Uhlmann Connection in Fermionic Systems Undergoing Phase Transitions

We study the behavior of the Uhlmann connection in systems of fermions undergoing phase transitions. In particular, we analyze some of the paradigmatic cases of topological insulators and superconductors in one dimension, as well as the BCS theory of superconductivity in three dimensions. We show that the Uhlmann connection signals phase transitions in which the eigenbasis of the state of the system changes. Moreover, using the established fidelity approach and the study of the edge states, we show the absence of thermally driven phase transitions in the case of topological insulators and superconductors. We clarify what is the relevant parameter space associated with the Uhlmann connection so that it signals the existence of order in mixed states. In addition, the study of Majorana modes at finite temperature opens the possibility of applications in realistic stable quantum memories. Finally, the analysis of the different behavior of the BCS model and the Kitaev chain, with respect to the Uhlmann connection, suggested that in realistic scenarios the gap of topological superconductors could also, generically, be temperature dependent.

Figure 1

The fidelity for thermal states $\rho$, when probing the parameter of the Hamiltonian that drives the topological PT $\delta M=M^{\prime }-M=0.01$ (left), and the temperature $\delta T=T^{\prime }-T=0.01$ (center), and the Uhlmann connection, when probing the parameter of the Hamiltonian $M$ (right), for the Creutz ladder model (representative of the symmetry class AIII). The plot for $\mathrm{\Delta }$ when deforming the thermal state along $T$ is omitted since it is equal to zero everywhere.

Figure 2

The fidelity for thermal states $\rho$ when probing the parameter of the Hamiltonian $\delta V=V^{\prime }-V={10}^{-3}$ (left) and the temperature $\delta T=T^{\prime }-T={10}^{-3}$ (center left), and the Uhlmann connection (center right and right, respectively), for BCS superconductivity.