Necessary and sufficient conditions for flat bands in $M$-dimensional $N$-band lattices with complex-valued nearest-neighbour hopping

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 $M$-dimensional tight-binding lattices with $N$ sites per unit cell and complex-amplitude nearest-neighbour tunneling between the lattice sites. The degrees of freedom $M$ can be traded for longer-range complex hopping in lattices with reduced dimensionality. We show the conditions explicitly for $(M=1,N\le 4)$, $(M=2,N=2,3)$, and $(M=3,N=2,3)$, and outline their systematic construction for arbitrary $N$, $M$. 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.

arXiv comment7 pages, 4 figures

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• Designing flat-band tight-binding models with tunable multifold band touching points
• Non-Hermitian flatband generator in one dimension
• Many-body localization transition from flatband fine-tuning
• Flat bands and entanglement in the Kitaev ladder
• Flat Band Induced Metal-Insulator Transitions for Weak Magnetic Flux and Spin-Orbit Disorder
• Methods for constructing parameter-dependent flat band lattices
• Disorder-induced two-body localised state in interacting quantum walks
• Non-Abelian Majorana fermions in topological $s$-wave Fermi superfluids