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Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory

Martin Eckstein·Marcus Kollar·Krzysztof Byczuk·Dieter Vollhardt
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Abstract

We derive an operator identity which relates tight-binding Hamiltonians with arbitrary hopping on the Bethe lattice to the Hamiltonian with nearest-neighbor hopping. This provides an exact expression for the density of states (DOS) of a non-interacting quantum-mechanical particle for any hopping. We present analytic results for the DOS corresponding to hopping between nearest and next-nearest neighbors, and also for exponentially decreasing hopping amplitudes. Conversely it is possible to construct a hopping Hamiltonian on the Bethe lattice for any given DOS. These methods are based only on the so-called distance regularity of the infinite Bethe lattice, and not on the absence of loops. Results are also obtained for the triangular Husimi cactus, a recursive lattice with loops. Furthermore we derive the exact self-consistency equations arising in the context of dynamical mean-field theory, which serve as a starting point for studies of Hubbard-type models with frustration.

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Martin Eckstein, Marcus Kollar, Krzysztof Byczuk, Dieter Vollhardt. 2005-07-23. Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory. https://doi.org/10.1103/physrevb.71.235119

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