Anomalous Mixed-State Floquet Topology in One-Dimensional Open Quantum Systems

We investigate the non-equilibrium topology of a periodically driven, dissipative Su-Schrieffer-Heeger chain using the ensemble geometric phase (EGP) $\phi_{\mathrm{EGP}}$-a generalisation of the Zak phase to open quantum systems. In contrast to earlier work, we use Floquet-Born-Markov theory to describe the coupling to thermal reservoirs microscopically. We show that the steady state can be characterised by a Hermitian purity spectrum, providing a direct analogue of band topology for mixed states. The periodic drive induces nontrivial winding and a quasienergy spectrum with distinct $0$ and $\pi$ band gaps, with protected edge modes in each gap. We identify a pair of topological invariants $(\phi^{0}_{\mathrm{EGP}}, \Delta \phi^{\pi}_{\mathrm{EGP}})$, revealing a structure consistent with a $\mathbb{Z}\times\mathbb{Z}$ classification known from isolated Floquet SSH systems, and show how it extends to a dissipative, finite-temperature setting in regimes where the steady-state structure remains well defined. Our results demonstrate when and how known Floquet topology survives in a driven-dissipative Gaussian steady state and establish Floquet topology as a robust concept beyond isolated zero-temperature systems. The underlying formalism provides a general framework for quadratic fermionic systems with linear bath couplings.