J. Phys. I France

**7**(1997) 1309-1368

## Non-Perturbative Many-Body Approach to the Hubbard Model and Single-Particle Pseudogap

**Y.M. Vilk**

^{1, 2}and A.-M.S. Tremblay^{2}^{1}Matérials Science Division, Bldg. 223, 9700 S. Case Ave., Argonne National Laboratory, Argonne IL 60439, USA

^{2}Département de Physique and Centre de Recherche en Physique du Solide, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1

(Received 7 July 1997, accepted 23 July 1997)

** Abstract **

A new approach to the single-band Hubbard model is described in the general context of many-body theories. It is based on
enforcing conservation laws, the Pauli principle and a number of crucial sum-rules. More specifically, spin and charge susceptibilities
are expressed, in a conserving approximation, as a function of two irreducible vertices whose values are found by imposing
the local Pauli principle
as well as the local-moment sum-rule and consistency with the equations of motion in a local-field approximation. The Mermin-Wagner
theorem in two dimensions is automatically satisfied. The effect of collective modes on single-particle properties is then
obtained by a paramagnon-like formula that is consistent with the two-particle properties in the sense that the potential
energy obtained from Tr
is identical to that obtained using the fluctuation-dissipation theorem for susceptibilities. Since there is no Migdal theorem
controlling the effect of spin and charge fluctuations on the self-energy, the required vertex corrections are included. It
is shown that the theory is in quantitative agreement with Monte Carlo simulations for both single-particle and two-particle
properties. The theory predicts a magnetic phase diagram where magnetic order persists away from half-filling but where ferromagnetism
is completely suppressed. Both quantum-critical and renormalized-classical behavior can occur in certain parameter ranges.
It is shown that in the renormalized classical regime, spin fluctuations lead to precursors of antiferromagnetic bands (shadow
bands) and to the destruction of the Fermi-liquid quasiparticles in a wide temperature range above the zero-temperature phase
transition. The upper critical dimension for this phenomenon is three. The analogous phenomenon of pairing pseudogap can occur
in the attractive model in two dimensions when the pairing fluctuations become critical. Simple analytical expressions for
the self-energy are derived in both the magnetic and pairing pseudogap regimes. Other approaches, such as paramagnon, self-consistent
fluctuation exchange approximation (FLEX), and pseudo-potential parquet approaches are critically compared. In particular,
it is argued that the failure of the FLEX approximation to reproduce the psuedogap and the precursors AFM bands in the weak
coupling regime and the Hubbard bands in the strong coupling regime is due to inconsistent treatment of vertex corrections
in the expression for the self-energy. Treating the spin fluctuations as if there was a Migdal's theorem can lead not only
to quantitatively wrong results but also to qualitatively wrong predictions, in particular with regard to the single-particle
pseudogap.

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*Les Editions de Physique 1997*