Micromotion area as proxy for anomalous Floquet topological systems

Abstract

Driven Floquet systems can realize topological phases with no static counterparts. These so-called anomalous Floquet topology breaks the bulk-boundary correspondence based on the Chern number. The number of edge modes in each band gap is instead determined by another integer index, a winding number, which is calculated from the time evolution operator of the bulk states within one driving period. While in the non-driven system, Chern markers provide a useful local proxy for the Chern number in the bulk, so far no such local bulk indicator is known for the winding number in Floquet systems. Here we consider two-band models and show that the area enclosed during a Floquet period by an initially localized particle signals the presence of an anomalous phase when it approaches half the unit cell area. In general, we show that at the fine-tuned point of dispersionless dynamics during the micromotion, the enclosed area is quantized and an exact proportionality relation exists between the area and the winding number. Direct detection of anomalous topology in real space could be realized in several quantum simulation platforms, and could be useful for systems with disorder or interactions. Building on the connection between area and winding number, we also show a way to realize arbitrarily high winding numbers.

Figures (3)

• Figure 1: Bulk observables for Chern systems and anomalous systems a) In static systems, the Chern number $C$ determines the number of chiral edge modes (right panel) and can be detected in the bulk by the quantized transverse conductance (left panel). b) In anomalous Floquet systems, the edge modes are given by the winding numbers $W_g$ for each band gap $g$. We show that - in the case of dispersionless micromotion dynamics - the micromotion area $A$ is also quantized and indicates the winding numbers while in the more general case, a finite micromotion area can be used to infer the presence of the anomalous phase.
• Figure 2: Micromotion area as a function of $\omega$ and $\Delta$. a) Tunneling in a honeycomb lattice is completely switched on and off in a cyclic way $J_1\xrightarrow{}J_2\xrightarrow{}J_3\xrightarrow{}...$ with frequency $\omega$. A constant sublattice offset $\Delta$ is present. We plot the area calculated across the phase diagram. Red dots mark topological phase transition points, and the brackets indicate $[W_0,W_\pi]$ (calculated in Martinez2023). b) Area (blue curve) as a function of the Floquet frequency for $\Delta=0$. An additional anomalous phase appears for even lower frequency drives Wintersperger2020. The anomalous phases are indicated by the purple shade. The red (magenta) curve shows the area for two (five) Floquet periods. c) Area as a function of the Floquet frequency (right) in the cyclic protocol (left) in the bipartite square lattice as introduced in Rudner2013. The $0$ gap is always closed for these protocols, and only the value of $W_\pi$ is noted. Also here anomalous phases for smaller driving frequencies are present (purple shades).
• Figure 3: Protocols for higher winding numbers. a) Dynamics in the bulk for a protocol where tunneling couplings $J_i$ are alternated according to 1313212. The unit of length is the lattice vector $a_{AA}$. The driving frequency $\omega$ is chosen to be perfectly resonant, and a deviation from the fine-tuned condition is given by a $\Delta=0.221J$. The red shading indicates the area enclosed during one Floquet period at the associated fine-tuned point. The right panel shows the spectrum in a cylinder geometry with gray color for bulk states and purple color for states residing on one of the edges (the states on the other edge of the system are not shown). The spectrum indicates the winding numbers $W_0=W_\pi=4$, which matches with the red area $A=2A_u=\frac{W_{0,\pi}}{2}A_u$. The green shading indicates the area enclosed during the second Floquet period. b) Same for the driving sequence 131321213232. Here $\omega$ is 0.92 of the resonant value, and $\Delta=0$. While the $0$ gap is closed, the spectrum exhibits $W_{\pi}=2\frac{A}{A_u}=6$ and a corresponding red shaded area $A=3A_u=\frac{W_\pi}{2}A_u$.