Floquet Möbius topological insulators

TL;DR

The paper investigates a nonequilibrium extension of Möbius topological insulators by constructing a 2D $\pi$-flux PQSSH model driven with periodic quenches. The edge spectrum hosts Möbius twists around $E=0$ and/or $E=\pi$, protected by chiral symmetry and $\mathbb{Z}_{2}$-PTS, and quantified by a pair of integers $$(\omega_{0},\omega_{\pi})$$ that dictate edge-band counts via $(N_{0},N_{\pi})=2(|\omega_{0}|,|\omega_{\pi}|)$. The study combines bulk topology analysis at a high-symmetry momentum, symmetry-based reductions to a 1D PQSSH problem, and numerical demonstrations of quasienergy spectra, entanglement spectra, and adiabatic edge-state dynamics, revealing gapped and gapless Floquet Möbius phases with unique zero/$\pi$-twisted edge states absent in static systems. The results broaden Möbius topology to nonequilibrium settings and point to experimental routes and potential quantum-information applications using adiabatic edge-state manipulation.

Abstract

Möbius topological insulators have dispersive edge bands with Möbius twists in momentum space, which are protected by the combination of chiral and $Z_2$-projective translational symmetries. In this work, we reveal a unique type of Möbius topological insulator, whose edge bands could twist around the quasienergy $π$ of a periodically driven system and are thus of Floquet origin. By applying time-periodic quenches to an experimentally realized Möbius insulator model, we obtain interconnected Möbius edge bands around zero and $π$ quasienergies, which can coexist with a gapped or gapless bulk. These Möbius bands are topologically characterized by a pair of generalized winding numbers, which are integer-quantized due to 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 Möbius topological phases. A protocol based on the adiabatic switching of edge-band populations is further introduced to dynamically characterize the topology of Floquet Möbius edge bands. Our findings thus extend the scope of Möbius topological phases to nonequilibrium settings and unveil a unique class of Möbius-twisted topological edge states without static counterparts.

Figures (10)

• Figure 1: Illustration of the PQSSH model and its 2D extension. (a) shows an SSH chain (lying along the $x$ direction at each time $t$) under time-periodic quenches over two driving periods ($t=0\rightarrow2T$). The $J_{1}$ and $J_{2}$ denote intracell and intercell hopping amplitudes. The red and green dots denote sublattices A and B. (b) shows a 2D extension of the SSH model with a $\pi$ magnetic flux per plaquette. The orange dashed line encircles a unit cell with four sublattices A, B, C and D. (c) and (d) show the 2D $\pi$-flux PQSSH model in its first and second half of a evolution period, respectively. We set the intracell and intercell hopping amplitudes along the $x$ direction in (b)--(d) to $J_{x}=J'_{x}=J$ throughout the paper.
• Figure 2: Topological phase diagram of the PQSSH model, with winding numbers $(w_{0},w_{\pi})$ denoted explicitly in each gapped phase. The nine data sets used in the calculation of Fig. \ref{['fig:PQSSH2']} are highlighted by different symbols in the phase diagram. The solid and dotted lines are trivial critical lines, with quasienergy gap closes at $E=0$ and $E=\pi$, respectively. The dashed and dash-dotted lines are topological critical lines, with quasienergy gap closes at $E=\pi$ and $E=0$, respectively.
• Figure 3: Floquet spectrum of the PQSSH model under the OBC. The values of $(J_{1},J_{2})$ are given in the caption of each panel. $20$ unit cells are used in each calculation. $n$ is the state index. The color of each data point is given by the inverse participation ratio of the corresponding state. The cases in (b), (d) and (f) correspond to Floquet topological insulator phases, with system parameters taken at the $\blacktriangle$, $\blacktriangleleft$ and $\blacktriangleright$ in Fig. \ref{['fig:PQSSH1']}. The case in (h) corresponds to a trivial insulator phase, with system parameters taken at the $\blacktriangledown$ in Fig. \ref{['fig:PQSSH1']}. The cases in (a) and (c) correspond to topologically nontrivial critical points, with system parameters taken at the $\bigstar$ and six-pointed star in Fig. \ref{['fig:PQSSH1']}. The cases in (e), (g) and (i) correspond to topologically trivial critical points, with system parameters taken at the $\CIRCLE$, $\blacksquare$ and $\blacklozenge$ in Fig. \ref{['fig:PQSSH1']}.
• Figure 4: The quasienergy spectrum $E(k_{x})$ of FMTIs, obtained under the PBC (OBC) along the $x$ ($y$) direction and with $N_{y}=50$ unit cells along $y$. Gray lines, blue stars and red circles denote bulk states, left-localized edge states and right-localized edge states along the $y$ direction. The hopping amplitude along the $x$ direction is $J=0.1\pi$ for all panels. The hopping amplitudes $(J_{1},J_{2})$ along the $y$ direction are given in each figure panel. (a) exemplifies a trivial phase, with no Floquet Möbius edge bands. (b) displays a Floquet $0$-Möbius topological insulator, with a pair of Möbius edge bands twisting around the quasienergy $E=0$. (c) showcases a Floquet $\pi$-Möbius topological insulator, with a pair of Möbius edge bands twisting around the quasienergy $E=\pi$. (d) represents a Floquet $0\pi$-Möbius topological insulator, with two pairs of Möbius edge bands twisting separately around the quasienergies $E=0$ and $E=\pi$.
• Figure 5: The quasienergy spectrum $E(k_{x})$ of the 2D $\pi$-flux PQSSH model at gapless critical points, obtained under the PBC (OBC) along the $x$ ($y$) direction and with $N_{y}=50$ unit cells along $y$. Gray lines, blue stars and red circles denote bulk states, left-localized edge states and right-localized edge states along the $y$ direction. The hopping amplitude along the $x$ direction is $J=0.1\pi$ for all panels. The hopping amplitudes $(J_{1},J_{2})$ along the $y$ direction are given in the caption of each figure panel. (a) and (b) represent trivial critical points, with Floquet bulk bands touching at $E=0$ and $E=\pi$, respectively. (c) displays a topological critical point, with bulk Floquet bands touching at $E=\pi$ and Möbius edge bands retaining around $E=0$. (d) shows another topological critical point, with bulk Floquet bands touching at $E=0$ and Möbius edge bands interconnecting around $E=\pi$.