This article introduces a general and experimentally feasible protocol to engineer dynamic higher-order topological phases by periodically driving its static lower-order counterparts with a suitable discrete symmetry breaking Dirac mass perturbation. Realization of dynamic corner modes in 2D, protected by a quantized Floquet quadrupole moment, exemplifies this general protocol. Additionally, the authors show that corner modes leave their signature even in a quantum spin Hall insulator for a long time after a sudden quench, manifesting their topological nature through periodic appearances of partial and complete revival of the survival probability.
This article introduces a general and experimentally feasible protocol to engineer dynamic higher-order topological phases by periodically driving its static lower-order counterparts with a suitable discrete symmetry breaking Dirac mass perturbation. Realization of dynamic corner modes in 2D, protected by a quantized Floquet quadrupole moment, exemplifies this general protocol. Additionally, the authors show that corner modes leave their signature even in a quantum spin Hall insulator for a long time after a sudden quench, manifesting their topological nature through periodic appearances of partial and complete revival of the survival probability.
Higher-order topological insulators have attracted significant interest in both static single-particle and many-body lattice systems. While periodically driven (Floquet) higher-order topological phases have been explored at the single-particle level, the role of interactions in such systems remains less understood. In this paper, we extend previous studies by investigating interaction-induced higher-order topological phases through Floquet engineering. To achieve this, we construct an extended Bose-Hubbard model on a square lattice subjected to periodic driving. We demonstrate the emergence of interaction-induced normal Floquet second-order topological corner states for doublons (i.e., bound boson pairs) from a trivial phase, which exhibit robustness against disorder. Notably, beyond the normal phase, we reveal an interaction-induced anomalous Floquet second-order topological phase, where in-gap corner states of doublons emerge within the $\ensuremath{\pi}/T$ gap (T being the driving period). Our model, accessible with state-of-the-art ultracold atom techniques, provides a platform for realizing interaction-driven higher-order topological phases uniquely enabled by periodic driving, with no direct counterparts in static or single-particle systems.
Periodic driving has a longstanding reputation for generating exotic phases of matter with no static counterparts. This work explores the interplay between periodic driving, interaction effects, and Z2 symmetry that leads to the emergence of Floquet symmetry protected second-order topological phases in a simple but insightful two-dimensional spin-1/2 lattice. Through a combination of analytical and numerical treatments, we verify the formation of corner-localized 0 and $\ensuremath{\pi}$ modes, i.e., Z2 symmetry broken operators that commute and anticommute, respectively, with the one-period time evolution operator, as well as establish the topological nature of these modes by demonstrating their presence over a wide range of parameter values and explicitly deriving their associated topological invariants under special conditions. Finally, we propose a means to detect the signature of such modes in experiments, and we discuss the effect of imperfections.
While periodically driven phases offer a unique insight into nonequilibrium topology that is richer than its static counterpart, their experimental realization is often hindered by ubiquitous decoherence effects. Recently, we have proposed a decoherence-free approach of realizing these Floquet phases. The central insight is that the reflection matrix, being unitary for a bulk insulator, plays the role of a Floquet time-evolution operator. We have shown that reflection processes off the boundaries of systems supporting higher-order topological phases (HOTPs) simulate nontrivial Floquet phases. So far, this method was shown to work for one-dimensional Floquet topological phases protected by local symmetries. Here, we extend the range of applicability by studying reflection off three-dimensional HOTPs with corner and hinge modes. We show that the reflection processes can simulate both first-order and second-order Floquet phases, protected by a combination of local and spatial symmetries. For every phase, we discuss appropriate topological invariants calculated with the nested scattering matrix method.
The complete characterization of a generic d-dimensional Floquet topological phase is usually hard for the requirement of information about the micromotion throughout the entire driving period. In a recent work [L. Zhang et al., Phys. Rev. Lett. 125, 183001 (2020)], an experimentally feasible dynamical detection scheme was proposed to characterize the integer Floquet topological phases using quantum quenches. However, this theory is still far away from completion, especially for free-fermion Floquet topological phases, where the states can also be characterized by Z2 invariants. Here we develop a full and unified dynamical characterization theory for the Z2 Floquet topological phases of different dimensionality and tenfold-way symmetry classes by quenching the system from a trivial and static initial state to the Floquet topological regime through suddenly changing the parameters and turning on the periodic driving. By measuring the minimal information of Floquet bands via the stroboscopic time-averaged spin polarizations, we show that the topological spin texture patterns emerging on certain discrete momenta of Brillouin zone called the 0 or $\ensuremath{\pi}$ gap highest-order band-inversion surfaces provide a measurable dynamical Z2 Floquet invariant, which uniquely determines the Floquet boundary modes in the corresponding quasienergy gap and characterizes the Z2 Floquet topology. The applications of our theory are illustrated via one- and two-dimensional models that are accessible in current quantum simulation experiments. Our work provides a highly feasible way to detect the Z2 Floquet topology and completes the dynamical characterization for the full tenfold classes of Floquet topological phases, which shall advance the research in theory and experiments.
Floquet engineering offers a unique approach to generate nonequilibrium topological phases in which the unbounded nature of a quasienergy band allows two kinds of topological edge modes, one of them traversing the 0 gap and another one traversing the $\ensuremath{\pi}$ gap. Characterizing of these two modes is the main topic of Floquet topological insulators, where they are usually characterized by different topological invariants. However, in this paper, for a specific protocol of Floquet engineering where the Dirac mass term of the Chern insulator is periodically kicked, its topological phases are characterized with the Floquet Chern number CF. Specifically, in our illustrative example, the periodically kicked Qi-Wu-Zhang model, there are six different topological phases in total, denoted as ${C}_{F}={\ensuremath{-}{1}_{0},\ensuremath{-}2,\ensuremath{-}{1}_{\ensuremath{\pi}},{1}_{\ensuremath{\pi}},2,{1}_{0}}$. Topological phases with larger topological number are observed, i.e., ${C}_{F}=\ifmmode\pm\else\textpm\fi{}2$, where the chiral edge modes traversing the 0 gap and those traversing $\ensuremath{\pi}$ gap have the opposite chirality. The mechanism of its topology is revealed by studying the corresponding low-energy effective Dirac Hamiltonian, and the phase boundaries between different topological phases are explicitly found. Additionally, we investigate the orders of phase transitions between different topological phases by studying the von Neumann entropy of the Floquet steady state (FSS), where the FSS corresponds to a stationary state of the Floquet system that is coupled to a Markovian environment. The hallmark of a periodically kicked Dirac mass term is uncovered in this paper, which may inspire further explorations of the physical effects of periodically kicked Dirac mass terms in other systems.
Dynamical quantum phase transitions occur in dynamically evolving quantum systems when nonanalyticities occur at critical times in the return rate, a dynamical analog of the free energy. This extension of the concept of phase transitions can be brought into contact with another, namely, that of topological phase transitions in which the phase transition is marked by a change in a topological invariant. Following a quantum quench dynamical quantum phase transitions can happen in topological matter, a fact which has already been explored in one-dimensional topological insulators and in two-dimensional Chern insulators. Additionally, in one-dimensional systems a dynamical bulk boundary correspondence has been seen, related to the periodic appearance of zero modes of the Loschmidt echo itself. Here we extend both of these concepts to two-dimensional higher-order topological matter in which the topologically protected boundary modes are corner modes. We consider a minimal model which encompasses all possible forms of higher-order topology in two-dimensional topological band structures. We find that dynamical quantum phase transitions can still occur, and can occur for quenches which cross both bulk and boundary gap closings. Furthermore, a dynamical bulk boundary correspondence is also found, which takes a different form to that in one dimension.