* Corresponding author. Tel.: #46-18-4713584; fax: #46- 18-4713524. E-mail address: urban.lundin@fysik.uu.se (U. Lundin) Physica B 281&282 (2000) 836}837 Mott}Hubbard transition in the N-orbital Hubbard model Urban Lundin*, Igor Sandalov, Bo K rje Johansson Condensed Matter Theory group, Department of Physics, Uppsala University, Box 530, 751 21 Uppsala, Sweden Kirensky Institute of Physics, RAS, 660036 Krasnoyarsk, Russia Abstract The Mott}Hubbard insulator}metal transition (MHT) is studied for the N-orbital symmetrical Hubbard model with diagonal (t ) and non-diagonal (t ) hopping matrix elements. In the paramagnetic state (PM) for an n"1 "lling the non-diagonal hopping gives rise to two wide Hubbard sub-bands with small spectral weight of the order of &1/N and 2N!2 narrow sub-bands with a large weight &(1!1/N). No orbital polarization arises in this solution. In some directions in the (t , t )-plane the insulating energy gap in the density of states is closed in the vicinity of a critical Hubbard repulsion of ; &z(t #(N!1)t )(z-coordination number), whereas the narrow bands in this region of parameters still correspond to a deeply correlated phase (with bandwidth z(t !t )). 2000 Elsevier Science B.V. All rights reserved. Keywords: Hubbard model; Mott}Hubbard transition; d systems The properties of the classical Hubbard model [1], describing an s-band of width = with an on-site Coulomb repulsion ;, are determined by two para- meters: ;/= and the electron density n (number of electrons per atom). The metal}insulator Mott}Hubbard transition (MHT) may occur only for a half-"lled band, n"1. For this case one can expect that a MHT takes place at ;/=K1, with slight variations depending on the lattice. This is indeed shown in the original work by Hubbard [2] ; "3 /2 ) =. In nature the strongly cor- related electron systems (SCES) occur for d- and f-sys- tems, where the number of orbitals N"2l#1 is 5 and 7. The generalization to many orbitals rather than s-shell introduces two additional parameters: (a) the number of orbitals N and (b) the matrix element of non-diagonal hopping. Therefore, the question arises, how these addi- tional parameters change the physics of the MHT in SCES in comparison with the Hubbard model for an s-band. We consider here the MHT physics with the perturbation theory from the atomic limit and show that the approximation, taking into account the #uctuations of the population numbers, contains a MHT. We will not address the magnetic aspect of the problem, but rather present an attempt to "rst get an answer for the easiest case, namely, for the paramagnetic state with n"1 "ll- ing. Let us consider the following symmetrical model: H";n ( n ( (1! )#t f f , where i, j are site indices, "(m , ) are orbitals, n ( "f f and t" t #(1! ) t . We will start from the limit ;<t , t and search for MHT by increasing the corresponding bandwidths w and w . If we put ; to zero the Hamiltonian gives rise to a degen- erate narrow band, with bandwidth z(t !t ), and a wide band, with bandwidth z(t #(N!1)t . Assuming that the transition takes place at ;/=&1 we have a simple estimation of the critical ; of z(t #(N!1)t . When ;O0 the bands are split. A convenient tool for developing a perturbation theory from atomic limit is the diagram technique for the Hubbard operators [3]. The latter was introduced by Hubbard [1,4] describing many electron intra-ion transitions in the following way: H(;)p"E p, X"pq, where in the present case of interest p"0, , , ; "(m , ), ", , ", , . Therefore, the 0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 9 8 0 - 1