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.
Highlights
Many interesting phases of matter exist when interactions are strong compared to the kinetic energy
To address the problem of finding flat bands, we provide here a general set of necessary and sufficient conditions for the occurrence of dispersionless bands in M-dimensional tight-binding lattices with N sites per unit cell and complex-amplitude nearest-neighbor tunneling between the lattice sites
Recent attempts at classifying and generating flat bands have concentrated on identifying the parameter U, the minimum number of unit cells that a compact localized state associated with the flat band occupies, or lattice-specific methods that lack generality
Summary
Many interesting phases of matter exist when interactions are strong compared to the kinetic energy. Known as flat bands, the kinetic energy is formally zero, and any interaction can be considered strong These special properties render flat band systems an excellent test bed for novel phases that appear once disorder and/or interactions are switched on: it has been shown that in principle FQHE states should be obtainable provided one can generate an approximately dispersionless band that is energetically isolated and topologically non-trivial, without the need for a magnetic field and its Landau levels [6,7,8,9,10]. We report a new general approach for constructing flat band Hamiltonians by using the momentum representation, which has received much less attention in the flat band literature [24]. In other words, knowing the new conditions answers the question of existence, and makes it possible, in principle at least numerically, to obtain the exhaustive set of points in the space of Hamiltonians that correspond to one or more flat bands
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