One- and two-particle properties of the weakly interacting two-dimensional Hubbard model in proximity to the van Hove singularity

TL;DR

This paper investigates the weak-coupling behavior of the 2D Hubbard model near van Hove singularities, revealing doping-dependent spin and charge excitations, reduced compressibility, and a splitting phenomenon linked to band structure.

Contribution

It provides a detailed perturbative analysis of dynamical susceptibilities in the Hubbard model near van Hove points, highlighting behaviors distinct from strong-coupling Mott physics.

Findings

Zero frequency spin and charge excitations peak at different densities.

Reduced compressibility near van Hove singularity, independent of Mott physics.

Distinct electron and hole doping excitation structures matching experimental observations.

Abstract

We study the weak-coupling limit of the $t-t^\prime-U$ Hubbard model on a two-dimensional square lattice using a direct perturbative approach. Aided by symbolic computational tools, we compute the longitudinal density-density correlation functions in the $\chi_{\uparrow \uparrow}$ and $\chi_{\uparrow \downarrow}$ basis from which we can obtain the dynamical spin and charge susceptibilities at arbitrary doping and temperature. We find that for non-zero $t^\prime$, the zero frequency commensurate $\mathbf{q} = (\pi, \pi)$ spin and charge excitations are each strongest at different densities and we observe a clear behavioral change that appears tied to the van Hove singularity of the non-interacting dispersion upon which the perturbative expansion is built. We find a strongly reduced compressibility in the vicinity of the van Hove singularity as well as a behavioral change in the double occupancy. For finite $t^\prime$, the observed van Hove singularity occurs away from half-filling leading us to conclude that that this reduction in compressibility is distinct from Mott insulating physics that one expects in the strong-coupling regime. We compute the full dynamical spin and charge excitations and observe distinct structure for electron and hole doped scenarios in agreement with experiments on cuprate materials. Finally, we observe a peculiar splitting in spin and charge excitations in the vicinity of the van Hove singularity, the origin of which is traced to a splitting near the bottom of the band.