Journal of Magnetism and Magnetic Materials 108 (1992) 170-172 North-Holland Luttinger and Hubbard ii sum rules" are they compatible? Konrad Matho C.N.R.S.-C.R.T.B.T. B.P. 166A; 38042 Grenoble C&tex, France 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 ground- state is marked by its resemblance to the single-site Kondo problem [4], iu which this separation first oc- curs. 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 chan- nel. 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 approx- imation to G(k, ~) [8,9] that obeyed these exact weights. This justifies to attach Hubbard's na:nc to the HSR in general. Also, the HP has the flavourless fermion property [10] that is a fundamental character- istic of the charge part in the total spectrum [1,2]. LSR and HSR as constraints. We assume that un- broken symmetry, such as characterizes the normal state experimentally, is also necessary and sufficient for the LSR and HSR to hold. Detailed information on the structure of the k-integrated propagator can be gained by imposing the sum rules as constraints on an "ad hoc" Ansatz. For brevity, the present discussion is restricted to N= 2 [8], beyond half-filling. Most as- pects carry over to the case of N flaw)urs [9,10]. The Ansatz for the total propagator neglects life- time broadening but allows for the separation of charge and spin by the existence of two 6-function peaks on the w-axis at fixed k, rather than only one. The spin peak of weight Q* is positioned at z=w-IX =Q*(Ek-l, t0(m)), (1) with #o(m)the bare chemical potential function of the free fermions with Bloch spectrum {Ek}. The dimen- sionality d enters through #o(m), e.g. its Van Hove singularities and band edges [11]. Equation (1) fulfils the LSR by construction. It is almost identical to the propagator of Gutzwiller- or slave boson-(mean field)"quasiparticles" [11-13]. However, the weight Q* is not set equal to its Gutzwiller value QGw = (2m- 1)/m, but rather left open as a parameter. As long as lifetime broadening is neglected can Q* be interpreted either as a quasipar- ticle residue or as the co-integrated weight of some more complicated singularity. The Ansatz for the charge peak is z =w - IX = Q,(E k- #,,(m,)), (2) leaviqg four frcc parameters, inc'.uding its weight, to be determined. The potential Ix0(m n) defines a surface in k-space on which the first moment of the charge excita- tion vanishes. The scale Qn = 1 fixes the overall disper- sion of the band of charge excitations, expected to be the same order as the bare bandwidth 2D. In addition, the observable band width will be lifetime broadened. 0312-S~53/92/$05.00 ,C~ 1992 - Elsevier Science Publishers B.V. All rights reserved