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Register a new userDynamical Symmetry Indicators for Floquet Crystals
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Various exotic topological phases of Floquet systems have been shown to arise from crystalline symmetries. Yet, a general theory for Floquet topology that is applicable to all crystalline symmetry groups is still in need. In this work, we propose such a theory for (effectively) non-interacting Floquet crystals. We first introduce quotient winding data to classify the dynamics of the Floquet crystals with equivalent symmetry data, and then construct dynamical symmetry indicators (DSIs) to sufficiently indicate the inherently dynamical Floquet crystals. The DSI and quotient winding data, as well as the symmetry data, are all computationally efficient since they only involve a small number of Bloch momenta. We demonstrate the high efficiency by computing all elementary DSI sets for all spinless and spinful plane groups using the mathematical theory of monoid, and find a large number of different nontrivial classifications, which contain both first-order and higher-order 2+1D anomalous Floquet topological phases. Using the framework, we further find a new 3+1D anomalous Floquet second-order topological insulator (AFSOTI) phase with anomalous chiral hinge modes.
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Although fragile topology has been intensely studied in static crystals, it is not clear how to generalize the concept to dynamical systems. In this work, we generalize the concept of fragile topology, and provide a definition of fragile topology for noninteracting Floquet crystals, which we refer to as dynamical fragile topology. In contrast to the static fragile topology defined for Wannier obstruction, dynamical fragile topology is defined for the nontrivial quantum dynamics characterized by obstruction to static limits (OTSL). Specifically, OTSL of a Floquet crystal is fragile if and only if the OTSL disappears after adding a symmetry-preserving static Hamiltonian in a direct-sum way preserving the relevant gaps (RGs). We further present a concrete 2+1D example for dynamical fragile topology, based on a slight modification of the model in [Rudner et al, Phys. Rev. X 3, 031005 (2013)]. The fragile OTSL in the 2+1D example exhibits anomalous chiral edge modes for a natural open boundary condition, and does not require any crystalline symmetries besides lattice translations. Our work paves the way to study fragile topology for general quantum dynamics.
The topological characterization of nonequilibrium topological matter is highly nontrivial because familiar approaches designed for equilibrium topological phases may not apply. In the presence of crystal symmetry, Floquet topological insulator states cannot be easily distinguished from normal insulators by a set of symmetry eigenvalues at high symmetry points in the Brillouin zone. This work advocates a physically motivated, easy-to-implement approach to enhance the symmetry analysis to distinguish between a variety of Floquet topological phases. Using a two-dimensional inversion-symmetric periodically-driven system as an example, we show that the symmetry eigenvalues for anomalous Floquet topological states, of both first-order and second-order, are the same as for normal atomic insulators. However, the topological states can be distinguished from one another and from normal insulators by inspecting the occurrence of stable symmetry inversion points in their microscopic dynamics. The analysis points to a simple picture for understanding how topological boundary states can coexist with localized bulk states in anomalous Floquet topological phases.
We propose a versatile framework to dynamically generate Floquet higher-order topological insulators by multi-step driving of topologically trivial Hamiltonians. Two analytically solvable examples are used to illustrate this procedure to yield Floquet quadrupole and octupole insulators with zero- and/or $pi$-corner modes protected by mirror symmetries. Furthermore, we introduce dynamical topological invariants from the full unitary return map and show its phase bands contain Weyl singularities whose topological charges form dynamical multipole moments in the Brillouin zone. Combining them with the topological index of Floquet Hamiltonian gives a pair of $mathbb{Z}_2$ invariant $ u_0$ and $ u_pi$ which fully characterize the higher-order topology and predict the appearance of zero- and $pi$-corner modes. Our work establishes a systematic route to construct and characterize Floquet higher-order topological phases.
We investigate an unconventional symmetry in time-periodically driven systems, the Floquet dynamical symmetry (FDS). Unlike the usual symmetries, the FDS gives symmetry sectors that are equidistant in the Floquet spectrum and protects quantum coherence between them from dissipation and dephasing, leading to two kinds of time crystals: the discrete time crystal and discrete time quasicrystal that have different periodicity in time. We show that these time crystals appear in the Bose- and Fermi-Hubbard models under ac fields and their periodicity can be tuned only by adjusting the strength of the field. These time crystals arise only from the FDS and thus appear in both dissipative and isolated systems and in the presence of disorder as long as the FDS is respected. We discuss their experimental realizations in cold atom experiments and generalization to the SU($N$)-symmetric Hubbard models.
We develop a systematic approach for constructing symmetry-based indicators of a topological classification for superconducting systems. The topological invariants constructed in this work form a complete set of symmetry-based indicators that can be computed from knowledge of the Bogoliubov-de Gennes Hamiltonian on high-symmetry points in Brillouin zone. After excluding topological invariants corresponding to the phases without boundary signatures, we arrive at natural generalization of symmetry-based indicators [H. C. Po, A. Vishwanath, and H. Watanabe, Nature Comm. 8, 50 (2017)] to Hamiltonians of Bogoliubov-de Gennes type.
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