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.

12 pages, 7 figures, some references added, printing errors corrected

References in corpus (13)

• Classification of topological insulators and superconductors in three spatial dimensions
• The space group classification of topological band insulators
• Thouless pumping and topology
• Topological invariants and interacting one-dimensional fermionic systems
• Spin pumping and measurement of spin currents in optical superlattices
• Splitting of topological charge pumping in an interacting two-component fermionic Rice-Mele Hubbard model
• Phase diagrams of one-dimensional half-filled two-orbital SU(N) cold fermions systems
• Scaling and renormalization in the modern theory of polarization: application to disordered systems
• Interacting second-order topological insulators in one-dimensional fermions with correlated hopping
• Interaction driven polarization shift in the $t-V-V^{\prime }$ lattice fermion model at half filling: emergent Haldane phase
• Ferrotoroidic Moment as a Quantum Geometric Phase
• Geometric cumulants associated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator transitions in crystalline electronic systems
• Phase diagram of a model for topological superconducting wires

Cited by in corpus (13)

• Interactions enable Thouless pumping in a nonsliding lattice
• 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
• Topological Phases in Fractals: Local Spin Chern Marker in the Sierpinski carpet Kane-Mele-Rashba Model
• Topological characterization of modified Kane-Mele-Rashba models via local spin Chern marker
• 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