Uhlmann Phase as a Topological Measure for One-Dimensional Fermion Systems

We introduce the Uhlmann geometric phase as a tool to characterize symmetry-protected topological phases in one-dimensional fermion systems, such as topological insulators and superconductors. Since this phase is formulated for general mixed quantum states, it provides a way to extend topological properties to finite temperature situations. We illustrate these ideas with some paradigmatic models and find that there exists a critical temperature $T_c$ at which the Uhlmann phase goes discontinuously and abruptly to zero. This stands as a borderline between two different topological phases as a function of the temperature. Furthermore, at small temperatures we recover the usual notion of topological phase in fermion systems.

Figure 1

Comparison of the Berry and Uhlmann approaches. The usual U(1)-gauge freedom is generalized to the $U(n)$-gauge freedom of the amplitudes in the Uhlmann approach. Thus, according to Berry, after a closed loop in the set $\mathcal{Q}$, a pure state carries a simple phase factor ${\mathrm{\Phi }}_B$. However, for mixed states, the amplitude carries a unitary matrix $\mathcal{P}{e}^{\oint A_{\mathrm{U}}}$.

Figure 2

Uhlmann topological phases for the CL (a), MC (b), and SSH (c). They are $\pi$ inside the green volume and zero outside. The FBPs are indicated with an arrow and are universal. Natural units have been taken. In addition, for the CL and the MC we have fixed the horizontal hopping $2R=1$ and the superconducting pairing $|M|=1$, respectively.