Topological Index for Periodically Driven Time-Reversal Invariant 2D Systems

Abstract

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].

Classification of Interacting Topological Floquet Phases in One Dimension

“…This noninteracting classification has since been shown to extend to higher dimensions, for the specific case of 2D time-reversal-invariant topological insulators [14], and has since been systematically generalized to other symmetry classes using K theory [49]. The focus of this section of our paper is instead on investigating how interactions modify results in 1D systems and, subsequently, on intrinsically interacting FSPT bosonic phases with no free-particle band-structure description.…”

“…Such Floquet engineering has led to various applications in quantum optical contexts, such as the engineering of artificial gauge fields [1], as well as in solid-state contexts, e.g., to produce new Floquet-Bloch band structures [2,3] or understand nonlinear optical phenomena [4]. In addition to providing new tools to engineer phases that could arise as ground states of a different static Hamiltonian, periodic driving also opens up the possibility of engineering entirely new phases with no equilibrium analog [5][6][7][8][9][10][11][12][13][14]. In the context of noninteracting particles, various examples of new topological phases that arise from driving are known, including dynamical Floquet analogs of Majorana fermions in 1D [6] and phases with chiral edge modes but vanishing Chern number in 2D [5,7].…”

Anomalous Floquet-Anderson Insulator as a Nonadiabatic Quantized Charge Pump

“…In addition to the phenomena above, driven systems may also host unique types of robust topological phenomena, without analogues in static systems [3,[26][27][28][29][30][31][32]. A prominent example is Thouless's quantized adiabatic pump [33]: a gapped one-dimensional system transmits a precisely quantized charge when subjected to a periodic modulation, in the adiabatic limit of slow driving.…”

Occupation probabilities and current densities of bulk and edge states of a Floquet topological insulator

“…In fact FTIs are extremely rich, showing a variety of topological phases as the amplitude, frequency, and polarization of the periodic drive is varied [17][18][19][20] .…”