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arXiv:1912.09078 · Physical Review A

Floquet topological phases with fourfold-degenerate edge modes in a driven spin-1/2 Creutz ladder

Longwen Zhou, Qianqian Du

Published Mar 10, 2020• 26 citations
arXivHTMLPDF

Abstract

Floquet engineering has the advantage of generating new phases with large topological invariants and many edge states by simple driving protocols. In this work, we propose an approach to obtain Floquet topological phases with fourfold-degenerate edge states and even-integer topological invariants in a spin-$1/2$ Creutz ladder model, which is realizable in current experiments. Putting the ladder under periodic quenches, we found rich Floquet topological phases in the system, which belong to the symmetry class $CII$. Each of these phases is characterized by a pair of even integer topological invariants $({w}_{0},{w}_{\ensuremath{\pi}})\ensuremath{\in}2\mathbb{Z}\ifmmode\times\else\texttimes\fi{}2\mathbb{Z}$, which can take arbitrarily large values with the increase of driving parameters. Under the open boundary condition, we further obtain multiple quartets of topological edge states with quasienergies zero and $\ensuremath{\pi}$ in the system. Their numbers are determined by the bulk topological invariants $({w}_{0},{w}_{\ensuremath{\pi}})$ due to the bulk-edge correspondence. Finally, we propose a way to dynamically probe the Floquet topological phases in our system by measuring a generalized mean chiral displacement. Our findings thus enrich the family of Floquet topological matter, and put forward the detection of their topological properties.

