Topological invariants based on generalized position operators and application to the interacting Rice-Mele model

Abstract

We discuss different properties and the potential of several topological invariants based on position operators to identify phase transitions, and compare with more accurate methods, such as crossing of excited energy levels and jumps in Berry phases. The invariants have the form $\text{Im}\phantom{\rule{4.pt}{0ex}}\text{ln}\ensuremath{\langle}exp[i(2\ensuremath{\pi}/L){\mathrm{\ensuremath{\Sigma}}}_{j}{x}_{j}({m}_{{}_{\ensuremath{\uparrow}}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{j\ensuremath{\uparrow}}+{m}_{\ensuremath{\downarrow}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{j\ensuremath{\downarrow}})]\ensuremath{\rangle}$, where $L$ is the length of the system, ${x}_{j}$ the position of site $j$, and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{j\ensuremath{\sigma}}$ the operator of the number of particles at site $j$ with spin $\ensuremath{\sigma}$. We show that ${m}_{\ensuremath{\sigma}}$ should be integers, and in some cases of magnitude larger than 1, to lead to well-defined expectation values. For the interacting Rice-Mele model (which contains the interacting Su-Schrieffer-Heeger and the ionic Hubbard model as specific cases), we show that three different invariants give complementary information and are necessary and sufficient to construct the phase diagrams in the regions where the invariants are protected by inversion symmetry. We also discuss the consequences for pumping of charge and spin, and the effect of an Ising spin-spin interaction or a staggered magnetic field.