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arXiv:2311.00114 · Physical review. B./Physical review. B

Dynamical characterization of Z2 Floquet topological phases via quantum quenches

Lin Zhang

Published Apr 2, 2024• 3 citations
arXivHTMLPDF

Abstract

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 ${\mathcal{Z}}_{2}$ invariants. Here we develop a full and unified dynamical characterization theory for the ${\mathcal{Z}}_{2}$ 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 ${\mathcal{Z}}_{2}$ Floquet invariant, which uniquely determines the Floquet boundary modes in the corresponding quasienergy gap and characterizes the ${\mathcal{Z}}_{2}$ 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 ${\mathcal{Z}}_{2}$ 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.

DOI
10.1103/physrevb.109.165110

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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 \ensuremathπ\ensuremath{\pi}\ensuremathπ 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.
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.
The conventional characterization of periodically driven systems usually necessitates the time-domain information beyond Floquet bands, hence lacking universal and direct schemes of measuring Floquet topological invariants. Here we propose a unified theory, based on quantum quenches, to characterize generic d-dimensional Floquet topological phases in which the topological invariants are constructed with only minimal information of the static Floquet bands. For a d-dimensional phase that is initially static and trivial, we introduce the quench dynamics by suddenly turning on the periodic driving. We show that the quench dynamics exhibits emergent topological patterns in (d-1)-dimensional momentum subspaces where Floquet bands cross, from which the Floquet topological invariants are directly obtained. This result provides a simple and unified characterization in which one can extract the number of conventional and anomalous Floquet boundary modes and identify the topologically protected singularities in the phase bands. These applications are illustrated with one- and two-dimensional models that are readily accessible in cold-atom experiments. Our study opens a new framework for the characterization of Floquet topological phases.
We investigate a generic dynamical theory to characterize by quantum quenches the topological quantum phases defined for the equilibrium ground state and study the emergent topology of quantum dynamics when the quenches start from a deep trivial (far from the phase boundary) or shallow trivial (close to the phase boundary) phase to topological regimes. Two dynamical schemes are examined: One is to characterize topological phases via quantum dynamics induced by a single quench along an arbitrary axis of (pseudo)spin polarization, and the other applies a set of quenches with respect to all (pseudo)spin axes. These two schemes are both built on the so-called dynamical bulk-surface correspondence, which shows that the ddd-dimensional (dDd\mathrm{D}dD) topological phases with integer invariants can be characterized by the dynamical topological pattern emerging on particular $(d\ensuremath{-}1)\mathrm{D}$ momentum subspaces called band inversion surfaces (BISs). We show that the first dynamical scheme works for both deep and shallow quenches, while for the second scheme, an emergent topological transition, associated with topological charges crossing BISs, is predicted in the joint quench dynamics. A generic criterion for the emergent transition is precisely obtained. Above the criterion (deep quench regime), quantum dynamics on BISs can directly characterize the topology of the postquench Hamiltonian. Below the criterion (shallow quench regime), the quench dynamics may depict a new dynamical topology; the postquench topology can be characterized by the emergent topological invariant plus the total charges moving outside the region enclosed by BISs. We illustrate our results by numerically calculating the 2D quantum anomalous Hall model, which has been realized in ultracold atoms. This work broadens the way to classify topological phases by nonequilibrium quantum dynamics and is feasible for experimental realization.
We propose a theory to characterize equilibrium topological phase with nonequilibrium quantum dynamics by introducing the concept of high-order topological charges, with novel phenomena being predicted. Through a dimension reduction approach, we can characterize a ddd-dimensional (dDd\mathrm{D}dD) integer-invariant topological phase with lower dimensional topological number quantified by high-order topological charges, of which the sths\mathrm{th}sth-order topological charges denote the monopoles confined on the $(s\ensuremath{-}1)\mathrm{th}$-order band inversion surfaces (BISs) that are $(d\ensuremath{-}s+1)\mathrm{D}$ momentum subspaces. The bulk topology is determined by the sths\mathrm{th}sth-order topological charges enclosed by the sths\mathrm{th}sth-order BISs. By quenching the system from trivial phase to topological regime, we show that the bulk topology of postquench Hamiltonian can be detected through a high-order dynamical bulk-surface correspondence, in which both the high-order topological charges and high-order BISs are identified from quench dynamics. This characterization theory has essential advantages in two aspects. First, the highest (dthd\mathrm{th}dth) order topological charges are characterized by only discrete signs of spin polarization in zero dimension (i.e., the 0th0\mathrm{th}0th Chern numbers), whose measurement is much easier than the first-order topological charges that are characterized by the continuous charge-related spin texture in higher dimensional space. Second, a more striking result is that a first-order high integer valued topological charge always reduces to multiple highest order topological charges with unit charge value, and the latter can be readily detected in experiments. The two fundamental features greatly simplify the characterization and detection of the topological charges and also topological phases, which shall advance the experimental studies in the near future.
Winding number and Zak phase, as topological invariants, play crucial roles in characterizing the topological phases of one-dimensional Floquet topological insulators. It is difficult to directly detect Floquet topological invariants with a unified scheme, especially for low driving frequencies. Here, by defining the dynamic winding number based on the time-averaged spin textures, we propose to use it to characterize one-dimensional Floquet topological phases with arbitrarily varying frequencies. In the high-frequency regime, both the conventional winding number and Zak phase can be fully determined by measuring the dynamic winding number of time-averaged spin textures, regardless of initial states and initial phases. In the low-frequency regime, only conventional winding number can accurately describe the system's topological phase due to the presence of large topological numbers, the Zak phase only captures the parity of the winding number and fails to characterize the number of topological edge states. However, one can still correctly characterize the Floquet topological phases through our dynamic winding number approach, in which stroboscopic measurement of spin textures is needed and the initial phases should guarantee chiral symmetry of the time-evolution operator. Our paper not only clarifies the necessity of stroboscopic measurements, but also provides a general dynamic approach for the detection of Floquet topological insulators.
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 $({w}_{0},{w}_{\ensuremath{\pi}})\ensuremath{\in}2\mathbb{Z}\ifmmode\times\else\texttimes\fi{}2\mathbb{Z}$, 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.