DOI
10.1103/physreva.101.033607

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Floquet engineering has the advantage of generating new phases with large topological invariants and many edge states by simple driving protocols. In this work, we propose an approach to obtain Floquet topological phases with fourfold-degenerate edge states and even-integer topological invariants in a spin-1/21/21/2 Creutz ladder model, which is realizable in current experiments. Putting the ladder under periodic quenches, we found rich Floquet topological phases in the system, which belong to the symmetry class CIICIICII. Each of these phases is characterized by a pair of even integer topological invariants (w0,w\ensuremathπ)\ensuremath∈2Z\ifmmode×\else\texttimes\fi2Z({w}_{0},{w}_{\ensuremath{\pi}})\ensuremath{\in}2\mathbb{Z}\ifmmode\times\else\texttimes\fi{}2\mathbb{Z}(w0​,w\ensuremathπ​)\ensuremath∈2Z\ifmmode×\else\texttimes\fi2Z, which can take arbitrarily large values with the increase of driving parameters. Under the open boundary condition, we further obtain multiple quartets of topological edge states with quasienergies zero and \ensuremathπ\ensuremath{\pi}\ensuremathπ in the system. Their numbers are determined by the bulk topological invariants (w0,w\ensuremathπ)({w}_{0},{w}_{\ensuremath{\pi}})(w0​,w\ensuremathπ​) due to the bulk-edge correspondence. Finally, we propose a way to dynamically probe the Floquet topological phases in our system by measuring a generalized mean chiral displacement. Our findings thus enrich the family of Floquet topological matter, and put forward the detection of their topological properties.
Abstract One-dimensional (1D) Floquet topological superconductors possess two types of degenerate Majorana edge modes at zero andπquasienergies, leaving more room for the design of boundary time crystals and quantum computing schemes than their static counterparts. In this work, we discover Floquet superconducting phases with large topological invariants and arbitrarily many Majorana edge modes in periodically driven Kitaev chains (KCs). Topological winding numbers defined for the Floquet operator and Floquet entanglement Hamiltonian are found to generate consistent predictions about the phase diagram, bulk-edge correspondence and numbers of zero andπMajorana edge modes of the system under different driving protocols. The bipartite entanglement entropy further shows non-analytic behaviors around the topological transition point between different Floquet superconducting phases. These general features are demonstrated by investigating the KC with periodically kicked pairing or hopping amplitudes. Our discovery reveals the rich topological phases and many Majorana edge modes that could be brought about by periodic driving fields in 1D superconducting systems. It further introduces a unified description for a class of Floquet topological superconductors from their quasienergy bands and entanglement properties.
The complete characterization of a generic ddd-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{\mathcal{Z}}_{2}Z2​ invariants. Here we develop a full and unified dynamical characterization theory for the Z2{\mathcal{Z}}_{2}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{\mathcal{Z}}_{2}Z2​ Floquet invariant, which uniquely determines the Floquet boundary modes in the corresponding quasienergy gap and characterizes the Z2{\mathcal{Z}}_{2}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{\mathcal{Z}}_{2}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.
M\"obius topological insulators have dispersive edge bands with M\"obius twists in momentum space, which are protected by the combination of chiral and Z2{\mathbb{Z}}_{2}Z2​-projective translational symmetries. In this work, we reveal a unique type of M\"obius topological insulator, whose edge bands can twist around the quasienergy $\ensuremath{\pi}$ of a periodically driven system and are thus of Floquet origin. By applying time-periodic quenches to an experimentally realized M\"obius insulator model, we obtain interconnected M\"obius edge bands around zero and $\ensuremath{\pi}$ quasienergies, which can coexist with a gapped or gapless bulk. These M\"obius bands are topologically characterized by a pair of generalized winding numbers, which are integer-quantized as a result of an emergent chiral symmetry at a high-symmetry point in momentum space. Numerical investigations of the quasienergy and entanglement spectra provide consistent evidence for the presence of such M\"obius topological phases. A protocol based on the adiabatic switching of edge-band populations is further introduced to dynamically characterize the topology of Floquet M\"obius edge bands. Our findings thus extend the scope of M\"obius topological phases to nonequilibrium settings and unveil a unique class of M\"obius-twisted topological edge states without static counterparts.
Symmetry-protected topological (SPT) phases in insulators and superconductors are known for their robust edge modes, linked to bulk invariants through the bulk-boundary correspondence. While this principle traditionally applies to gapped phases, recent advances have extended it to gapless systems, where topological edge-states persist even in the absence of a bulk gap. We extend this framework to periodically driven chains with chiral symmetry, revealing the existence of topological edge zero and $\ensuremath{\pi}$ modes despite the lack of bulk gaps in the quasienergy spectrum. By examining the half-period decomposition of chiral evolutions, we construct topological invariants that circumvent the need to define the Floquet Hamiltonian, making them more suitable for generalization to the gapless regime. We provide explicit examples, including generalizations of the Kitaev chain and related spin models, where localized $\ensuremath{\pi}$ modes emerge even when the bulk is gapless at the same quasienergy as the edge modes. We numerically study the effect of interactions, which give a finite lifetime to the edge modes in the thermodynamic limit with the decay rate consistent with Fermi's golden rule.
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.
We investigate the dynamical characterization theory for periodically driven systems in which Floquet topology can be fully detected by emergent topological patterns of quench dynamics in momentum subspaces called band-inversion surfaces. We improve the results of a recent work [Zhang et al., Phys. Rev. Lett. 125, 183001 (2020)] and propose a more flexible scheme to characterize a generic class of ddd-dimensional Floquet topological phases classified by Z\mathbb{Z}Z-valued invariants by applying a quench along an arbitrary spin-polarization axis. Our basic idea is that by disassembling the Floquet system into multiple static subsystems that are periodic in quasienergy, a full characterization of Floquet topological phases reduces to identifying a series of bulk topological invariants for time-independent Hamiltonians, which greatly enhances the convenience and flexibility of the measurement. We illustrate the scheme by numerically analyzing two experimentally realizable models in two and three dimensions, respectively, and adopting two different but equivalent viewpoints to examine the dynamical characterization. Finally, considering the imperfection of experiment, we demonstrate that the present scheme can also be applied to a general situation where the initial state is not completely polarized. This study provides an immediately implementable approach for dynamically classifying Floquet topological phases in ultracold atoms or other quantum simulators.