Proxy ensemble geometric phase and proxy index of time-reversal invariant topological insulators at finite temperatures

The ensemble geometric phase (EGP) has been proposed as a topological indicator for finite-temperatures systems. Wilson loop, or the transfer matrix, contains crucial information in EGP construction. We propose proxy index and directly from matrix apply them to time-reversal invariant insulators exemplified by Bernevig-Hughes-Zhang (BHZ) Kane-Mele (KM) models. quantized smoothly generalize ground-state finite temperatures. For BHZ model, comparison with another indicator, Uhlmann phase, shows different transition behavior temperature. KM have generalized previously, but does not require any splitting of contributions. thus offer an efficient means characterizing finite-temperature properties.