Topology of density matrices

We investigate the topological properties of density matrices, motivated by the question to what extent phenomena such as topological insulators and superconductors can be generalized to mixed states in the framework of open quantum systems. The notion of geometric phases has been extended from pure to mixed states by Uhlmann [Rep. Math. Phys. 24, 229 (1986)], who discovered an emergent gauge theory over the density matrices based on their pure state representation in a larger Hilbert space. However, since the uniquely defined square root $\sqrt{\rho }$ of a density matrix $\rho$ provides a global gauge, this construction is always topologically trivial. Here, we study a more restrictive gauge structure which can be topologically nontrivial and is capable of resolving homotopically distinct mappings of density matrices subject to various spectral constraints. Remarkably, in this framework, topological invariants can be directly defined and calculated for mixed states. In the limit of pure states, the well-known system of topological invariants for gapped band structures at zero temperature is reproduced. We compare our construction with recent approaches to Chern insulators at finite temperature.

Figure 1

Momentum dependence of the Uhlmann phase for a thermal state at $\beta =1.3$ for the model defined in Eq. (27), numerically obtained by discretizing the paths in momentum space into 500 equidistant points. Left panel: ${\varphi }_U^{k_x}$ as a function of $k_x$. Right panel: ${\varphi }_U^{k_y}$ as a function of $k_y$.