Local dynamics and strong correlation physics: One- and two-dimensional half-filled Hubbard models

We report on a nonperturbative approach to the one-dimensional (1D) and two-dimensional (2D) Hubbard models that is capable of recovering both strong- $\left(U\gg t\right)$ and weak-coupling $(U\ll t)$ limits, with U the on-site Coulomb repulsion and t the kinetic energy. Dynamical corrections to the electron self-energy in the single particle Green function are explicitly included by expanding in terms of the 16 eigenstates that characterize two nearest neighbor sites. We first show that even when U is much smaller than the bandwidth, the Mott-Hubbard gap never closes at half filling in both 1D and 2D. Consequently, the Hubbard model at half filling is always in the strong-coupling nonperturbative regime. For both large and small U, we find that the population of nearest-neighbor singlet states approaches a value of order unity as $\overrightarrow{T}0$ as would be expected for antiferromagnetic order. We also find that the double occupancy is a smooth monotonic function of U and approaches the anticipated noninteracting limit of $1/4$ as $\overrightarrow{U}0$ and vanishes as $\overrightarrow{U}\infty .$ Finally, we compute the heat capacity $\left[C\right(T,U\left)\mathrm{}\right]$ for both 1D and 2D. Our results for 1D at moderate to high temperatures are in quantitative agreement with those of the exact Bethe ansatz solution, differing by no more than $1\%.$ In addition, we find that in 2D, the $C(T,U)\mathrm{}$ curves vs T for different values of U exhibit a universal crossing point at two characteristic temperatures $T\approx 1.7t\pm 0.1t$ and $T\approx 0.4\pm 0.1t$ as is seen universally in Hubbard models and experimentally in a wide range of strongly correlated systems such as ${}^3\mathrm{He},$ ${\mathrm{UBe}}_3,$ and ${\mathrm{CeCu}}_{6-x}{\mathrm{Al}}_x.$ The success of this method in recovering well-established results that stem fundamentally from the Coulomb interaction suggests that local dynamics are at the heart of the physics of strongly correlated systems.