Local time-reversal-invariant topological markers in two dimensions

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 $\rho$. 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 $[S,\rho ]=0$, the local Chern-Simons descendant marker reduces to half the difference of the local Chern markers in the $S=\pm 1$ sectors. Constructing an $S$ that commutes exactly with $\rho$ generally requires fine-tuning, but this is not needed—the local Chern-Simon marker remains well defined whenever $\mathrm{\Gamma }=\{\rho ,S\}-S$ has a spectral gap around zero, even when $[S,\rho ]\mathrm{\ne }0$. 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 $S(\theta )$. We evaluate the optimal angle $\theta$ maximizing the spectral gap of $\mathrm{\Gamma }$ analytically assuming translation invariance, and find that the gap of $\mathrm{\Gamma }$ remains open for a broad range of angles around the optimum. The same value of $\theta$ 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.