Local time-reversal-invariant topological markers in two dimensions

Abstract

Local topological markers provide a real-space characterization of topological phases and are suited to noncrystalline models. We derive a local topological marker, the Chern-Simons descendant marker, for characterizing the topology of two-dimensional time-reversal-invariant near-Gaussian states from the one-particle density matrix \(ρ\). We construct an auxiliary three-dimensional state characterized by the local Chern-Simons marker, and use dimensional reduction to obtain the Chern-Simons descendant marker for the original two-dimensional state. This derivation introduces a local Hermitian involution \(S\) that is odd under time reversal. When \([S,ρ]=0\), the local Chern-Simons descendant marker reduces to half the difference of the local Chern markers in the \(S=\pm1\) sectors. Constructing an \(S\) that commutes exactly with \(ρ\) generally requires fine-tuning, but this is not needed---the local Chern-Simon marker remains well defined whenever \(Γ=\{ρ,S\}-S\) has a spectral gap around zero, even when \([S,ρ]\neq0\). This leaves the practical problem of identifying a suitable \(S\) for a given state, which, importantly, is straightforward even when spin is not a candidate. We show this by considering both strong Rashba and Dresselhaus spin-orbit coupling, for which knowledge of the physical origin of the state reduces the general problem of choosing $S$ to a one-parameter family \(S(θ)\). We evaluate the optimal angle $θ$ maximizing the spectral gap of $Γ$ analytically assuming translation invariance, and find that the gap of $Γ$ remains open for a broad range of angles around the optimum. The same value of \(θ\) remains valid in disordered systems, providing an easy way to obtain choice of \(S\) for evaluating the local Chern-Simons descendant marker of the state.