Phase structure of one-dimensional interacting Floquet systems. II. Symmetry-broken phases

Recent work suggests that a sharp definition of “phase of matter” can be given for periodically driven “Floquet” quantum systems exhibiting many-body localization. In this work, we propose a classification of the phases of interacting Floquet localized systems with (completely) spontaneously broken symmetries; we focus on the one-dimensional case, but our results appear to generalize to higher dimensions. We find that the different Floquet phases correspond to elements of $Z\left(G\right)$, the center of the symmetry group in question. In a previous paper [C. W. von Keyserlingk and S. L. Sondhi, preceding paper, Phys. Rev. B 93, 245145 (2016)], we offered a companion classification of unbroken, i.e., paramagnetic phases.

Figure 1

This shows the phase diagram for the binary drive in Eq. (1). The red and blue lines separate distinct Floquet phases. The lists involving $(u,p)$ summarize the protected multiplets in the spectrum for an open chain, e.g., in the $0\pi$ trivial phase, if there is a state with $U_f,P$ eigenvalues $(u,p)$, then there are guaranteed to be states at $(-u,p),(u,-p),(-u,-p)$ for the same $u$, up to exponentially small corrections in system size.