Local Topological Markers in Odd Spatial Dimensions and Their Application to Amorphous Topological Matter

Abstract: Local topological markers, topological invariants evaluated by local expectation values, are valuable for characterizing topological phases in materials lacking translation invariance. The Chern marker -- the Chern number expressed in terms of the Fourier transformed Chern character -- is an easily applicable local marker in even dimensions, but there are no analogous expressions for odd dimensions. We provide general analytic expressions for local markers for free-fermion topological states in odd dimensions protected by local symmetries: a Chiral marker, a local $\mathbb Z$ marker which in case of translation invariance is equivalent to the chiral winding number, and a Chern-Simons marker, a local $\mathbb Z_2$ marker characterizing all nonchiral phases in odd dimensions. We achieve this by introducing a one-parameter family $P_{\vartheta}$ of single-particle density matrices interpolating between a trivial state and the state of interest. By interpreting the parameter $\vartheta$ as an additional dimension, we calculate the Chern marker for the family $P_{\vartheta}$. We demonstrate the practical use of these markers by characterizing the topological phases of two amorphous Hamiltonians in three dimensions: a topological superconductor ($\mathbb Z$ classification) and a topological insulator ($\mathbb Z_2$ classification).