Analytic Treatment of Mott-Hubbard Transition in the Half-Filled Hubbard Model

The Mott-Hubbard transition in the half-filled Hubbard model is studied analytically for the paramagnetic ground state and the classical N\'{e}el state. The single-particle density of states is obtained by calculating the Green's function represented by the infinite continued fraction. The paramagnetic solution shows that the Mott-Hubbard transition is signaled by both collapsing Hubbard bands and appearing $\delta$-function peak at midgap, and the transition is second order at the ground state. We also provide specific heats in metallic regime to demonstrate that our results recover those of infinite dimension.