Self-consistent projection operator approach to nonlocal excitation spectra

We extend the projection operator method combined with the coherent potential approximation (CPA) to the nonlocal case in order to describe the momentum dependence of the single-particle self-energy. The inter-site excitations are taken into account successively by means of an incremental cluster expansion. The cluster memory functions embedded in a medium are obtained by extending a renormalized perturbation scheme. The medium is self-consistently determined from a CPA condition. In the weak Coulomb interaction limit the self-energy reduces to the second-order perturbation theory, but it is also correct in the atomic limit. Numerical calculations are performed for a Hubbard model at half filling on a simple-cubic lattice. It is demonstrated that the $k$-dependent self-energy reduces the quasiparticle density of states at the Fermi level, and broadens the band. Furthermore, in the metallic region it produces well-defined satellite bands around the $\Gamma$ and R points in the Brillouin zone. The momentum-dependent effective mass and the momentum distribution are also presented. The critical Coulomb interaction for the divergence of the effective mass is estimated to be at least twice as large as the one obtained in the single-site approximation.

Figure 1

Schematic representation of the incremental cluster expansion. On-site contribution (left), two-site contribution (middle), and three-site contribution (right) to the diagonal memory function ${\overline{G}}_{ii\sigma }\left(z\right)$. For the off-diagonal memory function ${\overline{G}}_{ij\sigma }\left(z\right)$, another site $j$ has to be added in the figure. In that case, the sites $(i,j)$ are not necessarily limited to nearest-neighbors.

Figure 2

Cluster system with on-site Coulomb interactions $U$ embedded in a medium (left), cavity state without $U$ (middle), and a uniform medium as an unperturbed state (right).

Figure 3

Excitation spectra of the half-filled single band Hubbard model on hyper-cubic lattice in infinite dimensions. The Coulomb interactions are taken to be $U=0.80U_{\mathrm{c}}$, $U_{\mathrm{c}}$ being the critical Coulomb interaction at which the effective mass diverges. The paramagnetic state is assumed for the ground state. The energy unit is taken so that the second moment of the noninteracting Gaussian density of states (DOS) becomes $1∕\sqrt{2}$. The results of a modified RPT-0 with parameters $(\lambda ,{\lambda }^{\prime })=(0.01,1)$ (solid curve) and of the MPT (Refs. 42, 43) ($(\lambda ,{\lambda }^{\prime })=(0,1)$, thin solid curve) are compared with the one of the numerical renormalization group theory (dashed curve) (Ref. 10).

Figure 4

Self-consistent coherent potentials for different values of the parameter $U$. The real (imaginary) parts are shown by the solid (dashed) curves. The result for the single-site approximation (SSA) is shown by the thin curves for $U=10$.

Figure 5

Imaginary part of the self-energy as a function of energy $\omega$ and momentum $k$ along the high-symmetry lines. $\Gamma$, X, M, and R denote the $k$ points at (0,0,0), $(\pi ,0,0)$, $(\pi ,\pi ,0)$, and $(\pi ,\pi ,\pi )$.

Figure 6

Real part of the self-energy as a function of energy and momentum.

Figure 7

Calculated self-energies at various $k$ points along the high-symmetry lines. The real (imaginary) parts are shown by solid (dotted) curves. They are obtained by projecting the curves in Figs. 5, 6 onto the $y\text{−}z$ plane. The results of the SSA are also shown by the thick solid and dotted curves.

Figure 8

Single-particle excitation spectra along the high-symmetry lines at $U=10$. The Fermi level is indicated by a bold line.

Figure 9

Single-particle excitation spectra in the SSA.

Figure 10

Single-particle excitation spectra in the SOPT.

Figure 11

Contour map of the excitation spectra at $U=10$. The spectrum for noninteracting system is shown by dashed curve.

Figure 12

Contour map on the excitation spectra in the SSA.

Figure 13

The total DOS for $U=2$, 6, 8, 10. The DOS for noninteracting system is shown by dotted curve.

Figure 14

The DOS in the present theory (solid curve), the SOPT (dashed curve), and the SSA (thin solid curve) at $U=10$. The result for $(\lambda ,{\lambda }^{\prime })=(0.024,1)$ in the present theory is shown by the dotted curve.

Figure 15

The momentum-dependent effective mass curves along the high-symmetry lines for various $U$ (0, 4, 6, 8, 10, and 12 from the bottom to the top). The curves in the SOPT are shown by dotted curves.

Figure 16

The average quasi-particle weight vs Coulomb interaction curves. Solid curve for the parameters $(\lambda ,{\lambda }^{\prime })=(0,1)$, the dashed curve for the parameters $(\lambda ,{\lambda }^{\prime })=(0.024,1)$. The corresponding results of the SSA are shown by thin solid curve and dashed curve, respectively.

Figure 17

Momentum distributions along the high-symmetry lines in the present theory (solid curves) and the SSA (dotted curves). The result of the SOPT for $U=12$ is also shown by dashed curve. Note that the numerical error $\pm 0.03$ is expected near the Fermi level because of the numerical integration of the quasiparticle energy.