Momentum structure of the self-energy and its parametrization for the two-dimensional Hubbard model

We compute the self-energy for the half-filled Hubbard model on a square lattice using lattice quantum Monte Carlo simulations and the dynamical vertex approximation. The self-energy is strongly momentum-dependent, but it can be parametrized via the noninteracting energy-momentum dispersion ${\varepsilon }_{\mathbf{k}}$, except for pseudogap features right at the Fermi edge. That is, it can be written as $\mathrm{\Sigma }({\varepsilon }_{\mathbf{k}},\omega )$, with two energylike parameters $(\varepsilon , \omega )$ instead of three ($k_x, k_y$, and $\omega$). The self-energy has two rather broad and weakly dispersing high-energy features and a sharp $\omega ={\varepsilon }_{\mathbf{k}}$ feature at high temperatures, which turns to $\omega =-{\varepsilon }_{\mathbf{k}}$ at low temperatures. Altogether this yields a $\mathcal{Z}$- and reversed-$\mathcal{Z}$-like structure, respectively, for the imaginary part of $\mathrm{\Sigma }({\varepsilon }_{\mathbf{k}},\omega )$. We attribute the change of the low-energy structure to antiferromagnetic spin fluctuations.

Figure 1

Imaginary part of the self-energy $\mathrm{\Sigma }(\mathbf{k},i{\omega }_n)$ from BSS-QMC for $U=4t$ and $\beta t=5.6$ at (a) the first three Matsubara frequencies and $k_x=0$; (b) at the first Matsubara frequency along the (brightness-coded) five momentum paths shown in the inset. The red points in (b) correspond to the nodal and antinodal point, which are emphasized alike in the inset by the red diagonal arrow.

Figure 2

Imaginary (a) and real (b) part of the self-energy $\mathrm{\Sigma }(\mathbf{k},i{\omega }_n)$ vs the noninteracting dispersion ${\varepsilon }_{\mathbf{k}}$ from BSS-QMC at $U=4t$ and $\beta t=5.6$. Different $(k_x,k_y)$ points with the same ${\varepsilon }_{\mathbf{k}}$ collapse onto a single curve.

Figure 3

Imaginary (a) and real (b) part of the self-energy $\mathrm{\Sigma }(\mathbf{k},i{\omega }_0)$ from BBS-QMC at $U=4t$ and $\beta t=5.6$. The different data points correspond to different lattice sizes and geometries.

Figure 4

Imaginary (a) and real (b) part of the self-energy $\mathrm{\Sigma }(\mathbf{k},i{\omega }_0)$ from BSS-QMC at $\beta t=5.6$ for different $U$ values.

Figure 5

Imaginary part of the self-energy $\mathrm{\Sigma }(\varepsilon ,\omega )$ and of the Green's function $G(\varepsilon ,\omega )$ at $U=4t$ and $\beta t=5.6$. Parts (a) and (b) contain the BSS-QMC data. Parts (c) and (d) represent continuous parametrizations, denoted in Eq. (9) and Table 1.

Figure 6

Same as Fig. 5 but at a higher temperature, $\beta t=2$.

Figure 7

The self-energy at three different (higher) Matsubara frequencies at $U=4t$ and $\beta t=5.6$. Together with the asymptotic behavior of the Matsubara self-energy, these BSS-QMC data are used to fix the model parameters in Eq. (9). The fitted parameters can be found in Table 1.

Figure 8

Self-energy and the Green's function calculated from second-order perturbation theory at $\beta t=1$ (left) and $\beta t=5.6$ (right) for $U=4t$.

Figure 9

Imaginary part of the self-energy $\mathrm{\Sigma }(\mathbf{k},i{\omega }_0)$ for $U=2t$ and two different temperatures: (a) $\beta t=5$ and (b) $\beta t=10$. The (light blue) continuum of data points represent all different $\text{D}\mathrm{\Gamma }\text{A}$ momenta for a given ${\varepsilon }_{\mathbf{k}}$. For a better comparison of $\text{D}\mathrm{\Gamma }\text{A}$ with BSS-QMC, we highlighted data points in the $\text{D}\mathrm{\Gamma }\text{A}$ calculation (dark-blue triangles) that correspond to the BSS-QMC data (red circles) with similar $\mathbf{k}$ points.

Figure 10

The self-energy for $U=4t, \beta t=2$, and the first (left) and second (right) Matsubara frequencies comparing BSS-QMC (red circles) and $\text{D}\mathrm{\Gamma }\text{A}$ (light blue; and dark blue triangles for similar momenta as the red circles).

Figure 11

Imaginary (a) and real (b) part of the self-energy $\mathrm{\Sigma }(\mathbf{k},i{\omega }_0)$ from BSS-QMC at $U=4t$ and $\beta t=5.6$ but for various degrees of anisotropies $\alpha$. Green and red lines are guides to the eye only, not fits to the parametrization model as in the other figures.

Figure 12

Imaginary (a) and real (b) part of the self-energy $\mathrm{\Sigma }(\mathbf{k},i{\omega }_0)$ from BSS-QMC at $U=4t, \beta t=3.6$, and $L=8\times 8$ for doped systems.