore the (quasi-)energy has a ``Brillouin zone". Surprisingly, this periodic structure of the energy space enables exotic band structures that are prohibited in static systems, such as the single chiral fermion [3,4]. In this talk, we first show that the exotic band structures are deeply related to topology of unitary time-evolution operators. Next, based on a topological classification of unitary time-evolution operators with symmetries, we demonstrate that a wide range of lattice-prohibited band structures are realizable using topological Floquet unitary operators. Specifically, we show that all gapless surface states of topological insulators and superconductors in ten Altland-Zirnbauer symmetry classes are realizable as bulk quasi-energy spectrum in Floquet systems.
 H. Nielsen and M. Ninomiya, Nucl. Phys. B 185, 20 (1981).
 H. Nielsen and M. Ninomiya, Nucl. Phys. B 185, 173 (1981).
 T. Kitagawa et al., Phys. Rev. B 82, 235114 (2010).
 S. Higashikawa, M. Nakagawa, and M. Ueda, in preparation.