Topology and Dynamical Liquid Crystallinity in Optically Driven Two-Dimensional Materials

“Floquet engineering” of band structures through the application of coherent time-periodic drives has recently emerged as a powerful tool for creating new types of topological phases. In this work, we show that this tool can also be used to induce non-equilibrium correlated states with dynamical spontaneously broken symmetry. We study how such phenomena can arise in periodically driven systems featuring a ring-like minimum in their Floquet dispersion. In particular, we focus on lightly doped semiconductors driven by a resonant driving field. We show that such a system can spontaneously develop quantum liquid crystalline order featuring extreme anisotropy whose directionality rotates as a function of time. The phase transition to this correlated state occurs in the steady state of the system achieved due to the interplay between the coherent external drive, electron-electron interactions, and dissipative processes arising from the coupling to phonons and the electromagnetic environment. Our results demonstrate how Floquet engineering can be used to induce novel non-equilibrium phases exhibiting an interplay of topology and dynamical symmetry breaking.