Necessary and sufficient conditions for flat bands in M-dimensional N-band lattices with complex-valued nearest-neighbour hopping. (10th December 2018)

• Record Type: Journal Article
• Authors: Toikka, L A; Andreanov, A
• Is Part Of: Journal of physics. Volume 52:Number 2(2019)
• Publication Date: 2018-12-10
• Subjects: lattice models in condensed matter -- electronic structure -- dispersionless band -- flat band generator; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/aaf25c
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We formulate the necessary and sufficient conditions for the existence of dispersionless energy eigenvalues (so-called 'flat bands') and their associated compact localized eigenstates in -dimensional tight-binding lattices with sites per unit cell and complex-amplitude nearest-neighbour tunneling between the lattice sites. The degrees of freedom can be traded for longer-range complex hopping in lattices with reduced dimensionality. We show the conditions explicitly for, , and, and outline their systematic construction for arbitrary, . If and only if the conditions are satisfied, then the system has one or more flat bands. By way of an example, we obtain new classes of flat band lattice geometries by solving the conditions for the lattice parameters in special cases.