Topological characterization of one-dimensional open fermionic systems

A topological measure characterizing symmetry-protected topological phases in one-dimensional open fermionic systems is proposed. It is built upon the kinematic approach to the geometric phase of mixed states and facilitates the extension of the notion of topological phases from zero-temperature to nonzero-temperature cases. In contrast to a previous finding that topological properties may not survive above a certain critical temperature, we find that topological properties of open systems, in the sense of the measure suggested here, can persist at any finite temperature and disappear only in the mathematical limit of infinite temperature. Our result is illustrated with two paradigmatic models of topological matter. The bulk topology at nonzero temperatures manifested as robust mixed edge state populations is examined via two figures of merit.