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

Abstract

We discuss different properties and the ability of several topological invariants based on position operators to identify phase transitions, and compare with more accurate methods, like crossing of excited energy levels and jumps in Berry phases. The invariants have the form $\text{Im}\text{ln}⟨\exp \left[i(2\pi /L){\Sigma }_j{x}_j\left(m_{{}_{\uparrow }}{\hat{n}}_{j\uparrow }+m_{\downarrow }{\hat{n}}_{j\downarrow }\right)\right]⟩$, where $L$ is the length of the system, $x_j$ the position of the site $j$, ${\hat{n}}_{j\sigma }$ the operator of the number of particles at site $j$ with spin $\sigma$. We show that $m_{\sigma }$ 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.

arXiv comment12 pages, 7 figures, some references added, printing errors corrected

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• Robustness of topological order against disorder
• Topological phase transitions at finite temperature
• Local Chern Marker for Periodic Systems
• Charge and spin gaps of the ionic Hubbard model with density-dependent hopping
• Phase diagram and topology of the XXZ chain with alternating bonds and staggered magnetic field
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• Quantum transport phenomena induced by time-dependent fields
• Identification of gapless phases by a twisting operator
• Absorbance marker: Detection of quantum geometry and spread of Wannier function in disordered 2D semiconductors
• Diagnosing energy gap in quantum spin liquids via polarization amplitude
• Anisotropy-driven topological quantum phase transition in magnetic impurities