Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers

We construct a topological invariant that classifies density matrices of symmetry-protected topological orders in two-dimensional fermionic systems. As it is constructed out of the previously introduced Uhlmann phase, we refer to it as the topological Uhlmann number $n_U$. With it, we study thermal topological phases in several two-dimensional models of topological insulators and superconductors, computing phase diagrams where the temperature $T$ is on an equal footing with the coupling constants in the Hamiltonian. Moreover, we find novel thermal-topological transitions between two nontrivial phases in a model with high Chern numbers. At small temperatures we recover the standard topological phases as the Uhlmann number approaches to the Chern number.

Figure 1

Topological Uhlmann phases for the $p$-wave superconductor model as a function of the chemical potential $\mu$ and the temperature $T$. At $T=0$, the system has $\text{Ch}=+1$ for $0<\mu <2$, $\text{Ch}=-1$ for $2<\mu <4$, and $\text{Ch}=0$ otherwise. As $T$ increases, the nontrivial phases remain up to some critical temperature $T_c$ at which Uhlmann number goes from $n_U=\pm 1$ to $n_U=0$.

Figure 2

Uhlmann topological phase diagram for the model of Eq. (10). The Uhlmann number is plotted for different values of $t_2$ and $T$. The dashed line highlights the purely thermal topological transitions between regimes of $n_U=2$, $n_U=1$, and $n_U=0$ by the sole effect of increasing $T$.

Figure 3

Uhlmann topological phase diagram for the Haldane model. Red color represents $n_U=1$ and blue $n_U=-1$. As we see, at $T=0$ the two well-known lobes of the Haldane model are obtained, and at a certain finite temperature $T_c$ the system goes to a trivial phase.