Self-energy and Fermi surface of the two-dimensional Hubbard model

We present an exact diagonalization study of the self-energy of the two-dimensional Hubbard model. To increase the range of available cluster sizes we use a corrected $t$-$J$ model to compute approximate Green’s functions for the Hubbard model. This allows to obtain spectra for clusters with 18 and 20 sites. The self-energy has several “bands” of poles with strong dispersion and extended incoherent continua with $k$-dependent intensity. We fit the self-energy by a minimal model and use this to extrapolate the cluster results to the infinite lattice. The resulting Fermi surface shows a transition from hole pockets in the underdoped regime to a large Fermi surface in the overdoped regime. We demonstrate that hole pockets can be completely consistent with the Luttinger theorem. Introduction of next-nearest-neighbor hopping changes the self-energy strongly and the spectral function with nonvanishing next-nearest-neighbor hopping in the underdoped region is in good agreement with angle-resolved photoelectron spectroscopy in the cuprates.

Figure 1

Spectral density $A(\mathbf{k},\omega )$ for a $4\times 4$ cluster with two holes computed with the true Hubbard model and with the strong-coupling model. The ratio $U/t=10$, $\eta =0.1$.

Figure 2

Spectral density $A(\mathbf{k},\omega )$ and imaginary part of the self-energy $\Sigma (\mathbf{k},\omega )$ at half-filling, $U/t=10$. The figure combines spectra from the $16$- and $18$-site clusters; momenta from the $16$-site cluster are marked by asterisks. The spectra are computed with $\eta =0.1$, $\Sigma (\mathbf{k},\omega )$ is multiplied by $1/50$. The dots indicate the position of ${\zeta }_{\mathbf{k}}=\mu +2t[\cos (k_x)+\cos (k_y)]$ for the respective momentum.

Figure 3

Spectral density $A(\mathbf{k},\omega )$ and imaginary part of the self-energy $\Sigma (\mathbf{k},\omega )$ at half-filling, $U/t=10$. The figure combines spectra from the $16$- and $18$-site clusters; momenta from the $16$-site cluster are marked by asterisks. The spectra are computed with $\eta =0.05$, and $\Sigma (\mathbf{k},\omega )$ is multiplied by $1/5$ and at $(\pi ,\pi )$ by an extra factor of $0.25$.

Figure 4

Dispersion of the central peak of $\Sigma (\mathbf{k},\omega )$ in the Hubbard gap for various doped systems. The lines are fits to tight binding harmonics with coefficients given in Table . The value $U/t=10$ for all systems.

Figure 5

Spectral density $A(\mathbf{k},\omega )$ and imaginary part of the self-energy $\Sigma (\mathbf{k},\omega )$ for the $16$-site and $18$-site clusters with two holes, $U/t=10$. The spectra are computed with $\eta =0.05$, and $\Sigma (\mathbf{k},\omega )$ is multiplied by $1/5$.

Figure 6

Spectral density $A(\mathbf{k},\omega )$ and imaginary part of the self-energy $\Sigma (\mathbf{k},\omega )$ for the $20$-site cluster with two holes, $U/t=10$. The spectra are computed with $\eta =0.05$, and $\Sigma (\mathbf{k},\omega )$ is multiplied by $1/5$.

Figure 7

Same as Fig. 6 but with $U/t=20$.

Figure 8

Spectral density $A(\mathbf{k},\omega )$ and imaginary part of the self-energy $\Sigma (\mathbf{k},\omega )$ for the $16$-site and $18$-site clusters with four holes, $U/t=10$. The spectra are calculated with $\eta =0.05$, and $\Sigma (\mathbf{k},\omega )$ is multiplied by $1/5$.

Figure 9

$\Sigma (\mathbf{k},\omega )$ for $U/t=10$ and two holes in the $16$- and $18$-site clusters compared to the fit (11). The spectra are computed with $\eta =0.05$, and $\Sigma (\mathbf{k},\omega )$ is multiplied by $1/5$. The dots indicate the dispersion of the poles $\nu =2$ and $\nu =3$.

Figure 10

Self-energy for $U/t=10$ and four holes in the $16$- and $18$-site clusters compared to the fit.

Figure 11

Single-particle spectral density $A(\mathbf{k},\omega )$ computed with the fitted self-energy (11) for the underdoped case $\delta =0.12$, the spectra are calculated with $\eta =0.05$.

Figure 12

Single-particle spectral density at $\mu$ computed with the fitted self-energy (11) for the underdoped case, $\delta =0.12$. The value of $\eta$ is $0.05$.

Figure 13

Single-particle spectral density at $\mu$ computed with the fitted self-energy for the overdoped case.

Figure 14

Bare dispersion ${\mathrm{\epsilon ̃}}_{\mathbf{k}}$ intersecting with a band of poles of the self-energy ${\zeta }_{\mathbf{k}}$.

Figure 15

Spectral density $A(\mathbf{k},\omega )$ and imaginary part of the self-energy $\Sigma (\mathbf{k},\omega )$ for the $16$-site and $18$-site clusters with two holes. The value $t^{\prime }=-0.4t$. The spectra are calculated with $\eta =0.05$, and $\Sigma (\mathbf{k},\omega )$ is multiplied by 1/5.

Figure 16

Imaginary part of the self-energy $\Sigma (\mathbf{k},\omega )$ for the $16$-site and $18$-site clusters with two holes and fit. The value $t^{\prime }=-0.4t$.

Figure 17

Single-particle spectral density $A(\mathbf{k},\omega )$ for $\delta =0.12$, $U/t=10$, and $t^{\prime }/t=-0.4$ obtained by using the interpolated self-energy for the infinite lattice.

Figure 18

Single-particle spectral density at $\mu$ for $\delta =0.12$, $U/t=10$, and $t^{\prime }/t=-0.4$ obtained by using the interpolated self-energy for the infinite lattice.