Upper bound of a band complex

The band structure for a crystal generally consists of connected components in energy-momentum space, known as band complexes. Here, we explore a fundamental aspect regarding the maximal number of bands that can be accommodated in a single band complex. We show that, in principle, a band complex can have no finite upper bound for certain space groups. This means infinitely many bands can entangle together, forming a connected pattern stable against symmetry-preserving perturbations. This is demonstrated by our developed inductive construction procedure, through which a given band complex can always be grown into a larger one by gluing a basic building block to it. As a by-product, we demonstrate the existence of arbitrarily large accordiontype band structures containing N-C = 4n bands, with n is an element of N.