Topological footprints of the Kitaev chain with long-range superconducting pairings at a finite temperature

We study the one-dimensional Kitaev chain with long-range superconductive pairing terms at a finite temperature where the system is prepared in a mixed state in equilibrium with a heat reservoir maintained at a constant temperature $T$. In order to probe the footprint of the ground-state topological behavior of the model at finite temperature, we look at two global quantities extracted out of two geometrical constructions: the Uhlmann and the interferometric phase. Interestingly, when the long-range effect dominates, the Uhlmann phase approach fails to reproduce the topological aspects of the model in the pure-state limit; on the other hand, the interferometric phase which has a proper pure state reduction, shows a behavior independent of the ambient temperature.

Figure 1

Phase diagram of the LRK chain in the $\mu \text{−}\alpha$ plane: for $\alpha >3/2$ the phase diagram of this model is topologically equivalent to the short-range Kitaev chain, whereas for $\alpha <1$ the model hosts massive Dirac edge modes for $\mu <1$ and is characterized by a half-integer winding number. There is a crossover phase in between (for $1<\alpha <3/2$) with no well-defined winding number.