Topological invariants in dissipative extensions of the Su-Schrieffer-Heeger model

TL;DR

This paper explores how topological invariants like the Zak phase can be extended to dissipative systems modeled by both Lindblad equations and complex potentials, revealing similarities in spectra and topological phases.

Contribution

It introduces a generalized Zak phase for dissipative systems and compares two frameworks for modeling dissipation in the SSH model, highlighting their spectral and topological relations.

Findings

Spectral similarities between non-Hermitian Hamiltonians and Liouvillean dynamics.

Generalization of the Zak phase to dissipative scenarios.

Relation between dissipative topological phases and Hermitian counterparts.

Abstract

We investigate dissipative extensions of the Su-Schrieffer-Heeger model with regard to different approaches of modeling dissipation. In doing so, we use two distinct frameworks to describe the gain and loss of particles, one uses Lindblad operators within the scope of Lindblad master equations, the other uses complex potentials as an effective description of dissipation. The reservoirs are chosen in such a way that the non-Hermitian complex potentials are $\mathcal{PT}$-symmetric. From the effective theory we extract a state which has similar properties as the non-equilibrium steady state following from Lindblad master equations with respect to lattice site occupation. We find considerable similarities in the spectra of the effective Hamiltonian and the corresponding Liouvillean. Further, we generalize the concept of the Zak phase to the dissipative scenario in terms of the Lindblad description and relate it to the topological phases of the underlying Hermitian Hamiltonian.