Quantum collision theory in flat bands

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.

7 pages, 0 figures. Updated references

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Cited by in corpus (7)

• Quantum metric and effective mass of a two-body bound state in a flat band
• Bose-Fermi dualities for arbitrary one-dimensional quantum systems in the universal low energy regime
• Flat bands and dynamical localization of spin-orbit coupled Bose-Einstein condensates
• Flat bands and compactons in mechanical lattices
• Pairs, trimers and BCS-BEC crossover near a flat band: the sawtooth lattice
• K-matrix formulation of two-particle scattering in a wave guide in the presence of one-dimensional spin-orbit coupling
• From few to many body degrees of freedom