Probing the Topology of Fermionic Gaussian Mixed States with {U(1)} symmetry by Full Counting Statistics

Topological band theory has been studied for free fermions for decades, and one of the most profound physical results is the bulk-boundary correspondence. Recently a focus in topological physics is extending topological classification to mixed states. Here, we focus on Gaussian mixed states where the modular Hamiltonians of the density matrix are quadratic free fermion models with {U(1)} symmetry and can be classified by topological invariants. The bulk-boundary correspondence is then manifested as stable gapless modes of the modular Hamiltonian and degenerate spectrum of the density matrix. In this article, we show that these gapless modes can be detected by the full counting statistics, mathematically described by a function introduced as {F(\theta)}. A divergent derivative at {\theta=\pi} can be used to probe the gapless modes in the modular Hamiltonian. Based on this, a topological indicator, whose quantization to unity senses topologically nontrivial mixed states, is introduced. We present the physical intuition of these results and also demonstrate these results with concrete models in both one- and two-dimensions. Our results pave the way for revealing the physical significance of topology in mixed states.