Quality Control for Low Temperature Strong Correlation Solutions to the Hubbard Model

Finally a rigorous (exact!) way to qualify solutions to the Hubbard model in the important low temperature strong correlation region, for any dimension, lattice structure and electron density. Considering the solution space axes as U/t and kT/t, the small U/t and large kT/t solutions are simple and available. To qualify solutions in the physically important region of low to moderate temperature and moderate to large U, a reliable quantitative test is presented for low T and large U that can distinguish how well the many proposed solutions are faithful to the model itself. Exact expressions for the first (quasiparticle energy) and second (quasiparticle lifetime) moments, involving higher order correlation functions, of the general spectral weight function (SWF) are obtained for zero temperature, infinite U and finite t, in terms of the equal time single particle correlation function (SPCF). Importantly, the relations are exact and do not involve simplistic decoupling approximations. Any proposed solution and related SWF can then compute the quasiparticle energy and lifetime both directly (from frequency-weighted integrations over the SWF) and indirectly (using the exact expressions in terms of the SPCF). The deviations between the two provides a quantitative insight into the accuracy of the quasiparticle energy and lifetime (in a perfect solution, the direct and indirect results would coincide). The venerable Hubbard III solution is presented as a demonstration. Interestingly, that solution performs quite well.

In addition, the almost universal claim of solutions becoming "exact in the atomic limit" is shown to be clearly wrong for the physically relevant case of low temperature (keeping kT < t as t goes to zero) solutions.