Quantum collision theory in flat bands

TL;DR

This paper investigates quantum scattering in flat and linearly dispersed bands, revealing that scattering is suppressed and replaced by projections onto eigenstates, with implications for localized states and special solutions.

Contribution

It introduces a general framework for understanding scattering in flat bands, showing that transition matrices vanish and scattering is replaced by projections, with applications to localized states and specific lattice models.

Findings

Scattering does not occur in flat bands due to vanishing transition matrices.

Localized flat band eigenstates can be constructed with impurity potentials.

Relations between 'strange' solutions and linearly dispersed systems are established.

Abstract

We consider quantum scattering of particles in media exhibiting strong dispersion degeneracy. In particular, we study flat-banded lattices and linearly dispersed energy bands. The former constitute a prime example of single-particle frustration while the latter show degeneracy at the few- and many-particle level. We investigate both impurity and two-body scattering and show that, quite generally, scattering does not occur, which we relate to the fact that transition matrices vanish on the energy shell. We prove that scattering is instead replaced by projections onto band-projected eigenstates of the interaction potential. We then use the general results to obtain localised flat band states that are eigenstates of impurity potentials with vanishing eigenvalues in one-dimensional flat bands and study the particular case of a sawtooth lattice. We also uncover the relation between certain solutions of one-dimensional systems that have been categorised as "strange", and the scattering states in linearly dispersed continuum systems.