Quantum impurity model

The main goal of the iQIST software package is to provide a comprehensive tool for solving the quantum impurity models, especially the Anderson impurity model. Generally speaking, the multi-orbital Anderson impurity model (AIM) can be written as

$$H_{\text{imp}} = H_{\text{loc}} + H_{\text{bath}} + H_{\text{hyb}},$$

where

$$\begin{align} & H_{\text{loc}} = \sum_{\alpha\beta} E_{\alpha\beta} d_{\alpha}^{\dagger} d_{\beta}+\sum_{\alpha\beta\gamma\delta} U_{\alpha\beta\gamma\delta} d^{\dagger}_{\alpha}d^{\dagger}_{\beta} d_{\gamma} d_{\delta}, \\ & H_{\text{hyb} } = \sum_{\textbf{k}\alpha\beta} V^{\alpha\beta}_{\textbf{k}} c_{\textbf{k}\alpha}^{\dagger} d_{\beta} + h.c., \\ & H_{\text{bath}} = \sum_{\textbf{k}\alpha} \epsilon_{\textbf{k}\alpha} c_{\textbf{k}\alpha}^{\dagger} c_{\textbf{k}\alpha}. \end{align}$$

In these equations, Greek letters in the subscripts denote a combined spin-orbital index, the fermion operator $d_\alpha^{\dagger}$ ($d_\alpha$) is creating (annihilating) an electron with index $\alpha$ on the impurity site, while $c_{\textbf{k}\alpha}^{\dagger}$ ($c_{\textbf{k}\alpha}$) is the creation (annihilation) operator for conduction band (bath) electron with spin-orbital index $\alpha$ and momentum $\textbf{k}$.

The first term in $H_{\text{loc}}$ is the general form of the impurity single particle term with energy level splitting and inter-orbital hybridization. This term can be generated by crystal field (CF) splitting or spin-orbit coupling (SOC), etc. The second term in $H_{\text{loc}}$ is the Coulomb interaction term which can be parameterized by intra(inter)-band Coulomb interactions $U$ $(U')$ and Hund's rule coupling $J$ or Slater integral parameters $F^{k}$. The hybridization term $H_{\text{hyb}}$ describes the process of electrons hopping from the impurity site to the environment and back. $H_{\text{bath}}$ describes the non-interacting bath. This Anderson impurity model is usually solved self-consistently in the DMFT calculations.