Self-energy dispersion in the Hubbard model

We introduce the concept of self-energy dispersion as an error bound on local theories and apply it to the two-dimensional Hubbard model on the square lattice at half filling. Since the self-energy has no single-particle analog and is not directly measurable in experiments, its general behavior as a function of momentum is an open question. In this paper, we benchmark the momentum dependence with the two-particle self-consistent approach together with analytical and numerical considerations and we show that through the addition of a local single-particle potential to the Hubbard model the self-energy can be flattened, such that it is essentially described by only a frequency-dependent term. We use this observation to motivate that local theories, such as the dynamical mean-field theory, should be expected to give very accurate results in the presence of a potential of this kind. This is especially interesting in the context of topology, where such a term is present in many popular models, e.g., the Haldane and Kane-Mele models. Finally, we propose a simple energy argument as an estimator for the crossover from nonlocal to local self-energies, which can be computed even by local theories such as dynamical mean-field theory.

Figure 1

Illustration of the electron density in a shallow and deep on-site potential with amplitude $\mathrm{\Delta }/t$ [cf. Eq. (4)]. For small $\mathrm{\Delta }/t$ (lower left) the average densities are near unity on $A$ and $B$ sites. For large $\mathrm{\Delta }/t$ (upper right) $A$ sites will be filled almost completely while $B$ sites are almost empty.

Figure 2

Frequency-resolved self-energy dispersion strength $d_a\left(i{\omega }_n\right)$ for diagonal matrix elements (lines) and off-diagonal matrix elements (dotted) at $U/t=0.6$. The dispersion strength decays rather quickly towards higher Matsubara frequencies and is maximal at ${\omega }_{n=0}=\pi /\beta$. Off-diagonal elements are only weakly dependent on $\mathrm{\Delta }$.

Figure 3

Relative dispersion strength $d_r\left(i{\omega }_{n=0}\right)$ for the square lattice at inverse temperature $\beta t=10$; for better visibility of small dispersion amplitudes we have fixed the upper limit of the color scale to 0.1. In addition, we plot the critical estimation of Eq. (8) obtained for $t=0$ (white line) and in gray: TPSC (solid), ED (dashed), DMFT (dotted). Here, we included the chemical potential $\mu$ in the self-energy.

Figure 4

We compare TPSC (lines) with DMFT (dotted) at $U/t=2$. Top: Densities and double occupancies on $A$ and $B$ sites of the square lattice. Both methods agree at small and large $\mathrm{\Delta }/t$, and at moderate $\mathrm{\Delta }/t$ the values for $D_A$ differ. Bottom: Imaginary part of the Green's function in units of $t^{-1}$ for the $A$ site at the lowest Matsubara frequency at different momenta. At $(0,0)$ and $(0,\pi )$ the momentum dependence leads to a reduced spectral weight, and at $\mathrm{\Delta }/t\gtrsim 1$ there is no significant difference.