Conductivity in the Square Lattice Hubbard Model at High Temperatures: Importance of Vertex Corrections

Recent experiments on cold atoms in optical lattices allow for a quantitative comparison of the measurements to the conductivity calculations in the square lattice Hubbard model. However, the available calculations do not give consistent results, and the question of the exact solution for the conductivity in the Hubbard model remained open. In this Letter, we employ several complementary state-of-the-art numerical methods to disentangle various contributions to conductivity and identify the best available result to be compared to experiment. We find that, at relevant (high) temperatures, the self-energy is practically local, yet the vertex corrections remain rather important, contrary to expectations. The finite-size effects are small even at the lattice size $4\times 4$, and the corresponding Lanczos diagonalization result is, therefore, close to the exact result in the thermodynamic limit.

Figure 1

(a) Illustration of the type of self-energy diagrams that are captured by isolated cluster and embedded cluster (in particular cellular DMFT), and the respective difference in the Brillouin zone (discrete vs continuous). (b) Separation of a susceptibility into the bubble and the vertex corrections part.

Figure 2

Charge susceptibility (upper) and dc resistivity (lower) as a function of temperature, at different levels of doping. The color between the curves denotes the physical origin of the difference. Dashed curves denote just the bubble contribution, solid lines the full result.

Figure 3

All panels: $p=0.1$, $T=0.5D$. a) Benchmark of self-energy and inspection of its leading non-local component. b) Comparison of the optical conductivity between various methods. c) See text. d) Real-space resolution of the vertex corrections along two spatial directions (CTINT $8\times 8$ result).