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

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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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          Author and article information

          Journal
          29 September 2004
          2005-07-23
          Article
          10.1103/PhysRevB.71.235119
          cond-mat/0409730
          Custom metadata
          Phys. Rev. B 71, 235119 (2005)
          14 pages, 9 figures; introduction expanded, references added; published version
          cond-mat.str-el

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