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arXiv:1601.04437 · Physical review. B./Physical review. B

Towards large-Chern-number topological phases by periodic quenching

Tian-Shi Xiong, Jiangbin Gong, Jun‐Hong An

Published May 31, 2016• 90 citations
arXivHTMLPDF

Abstract

Topological phases with large Chern numbers have important implications. They were previously predicted to exist by considering fabricated long-range interactions or multilayered materials. Stimulated by recent wide interests in Floquet topological phases, here we propose a scheme to engineer large-Chern-number phases with ease by periodic quenching. Using a two-band system as an example, we theoretically show how a variety of topological phases with widely tunable Chern numbers can be generated by periodic quenching between two simple Hamiltonians that otherwise give low Chern numbers. The obtained large Chern numbers are explained through the emergence of multiple Dirac cones in the Floquet spectra. The transition lines between different topological phases in the two-band model are also explicitly found, thus establishing a class of easily solvable but very rich systems useful for further understandings and applications of topological phases in periodically driven systems.

DOI
10.1103/physrevb.93.184306

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Topological phases with large Chern numbers have important implications. They were previously predicted to exist by considering fabricated long-range interactions or multilayered materials. Stimulated by recent wide interests in Floquet topological phases, here we propose a scheme to engineer large-Chern-number phases with ease by periodic quenching. Using a two-band system as an example, we theoretically show how a variety of topological phases with widely tunable Chern numbers can be generated by periodic quenching between two simple Hamiltonians that otherwise give low Chern numbers. The obtained large Chern numbers are explained through the emergence of multiple Dirac cones in the Floquet spectra. The transition lines between different topological phases in the two-band model are also explicitly found, thus establishing a class of easily solvable but very rich systems useful for further understandings and applications of topological phases in periodically driven systems.
Topological phases with a widely tunable number of edge modes have been extensively studied as a typical class of exotic states of matter with potentially important applications. Although several models have been shown to support such phases, they are not easy to realize in solid-state systems due to the complexity of various intervening factors. Inspired by the realization of synthetic spin-orbit coupling in a cold-atom system [Z. Wu et al., Science 354, 83 (2016)], we propose a periodic quenching scheme to realize large-topological-number phases with multiple edge modes in optical lattices. Via introducing the periodic quenching to the Raman lattice, it is found that a large number of edge modes can be induced in a controllable manner from the static topologically trivial system. Our result provides an experimentally accessible method to artificially synthesize and manipulate exotic topological phases with large topological numbers and multiple edge modes.
Recently, Floquet systems have attracted a great deal of interest as they offer the unprecedented ability to engineer topological states through the tuning of an external time-periodic drive. Consequentially, seeking new driving protocols that allow for more exotic topological phases and transitions becomes imperative for the Floquet engineer. In this paper, we study the Su-Schrieffer-Heeger model driven by two time-dependent periodic sources with commensurate frequencies and an amplitude modulation. Imposing more than one driving frequency allows us to realize even more exotic topological phases resulting from new couplings appearing in the Fourier space representation. Moreover, we find an experimentally practical method for sweeping the system through a topological phase transition by varying the amplitude mixture of the commensurate sources. We employ the local Chern marker, a real-space representation of the Chern number, to simulate topological phase diagrams of the two-drive Floquet Hamiltonian in a variety of driving cases.
Coherent control via periodic modulation, also known as Floquet engineering, has emerged as a powerful experimental method for the realization of novel quantum systems with exotic properties. In particular, it has been employed to study topological phenomena in a variety of different platforms. In driven systems, the topological properties of the quasienergy bands can often be determined by standard topological invariants, such as Chern numbers, which are commonly used in static systems. However, due to the periodic nature of the quasienergy spectrum, this topological description is incomplete and new invariants are required to fully capture the topological properties. Most prominently, there exist two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers are equal to zero. Here, we realize such a system with bosonic atoms in a periodically-driven hexagonal lattice and infer the complete set of topological invariants from energy gap measurements and local Hall deflections.
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.
Different topological phases of quantum systems have become areas of increased focus in recent decades. In particular, the question of how to realize and manipulate systems with a nontrivial first Chern number is pursued both experimentally and theoretically. Here we go beyond typical spin-1/2 systems and consider both single and coupled spin-1 systems as a means of realizing higher first Chern numbers and studying the emergence of different topological phases. We show that rich topological phase diagrams can be realized by coupling two spin-1 systems using both numerical and analytical methods. Furthermore, we consider a concrete realization of spin-1 systems using a superconducting circuit. This realization includes nonstandard spin-spin interaction terms that may endanger the topological properties. We argue, however, that the realistic circuit Hamiltonian including all terms should be expected to show the rich phase structure as well. This puts our theoretical predictions within reach of state-of-the-art experimental setups.
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.