Almost Strong Zero Modes at Finite Temperature

TL;DR

This paper investigates the behavior of Almost Strong Zero Modes in interacting fermionic chains at finite temperatures, revealing an exponential dependence of decay rates on inverse temperature and introducing advanced simulation techniques.

Contribution

It introduces a novel approach combining Lanczos series expansion and tensor network methods to study operator dynamics at finite temperature.

Findings

Decay rate depends exponentially on inverse temperature

Effective energy scale exceeds the thermodynamic gap

Method enables simulation of large systems over long times

Abstract

Interacting fermionic chains exhibit extended regions of topological degeneracy of their ground states as a result of the presence of Majorana or parafermionic zero modes localized at the edges. In the opposite limit of infinite temperature, the corresponding non-integrable spin chains, obtained via generalized Jordan-Wigner mapping, are known to host so-called Almost Strong Zero Modes, which are long-lived with respect to any bulk excitations. Here, we study the fairly unexplored territory that bridges these two extreme cases of zero and infinite temperature. We blend two established techniques for states, the Lanczos series expansion and a tensor network ansatz, uplifting them to the level of operator algebra. This allows us to efficiently simulate large system sizes for arbitrarily long timescales and to extract the temperature-dependent decay rates. We observe that for the Kitaev-Hubbard model, the decay rate of the edge mode depends exponentially on the inverse temperature $\beta$, and on an effective energy scale $\Delta_{\rm eff}$ that is greater than the thermodynamic gap of the system $\Delta$.