Band connectivity#

Refs. [BEV+18, VEW+17]

We consider to “sew” a band structure between neighboring \(\mathbf{k}\)-vectors.

Problem statement#

Let \(\mathbf{G}\) be a space group. We call the largest continuous submanifold of reciprocal space that the little co-group of every point in the manifold is isomorphic to each other as \(\mathbf{k}\)-manifold. We say that two \(\mathbf{k}\)-manifolds are connected if the two \(\mathbf{k}\)-manifolds have a common \(\mathbf{k}\)-vector with some specific free parameters 1. We call a manifold of \(\mathbf{k}\)-vector is of maximal symmetry if its little co-group is not a proper subgroup of \(\mathbf{k}\)-vectors in the connected manifolds. We refer to a \(\mathbf{k}\)-vector in \(\mathbf{k}\)-manifold of maximal symmetry as maximal \(\mathbf{k}\)-vector.

For a \(\mathbf{k}\)-vector, we consider a pair of irrep formed by eigenvectors and corresponding eigenvalue as a node of a graph:

\[ \mathcal{N}_{\mathbf{k}} = \{ (\Gamma^{\mathbf{k}\alpha}, \lambda) \mid \mbox{some eigenvectors with eigenvalue $\lambda$ form irrep $\Gamma^{\mathbf{k}\alpha}$} \}.\]

Given two sets of nodes \(\mathcal{N}_{\mathbf{k}_{1}}\) and \(\mathcal{N}_{\mathbf{k}_{2}}\) from two connected \(\mathbf{k}\)-manifolds, these nodes are linked via \(\mathcal{N}_{\mathbf{k}_{p}}\) where a \(\mathbf{k}\)-manifold containing \(\mathbf{k}_{p}\) is connected to both \(\mathbf{k}_{1}\) and \(\mathbf{k}_{2}\).

Two nodes \((\Gamma^{\mathbf{k}_{i}\alpha}, \lambda) \in \mathcal{N}_{\mathbf{k}_{i}}\) and \((\Gamma^{\mathbf{k}_{p}\beta}, \lambda') \in \mathcal{N}_{\mathbf{k}_{p}}\) are connected if the subduced representation of \(\Gamma^{\mathbf{k}_{i}\alpha}\) contains \(\Gamma^{\mathbf{k}_{p}\beta}\),

\[ \Gamma^{\mathbf{k}_{i}\alpha} \downarrow \mathcal{G}^{\mathbf{k}_{p}} \simeq \bigoplus_{\beta} m_{\beta} \Gamma^{\mathbf{k}_{\mathbf{p}}\beta} \quad (i=1,2).\]

References#

BEV+18(1,2)

Barry Bradlyn, L. Elcoro, M. G. Vergniory, Jennifer Cano, Zhijun Wang, C. Felser, M. I. Aroyo, and B. Andrei Bernevig. Band connectivity for topological quantum chemistry: band structures as a graph theory problem. Phys. Rev. B, 97:035138, Jan 2018. URL: https://link.aps.org/doi/10.1103/PhysRevB.97.035138, doi:10.1103/PhysRevB.97.035138.

VEW+17

M. G. Vergniory, L. Elcoro, Zhijun Wang, Jennifer Cano, C. Felser, M. I. Aroyo, B. Andrei Bernevig, and Barry Bradlyn. Graph theory data for topological quantum chemistry. Phys. Rev. E, 96:023310, Aug 2017. URL: https://link.aps.org/doi/10.1103/PhysRevE.96.023310, doi:10.1103/PhysRevE.96.023310.

1

For our aim, this definition does not consider reciprocal lattice translations unlike Ref. [BEV+18].