By Nils Blumer
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The striking feature of the Bethe DOS that the full bandwidth (including all states) is O(Z 1/2 ) instead of O(Z) (which would be expected from Perron’s theorem) can be attributed to the absence of long-wavelength excitations; see Thorpe (1981) and references therein. The spectral weight, concentrated at the band edges for low Z, shifts toward the center with increasing Z until a semi-elliptic form, 1√ ρBethe ( ) = 4− 2 . 24) 2π is obtained for Z = ∞ which we will label as “Bethe DOS” in the remainder of this work.
Furthermore, nonlocal spectral functions and the optical conductivity σ(ω) can, within the DMFT, be calculated from A(ω) (see chapter 4). In QMC calculations, however, the Green function G (and thus the spectral function A(ω)) cannot be directly computed on the real axis. Instead, real-time dynamical information has to be extracted from imaginary-time data G(τ ) (or, equivalently, from the Fourier transformed Matsubara-frequency data G(iωn )) via analytic continuation. 56) becomes exponentially small for generic 24 1.
We will also consider a Bethe lattice with NNN hopping and derive some more general results from an expansion in terms of self-avoiding loops, the RPE. 13 While this construct coincides with the usual d = 1 lattice for Z = 2, it is not a regular lattice for Z > 2 as is visualized in Fig. , Z = 5. Here it is shown as a directed graph where a level number i ∈ Z can be assigned to each site, so that a site at level i is connected by one edge to a site in the lower level i − 1 and by K edges to sites in the higher level i + 1.
