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

TL;DR

This paper derives exact conditions for the existence of flat bands in complex-valued tight-binding lattices across various dimensions and sites per unit cell, enabling systematic design of flat band geometries.

Contribution

It provides necessary and sufficient mathematical conditions for flat bands in multi-dimensional, multi-band lattices with complex hopping, extending previous understanding and enabling new lattice designs.

Findings

Explicit conditions for flat bands in 1D, 2D, and 3D lattices.

Systematic construction method for arbitrary N and M.

Discovery of new flat band lattice geometries.

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 $(M = 1, N\leq4)$, $(M = 2, N = 2,3)$, 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.