Mean-field approximation for thermodynamic and spectral functions of correlated electrons: Strong coupling and arbitrary band filling

We present a construction of a mean-field theory for thermodynamic and spectral properties of correlated electrons reliable in the strong-coupling limit. We introduce an effective interaction determined self-consistently from the reduced parquet equations. It is a static local approximation of the two-particle irreducible vertex, the kernel of a potentially singular Bethe-Salpeter equation. The effective interaction enters the Ward identity from which a thermodynamic self-energy, renormalizing the one-electron propagators, is determined. The dynamical Schwinger-Dyson equation with the thermodynamic propagators is then used to calculate the spectral properties. The thermodynamic and spectral properties of correlated electrons are in this way determined on the same footing and in a consistent manner. Such a mean-field approximation is analytically controllable and free of unphysical behavior and spurious phase transitions. We apply the construction to the asymmetric Anderson impurity and the Hubbard models in the strong-coupling regime.

Figure 1

Spectral function of SIAM at zero temperature and $U=12$. We used Lorentzian density of states with $\mathrm{\Delta }$ taken as the energy unit. Doping the state with holes leads to merging of the central quasiparticle peak with the lower Hubbard satellite band.

Figure 2

Spectral function of SIAM for $U=8$ compared with the NRG result.

Figure 3

Band fillings $n$ and $n^T$ as a function of the doping parameter $x=-E_d-U/2$ from the static local approximation in the reduced parquet equations compared with the NRG result.

Figure 4

Kondo scale calculated from the denominator of the vertex function $K\left(\omega \right)$ at the Fermi energy $\omega =0$ as a function of the interaction strength $U$ compared with the exact Bethe-ansatz result for two doping parameters $x$.

Figure 5

Kondo scale calculated from the denominator of the vertex function $K\left(\omega \right)$ at the Fermi energy $\omega =0$ as a function of the doping parameter $x$ compared with the exact Bethe-ansatz result for a few interaction strengths.

Figure 6

Spectral function in the mean-field approximation for the spectral self-energy of the Hubbard model at zero temperature with the semielliptic density of states at half-filling.

Figure 7

Spectral function in the mean-field approximation for the spectral self-energy of the Hubbard model at zero temperature and off half-filling for $U=2$ (left panel) and $U=4$ (right panel).

Figure 8

Dependence of the local magnetic susceptibilities of SIAM $\chi$ and ${\chi }^T$ on the doping parameter $x$ for two interaction strengths. The inset shows the corresponding occupation numbers.

Figure 9

Local magnetic susceptibilities of SIAM $\chi$ and ${\chi }^T$ as a function of the interaction strength $U$ for fixed doping parameters $x$ in a logarithmic scale.

Figure 10

Local magnetic susceptibilities of SIAM $\chi$ and ${\chi }^T$ as a function of the interaction strength $U$ for fixed occupation numbers $n$ in a logarithmic scale.

Figure 11

Mean-field phase diagram of the Hubbard model at zero temperature for the paramagnet-antiferromagnet transition in $n\text{−}U$ plane. Solid (blue) line for the cubic DOS, the dashed (red) line for the semielliptic DOS.