Optimal quasi-free approximation:reconstructing the spectrum from ground state energies

The sequence of ground state energy density at finite size, e_{L}, provides much more information than usually believed. Having at disposal e_{L} for short lattice sizes, we show how to re-construct an approximate quasi-particle dispersion for any interacting model. The accuracy of this method relies on the best possible quasi-free approximation to the model, consistent with the observed values of the energy e_{L}. We also provide a simple criterion to assess whether such a quasi-free approximation is valid. As a side effect, our method is able to assess whether the nature of the quasi-particles is fermionic or bosonic together with the effective boundary conditions of the model. When applied to the spin-1/2 Heisenberg model, the method produces a band of Fermi quasi-particles very close to the exact one of des Cloizeaux and Pearson. The method is further tested on a spin-1/2 Heisenberg model with explicit dimerization and on a spin-1 chain with single ion anisotropy. A connection with the Riemann Hypothesis is also pointed out.