Characterizing topology at nonzero temperature: Topological invariants and indicators in the extended SSH model

TL;DR

This paper compares various topological diagnostics for the SSH model at nonzero temperature, introducing local indicators and a generalized chiral marker to effectively identify topological phases in mixed states.

Contribution

It introduces local twist operators and a generalized local chiral marker as new tools for detecting topology in mixed Gaussian states at finite temperature.

Findings

The ensemble geometric phase remains well defined but less practical at large scales.

Local twist operators serve as effective local indicators of topological phases.

The generalized chiral marker aligns with the winding number at zero temperature.

Abstract

We compare three complementary diagnostics for mixed Gaussian states at nonzero temperature, focusing on the Su-Schrieffer-Heeger (SSH) chain and its inversion-symmetric extension. Whilst the ensemble geometric phase, a mixed-state generalization of the Zak phase, remains well defined at nonzero temperature, the modulus of the corresponding expectation value vanishes in the thermodynamic limit, limiting its practical use. To develop diagnostics suitable for large systems, we introduce local twist operators acting on neighboring sites, whose expectation values provide local indicators of the underlying topological phase. The topological phase is identified from the relative magnitude of these expectation values, which only requires measuring two local expectation values at nonzero temperature, together with one additional nonlocal expectation value when next-nearest-neighbor hopping is included. In addition, we generalize the local chiral marker to mixed Gaussian states, fully determined by its single-particle correlation matrix, with a nonzero purity gap in their effective single-particle Hamiltonian. The presence of a purity gap ensures that the correlation matrix can be flattened to an effective projector. Evaluating the chiral marker with respect to the band-flattened correlation matrix yields a real-space topological invariant that coincides with the winding number in the zero-temperature limit. The ensemble geometric phase, the local twist operators, and the local chiral marker provide complementary methods to characterize topology in the SSH chain beyond pure states.