Luttinger and Hubbard sum rules: are they compatible? (original) (raw)

1992, Journal of Magnetism and Magnetic Materials

A so-called Hubbard sum rule determines the weight of a satellite in termionic single-particle excitations with strong local repulsion (U ~). Together with the Luttinger sum rule, this imposes two different energy scales on the remaining finite excitations. In the Hubbard chain, this has been identified microscopically as being due to a separation of spin and charge. Lutthzger sum rule. The separation o[ charge and spin degrees of freedom is a ubiquitous phenomenon in the single-particle propagator G(k, co) of lattice fermions with strong local repulsion U. From space dimension d = I [1,2] to d = ~ [3], the total spectrum of excitations above the "normal" electronic groundstate is marked by its resemblance to the single-site Kondo problem [4], iu which this separation first occurs. As the property defining a "normal" state, wc retain the unbroken symmetry among spin channels. In the Hubbard chain [1] and t-J chain [2] the analyticity properties of a l~ermi-hqund state according to Luttinger [5] break down [1,2,6]. Nevertheless, the spin spectrum continues to have a Fermi surface (FS) that follows the Luttinger sum rule (LSR) in k-space. Here, wc assume that the LSR holds, beyond the Fermi liquid fixed point, also in dimcnsions d > 1, whenever there is unbroken symmetry. Hubbard sum rule. Hubbard satellites occur in all modcls with a "configurational crossovcr" [7]. Thc weight of a satellite is givcn by an exact sum rulc, which wc call ttubbard sum rule (HSR). Thc only input nccdcd to derive the HSR, for crossovers of arbitrary configurations, is precisely the unbroken symmetry in their flavours [7]. Electrons in the local configurations participate in the FS. Anderson [6] pointed out that the expulsion of one new excitation into the the satellite for each added barc particle causes the vanishing of the quisiparticle residue in the propagator. As we shall demonstrate, this intimate link between a low-energy phenomenon and a high-energy phenomenon is already present on the crudest level of global sum rules. Considcr thc N = 2 Hubbard model [8] in the filling interval ! <n d < 2, n a= 2m. Thc satellite is then holc-likc and has a wcight Q = i-m per spin channel. This reduccs the weight of holes with finite energy (co<~) to Q~, =2m-1. The particle sector (~o>#) retains its full free-fermion weight Q~, + = 1-m. The total weight for excitations with finite energy, as counted from Ix, is Q~, = Q~, + + Q~, = m, per channel. The Hubbard propagator (HP)was the fir,,;t approximation to G(k, ~) [8,9] that obeyed these exact weights. This justifies to attach Hubbard's na:nc to the