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.

中文翻译：

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Rice-Mele 模型的丢番图方程：填充数和相关空间泵的拓扑方面

我们将长周期通用空间调制引入到 Thouless 泵的典型模型中，即 Rice-Mele (RM) 模型，以检查量子场论中费米子电荷的晶格模拟。我们推导出与费米子电荷和泵浦电荷相关的丢番图方程，这导致量子霍尔效应 (QHE) 中 Streda 公式的一维 (1D) 模拟。该公式意味着空间调制周期性的绝热变化产生空间电荷泵，使得与最左边的电荷相比，最右边的电荷通过陈数向右泵送。这会导致费米子链的长度发生一个整数变化，从而为在 1D 系统中直接测量 Streda 公式提供了机会。