Diophantine equation for the Rice-Mele model: Topological aspect of filling numbers and associated spatial pump

TL;DR

This paper extends the Rice-Mele model with a long-period spatial modulation to explore topological charge pumping, deriving a Diophantine equation linking fermion and pumped charges, and demonstrating a 1D analog of the quantum Hall Streda formula.

Contribution

It introduces a novel long-period modulation in the Rice-Mele model and derives a Diophantine equation connecting fermion charge and pumped charge, revealing a 1D topological charge pump.

Findings

Derivation of a Diophantine equation relating fermion and pumped charges.

Identification of a 1D analog of the Streda formula in the Rice-Mele model.

Proposal for direct measurement of the Streda formula in 1D systems.

Abstract

We introduce a long-period generic spatial modulation into a typical model of the Thouless pump, namely, the Rice--Mele (RM) model, to examine the lattice analog of the fermion charge in quantum field theory. We derive a Diophantine equation relating the fermion charge and the pumped charge, which leads to the one-dimensional (1D) analog of the Streda formula in the quantum Hall effect (QHE). This formula implies that an adiabatic change of the periodicity of the spatial modulation yields a spatial charge pump such that the rightmost charge is pumped to the right by the Chern number compared with the leftmost charge. This causes a change in the length of the fermion chain by an integer, thus providing the opportunity for direct measurement of the Streda formula in 1D systems.