Floquet quantum geometry in periodically driven topological insulators

Abstract

Quantum geometry plays a fundamental role across many branches of modern physics, yet its full characterization in nonequilibrium systems remains a challenge. Here, we propose a framework for quantum geometry in Floquet topological insulators by introducing a time-resolved quantum metric tensor, defined via the trace distance between micromotion operators in momentum-time space. For class A in two spatial dimensions, we find a general inequality linking the Floquet quantum metric tensor and the Floquet topology: the associated quantum volume is bounded below by the Floquet topological invariant. This relation is found to also hold in class AIII in one dimension, where the Floquet geometric tensor may be notably reduced due to time-reflection symmetry. This work will be useful in digesting the general aspects of quantum geometry in periodically driven systems in connection with their topological characterization.

Figures (3)

• Figure 1: (a)-(c) Quasienergy band structures of the Floquet Qi-Wu-Zhang model in the first Brillouin zone in the topological 0 phase (a), $\pi$ phase (b) and $0$-$\pi$ phase (c). (d)-(e) The snapshots of the determinants of the FQMT at different times, with (d) in the topological 0 phase and (e) in the topological $\pi$ phase with the same system parameters as in (a) and (b). Here $v_x=v_y=v_z=1$ are used as energy units and the mass terms is (a) $m=5$, (b) $m=15$ and (c) $m=15$. The driving frequency is set as (a) $\omega=8$, (b) $\omega=8$ and (c) $\omega=4$, and $T_1=0.6T$, $T_2=0.4T$.
• Figure 2: (a) The quasienergy gap for Floquet Qi-Wu-Zhang model as the function of mass term $m$. (b) The winding numbers $W_0$ and $W_\pi$ as the function of mass term $m$. (c) The Floquet quantum volume $\mathrm{vol}_0/(2\pi^2)$ and $\mathrm{vol}_\pi/(2\pi^2)$ as the function of mass term $m$. (d) The quantum volume of the effective Hamiltonian $\mathrm{vol}_{\mathrm{eff}}$ as the function of mass term $m$. Here $v_x=v_y=v_z=1$, $\omega=8$, and $T_1=0.6T$, $T_2=0.4T$.
• Figure 3: (a) The winding numbers $W_0$ and $W_\pi$ for the Floquet SSH model vs the system parameter $J/J_1$. (b) The FQV $\mathrm{vol}_0/(2\pi)$ and $\mathrm{vol}_\pi/(2\pi)$ vs $J/J_1$. (c) A typical quasienergy spectrum in the topological 0-$\pi$ phase with $J/J_1=0.75$. (d) The FQV corresponding to (c). Here other system parameters are chosen to be $q=3$, $\omega=2\pi$, $T_1=T_2$.