# Finite-temperature topological invariant for interacting systems

## TL;DR

This paper extends the Ensemble Geometric Phase to finite-temperature interacting 1D systems, establishing a topological invariant that remains quantized below a critical temperature and demonstrating its robustness through numerical simulations of a Bose-Hubbard model.

## Contribution

It introduces a finite-temperature topological invariant for interacting 1D systems that generalizes previous zero-temperature invariants, applicable even with degeneracies and fractional fillings.

## Key findings

- The finite-temperature invariant matches the zero-temperature invariant below a critical temperature.
- Numerical simulations show the invariant remains quantized despite non-quantized particle transport at higher temperatures.
- The approach applies to systems with degenerate ground states and fractional fillings.

## Abstract

We generalize the Ensemble Geometric Phase (EGP), recently introduced to classify the topology of density matrices, to finite-temperature states of interacting systems in one spatial dimension (1D). This includes cases where the gapped ground state has a fractional filling and is degenerate. At zero temperature the corresponding topological invariant agrees with the well-known invariant of Niu, Thouless and Wu. We show that its value at finite temperatures is identical to that of the ground state below some critical temperature $T_c$ larger than the many-body gap. We illustrate our result with numerical simulations of the 1D extended super-lattice Bose-Hubbard model at quarter filling. Here a cyclic change of parameters in the ground state leads to a topological charge pump with fractional winding $\nu=1/2$. The particle transport is no longer quantized when the temperature becomes comparable to the many-body gap, yet the winding of the generalized EGP is.