  Complex band structure generalizes conventional band structure by also
considering wavevectors with complex components. In this way, complex band
structure describes both the bulk-propagating states from conventional band
structure and the evanescent states that grow or decay from one unit cell to
the next. Even though these latter states are excluded by translational
symmetry, they become important when translational symmetry is broken via, for
example, a surface or impurity.
  Many studies over the last 80 years have directly or indirectly developed
complex band structure for an impressive range of applications, but very few
discuss its fundamentals or compare its various results. In this work we build
upon these previous efforts to expose the physical foundation of complex band
structure, which mathematically implies the existence of complex band
structure. We find that a material's static and dynamic electronic structure
are both completely described by complex band structure. Furthermore, we show
that complex band structure reflects the minimal, intrinsic information
contained in the material's Hamiltonian. These realizations then provide a
context for discussing, comparing, and unifying the different formulations and
applications of complex band structure that have been developed over the years.
Ultimately, this discussion introduces the idea of examining the amount of
information contained in a material's Hamiltonian so that we can find and
exploit the minimal information necessary for understanding a material's
properties.
