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

TL;DR

This paper derives exact analytical expressions for the density of states of particles on the Bethe lattice with various hopping terms and provides self-consistency equations for dynamical mean-field theory applicable to frustrated Hubbard models.

Contribution

It introduces an operator identity linking arbitrary hopping Hamiltonians to nearest-neighbor models on the Bethe lattice, enabling exact DOS calculations and DMFT equations.

Findings

Exact DOS expressions for various hopping amplitudes.

Ability to construct Hamiltonians for any given DOS.

Derived self-consistency equations for DMFT in frustrated systems.

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.