For time-reversal symmetric cold atomic insulating systems, it is found that the usual driving approach based on electromagnetic field used in solid state systems loses its power to drive them from trivial regimes to topological regimes if the driven systems still hold time-reversal symmetry (TRS). For such systems, we point out that simply varying the optical lattice potential periodically provides a general and effective way to drive them into topological regimes without breaking their symmetries. Based on this approach, we find that the time-reversal symmetric Kane-Mele model can be effectively driven from the trivial phase to topological phases named as Floquet Quantum Spin Hall insulator. Due to the existence of two gaps in the Floquet system, this novel state of matter can stably host one or two pair of gapless helical states on the same boundary, which suggests this state is not a simple analog of the Quantum Spin Hall insulator. This new driving approach to a system without TRS is also investigated.
For time-reversal symmetric cold atomic insulating systems, it is found that the usual driving approach based on electromagnetic field used in solid state systems loses its power to drive them from trivial regimes to topological regimes if the driven systems still hold time-reversal symmetry (TRS). For such systems, we point out that simply varying the optical lattice potential periodically provides a general and effective way to drive them into topological regimes without breaking their symmetries. Based on this approach, we find that the time-reversal symmetric Kane-Mele model can be effectively driven from the trivial phase to topological phases named as Floquet Quantum Spin Hall insulator. Due to the existence of two gaps in the Floquet system, this novel state of matter can stably host one or two pair of gapless helical states on the same boundary, which suggests this state is not a simple analog of the Quantum Spin Hall insulator. This new driving approach to a system without TRS is also investigated.
Nonequilibrium systems under periodic driving realize novel topological phases that cannot be achieved in equilibrium systems. One unique feature of periodically driven systems is that they can host a purely dynamical symmetry that involves time translation. This work explores a new class of Floquet topological phases protected by one realization of such dynamical symmetry, i.e., ``time glide symmetry'', which is defined by a combination of reflection and time translation. Lattice models with time glide symmetric driving that are introduced show stable gapless surface states along with nontrivial topological numbers defined with time glide symmetry. In addition, a general classification theory of time glide symmetric topological phases is obtained by using a Clifford algebra approach.
Time-periodic driving enables new nonequilibrium quantum phases of matter with topologically protected and quantum coherent motion that would be forbidden in thermal equilibrium. This work uncovers two new classes of driven topological phases in driven two-dimensional spin systems: i) Floquet symmetry protected topological phases, which are driven and interacting analogs of topological insulators whose edge states are protected against disorder and localization; and ii) Floquet enriched topological orders, in which time-dependent driving ``breaks'' the spins into anyonic particles with fractional statistics and oscillating topological charge. The physics of these phases is explored by constructing toy models that can be explicitly solved despite the presence of strong interactions.
This thesis reports on the realization of Floquet topological systems with ultracold atoms in an optical honeycomb lattice. Using periodically driven quantum systems, topological phases of matter can be simulated by an effective, static Hamiltonian related to the time-evolution at integer multiples of the driving period, which is known as Floquet engineering. However, periodic driving can also give rise to genuinely time-dependent settings without static counterparts. One example is the anomalous Floquet phase in two dimensions, in which all bulk bands have a Chern number of zero but nevertheless robust chiral edge modes appear, which would be precluded by the bulk-edge correspondence in a static system. In an optical honeycomb lattice, anomalous Floquet systems can be created by continuous, periodic modulation of the laser intensities. This driving scheme results in different topological regimes, three of which are investigated, including the anomalous Floquet phase and a Haldane-like phase. Periodically driven systems feature periodic quasienergies and can be characterized in terms of winding numbers, which count the number of chiral edge modes in each quasienergy gap. By interferometric measurements of the quasienergy gaps between the first two Floquet bands, the topological phase transitions, emerging as gap closings, are located. For periodically driven systems, a modified bulk-edge correspondence can be formulated. In particular, the change of the winding number at a phase transition is related to the sign change of the local Berry curvature. The Berry curvature at the gap closing points is probed by Hall deflection measurements to obtain the winding numbers in each of the topological regimes, revealing the existence of chiral edge modes also in a setting with smooth boundaries as used in the experiments. The measured quasienergy gaps and transverse deflections are quantitatively well described by a numerical calculation of the Floquet bandstructure that includes coupling to higher bands during the driving period. To derive the spectrum of the modulated lattice in a geometry with edges, a tight-binding description of the system is discussed. Circular phase modulation of the honeycomb lattice can also give rise to an anomalous Floquet system when inversion symmetry is broken. The topological phase diagrams and the experimental feasibility are compared for both modulation schemes. Due to the vanishing Chern numbers, the bulk states in the anomalous Floquet regime can be fully localized by a disorder potential. This could prevent heating in interacting, periodically driven systems, resulting in a many-body localized bulk coexisting with thermalizing edge states. The experimental realization of disorder and strong interactions, as well as independent probes of the edge states are investigated. Bloch band geometry can be extended to multiband systems, using Wilson lines. The possible symmetry protection of their eigenvalues is discussed, along with measurements in the optical honeycomb lattice. Moreover, the onset of heating in weakly-interacting, periodically driven systems, triggered by parametric instabilities, is studied experimentally in a one-dimensional lattice.
Recently, anomalous Floquet topological phases without static counterparts have been observed in different systems, where periodically driven models are realized to support a winding number of 1 and a pair of edge modes in each quasienergy gap. Here, we focus on cold atomic gases in optical lattices and propose a driving scheme that breaks rotation symmetry but maintains inversion symmetry of the instantaneous Hamiltonian, and discover a type of anomalous Floquet topological phase with a winding number larger than 1. By analyzing the condition of band touching under the constraint of symmetries, we map out the phase diagram exactly by varying the driving parameters and discuss the quasienergy spectra of typical topological phases, which can present multiple pairs of edge modes within a single gap. Finally, we suggest to characterize the topology of such phases by detecting the band inversion surfaces via quench dynamics.
Time-periodic (Floquet) topological phases of matter exhibit bulk-edge relationships that are more complex than static topological insulators and superconductors. Finding the edge modes unique to driven systems usually requires numerics. Here we present a minimal two-band model of Floquet topological insulators and semimetals in two dimensions where all the bulk and edge properties can be obtained analytically. It is based on the extended Harper model of quantum Hall effect at flux one-half. We show that periodical driving gives rise to a series of phases characterized by a pair of integers. The model has a most striking feature: the spectrum of the edge modes is always given by a single cosine function, $\ensuremath{\omega}({k}_{y})\ensuremath{\propto}cos{k}_{y}$ where ky is the wave number along the edge, as if it is freely dispersing and completely decoupled from the bulk. The cosine mode is robust against the change in driving parameters. It also persists in the semimetallic phases with Dirac points.
We propose to synthesize tunable periodic gauge fields via Floquet engineering cold-alkaline-earth-metal atoms in one-dimensional optical lattice. The artificial magnetic flux is designed to emerge during the combined process of Floquet photon-assisted tunneling and internal state transitions. By varying the initial phases of the driving protocol, our proposal presents the ability to smoothly tune the periodic flux. Moreover, we demonstrate that the effective two-leg flux ladder model can simulate one typical real topological insulator, which is described by the first Stiefel-Whitney class and protected by the space-inversion--time-reversal symmetry. Benefiting from the long lifetime of excited states of alkaline-earth-metal atoms, our work opens new possibilities for exploiting the physics related to gauge fields, such as topological phases, in the current cold-atom platform.