Real-space renormalisation-group study of the one-dimensional Hubbard model

The one-dimensional Hubbard model is studied using a real-space renormalisation-group block method previously applied to other quantum systems. The authors primarily consider the half-filled band for which the ground state is found to be insulating for all non-zero values of the Hubbard interaction, in agreement with the exact solution. When the method is applied to first order in perturbation, they find that the insulating gap for small U/t behaves like exp (-At2/U2) where U is the Hubbard interaction and t is the nearest-neighbour transfer integral. When the method is extended to second order, the authors find that for small U/t the gap behaves like (U/t)alpha with alpha approximately=11.2. The exact asymptotic behaviour of the gap ( approximately (U/t)1/2 exp(-At/U)) is not recovered, but a good semiquantative agreement with the exact solution is obtained for both the gap and the ground state energy. Extensions to non-half-filled bands and to higher dimensions are discussed. The validity of the renormalisation group method is also discussed.