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arXiv:2210.08525 · Physical review. B./Physical review. B

Topological phase transition in the commensurate multifrequency Floquet Su-Schrieffer-Heeger model

Sam Olin, Wei‐Cheng Lee

Published Mar 22, 2023• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.107.094310

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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.
| > 1 are observed in this study of a periodically driven extended Su-Schrieffer-Heeger (E-SSH) model with a cyclic parameter. Besides the standard intra-dimer and the nearest-neighbor inter-dimer hopping of the SSH model, an additional next-nearest-neighbor hopping is considered in the E-SSH model. The cyclic parameter, which plays the role of a synthetic dimension, is invoked as a modulation of the hopping strengths. A rigorous analysis of different phase diagrams has shown multiple Floquet topological phase transitions among the high Chern number phases. These phase transitions can be controlled by the strength and frequency of the periodic driving. Instead of applying perturbation theory, the whole analysis is done by Floquet replica technique. This gives a freedom to study high as well as low-frequency effects on the system by considering less or more number of photon sectors. This system can be experimentally realized through a pulse sequence scheme in the optical lattice setup.
Floquet dynamical quantum phase transitions (FDQPTs) are signified by recurrent nonanalytic behaviors of observables in time. In this work, we introduce a quench-free and generic approach to engineer and control FDQPTs for both pure and mixed Floquet states. By applying time-periodic modulations with two driving frequencies to a general class of spin chain model, we find multiple FDQPTs within each driving period. The model is investigated with equal, commensurate and incommensurate driving frequencies. The nonanalytic cusps of return probability form sublattice structures in time domain. Notably, the number and time locations of these cusps can be flexibly controlled by tuning the Hamiltonian parameter and the frequencies of the drive. We further employ the dynamical topological order parameter, which shows a quantized jump whenever a DQPT happens, to identify the topological feature of FDQPTs. Our findings reveal the advantage of engineering nonequilibrium phase transitions with multifrequency driving fields.
Periodically driven quantum systems, known as Floquet systems, provide a versatile platform for engineering novel topological phases absent in static settings. However, dynamically characterizing these non-equilibrium topological invariants remains a challenge. Here, we develop a Floquet perturbation theory in the extended Hilbert space to analytically describe the center-of-mass (CoM) dynamics of a wave packet. When applied to the driven Su-Schrieffer-Heeger model, our theory reveals that the CoM exhibits multi-frequency Zitterbewegung oscillations, whose spectral composition and phase are directly tied to the system's Floquet band structure. Crucially, we find that band inversions at topological phase transitions imprint distinct signatures in the CoM dynamics, including the emergence of low-frequency modes and phase shifts of the oscillatory trajectory. These dynamical signatures offer a practical protocol for detecting Floquet topological invariants, which we demonstrate for both high-frequency and strongly driven regimes. Our work establishes CoM dynamics as a simple and experimentally accessible probe for exploring topological phase transitions in Floquet systems.
We explore the impact of commensurate multifrequency driving protocols on the stability of topological edge modes in topological one-dimensional systems of spinless fermions. Using Floquet theory, we show that all the topological phase transitions can be mapped to their single-frequency counterparts in terms of effective driving parameters. This drastic simplification can be explained by considering the gap closures in the quasienergy dispersion. These gap closures are pinned to high-symmetry points, where all the effective Floquet Hamiltonians collapse to the same form. While for all protocols all topological phase transitions coincide, the gap size and the number of edge modes in the quasienergy spectra vary considerably depending on the chosen driving pattern. This gives rise to a full range of edge states with different degrees of localization, from highly localized to completely delocalized. Switching between different driving protocols then suggests a dynamical control of the localization and stability of topological edge modes, with possible applications in quantum computation. We illustrate our findings on three paradigmatic fermionic systems, namely the Kitaev chain, the Su-Schrieffer-Heeger model, and the Creutz ladder, and demonstrate how to control the localization length of the corresponding edge states.
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.
Floquet insulators are periodically driven quantum systems that can host novel topological phases as a function of the drive parameters. These new phases exhibit features reminiscent of fermion doubling in discrete-time lattice fermion theories. We make this suggestion concrete by mapping the spectrum of a noninteracting (1+1)D Floquet insulator for certain drive parameters onto that of a discrete-time lattice fermion theory with a time-independent Hamiltonian. The resulting Hamiltonian is distinct from the Floquet Hamiltonian that generates stroboscopic dynamics. It can take the form of a discrete-time Su-Schrieffer-Heeger model with half the number of spatial sites of the original model, or of a (1+1)D Wilson-Dirac theory with one quarter of the spatial sites.