Universal topological marker

TL;DR

This paper introduces a universal topological marker for topological insulators and superconductors, enabling lattice site mapping of topological order across various models and symmetry classes, with implications for understanding phase transitions.

Contribution

It develops a universal topological operator and marker applicable to all symmetry classes and dimensions, connecting momentum-space invariants to real-space lattice site characterization.

Findings

Topological marker explicitly constructed for all symmetry classes from 1D to 3D.

Nonlocal topological marker decays with a correlation length diverging at phase transitions.

Demonstrated on multiple models including SSH, Majorana chain, and 3D topological insulators.

Abstract

We elaborate that for topological insulators and topological superconductors described by Dirac models in any dimension and symmetry class, the topological order can be mapped to lattice sites by a universal topological marker. Deriving from a recently discovered momentum-space universal topological invariant, we introduce a topological operator that consists of alternating projectors to filled and empty lattice eigenstates and the position operators, multiplied by the Dirac matrices that are omitted in the Hamiltonian. The topological operator projected to lattice sites yields the topological marker, whose form is explicitly constructed for every topologically nontrivial symmetry class from 1D to 3D. The off-diagonal elements of the topological operator yields a nonlocal topological marker, which decays with a correlation length that diverges at topological phase transitions, and represents a Wannier state correlation function. Various prototype examples, including Su-Schrieffer-Heeger model, Majorana chain, Chern insulators, Bernevig-Hughes-Zhang model, 2D chiral and helical $p$-wave superconductors, lattice model of $^{3}$He B-phase, and 3D time-reversal symmetric topological insulators, etc, are employed to demonstrate the ubiquity of our formalism.