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

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 noninteracting 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.

Figure 1

Part of the Bethe lattice with coordination number $Z=4$. Any two sites are connected by a unique shortest path of bonds. Starting from the site marked by the open circle, horizontally shaded circles can be reached by one lattice step (NN), vertically shaded circles by two lattice steps (NNN), and doubly shaded circles by three lattice steps. Note that the lattice is infinite and that all sites are equivalent; the shading appears only for visualization of hopping processes.

Figure 2

(Color online) Density of states (30) for pure NNN hopping $t_2=t_2^{*}∕(Z-1)\ne 0$ on the Bethe lattice, for several numbers of nearest neighbors $Z$; $t_2^{*}=1$ sets the energy scale. Divergences are marked on the horizontal axis.

Figure 3

(Color online) Density of states (26) for $t_1\text{−}{t}_2$ hopping on the Bethe lattice with $Z=4$ for selected values of $x=t_2^{*}∕t^{*}$, where $t_1^{*},t_2^{*}⩾0,t^{*}=t_1^{*}+t_2^{*}$ sets the energy scale. Divergences are marked on the horizontal axis.

Figure 4

(Color online) Same as Fig. 3, but for $Z\to \infty$ [Eq. (28)].

Figure 5

(Color online) Density of states for exponentially decreasing long-range hopping on the Bethe lattice [Eq. (36)], for $t^{*}=1$ and $Z\to \infty$.

Figure 6

(Color online) Density of states for exponentially decreasing long-range hopping between different sublattices (“odd hopping”) for $t^{*}=1,Z\to \infty$, and $w<w_{*}$. Divergences are marked on the horizontal axis.

Figure 7

(Color online) Density of states for exponentially decreasing long-range hopping between same sublattices (“even hopping”) for $t^{*}=1$, $Z\to \infty$, and $w⩽0$. Divergences are marked on the horizontal axis. For $w=-0.9$ the finite peak at the upper band edge, as well as the singularity at the lower band edge, are outside the plotting range.

Figure 8

(Color online) Hopping amplitudes $t_d^{*}$ on the Bethe lattice (with $Z\to \infty$) corresponding to the DOS (44) (see inset) for several values of the asymmetry parameter $a$. The odd hopping amplitudes $t_1^{*},t_3^{*}$, etc., are independent of $a$; $t_0^{*}$ gives the center of the mass of the band.

Figure 9

Part of the triangular Husimi cactus with $Z$ triangles connected to each site; here $Z=2$. Any two sites are connected by a unique shortest path of bonds. The lattice is infinite and all sites are equivalent. The shading of sites has the same meaning as in Fig. 1.