Efficient variational approach to the impurity problem and its application to the dynamical mean-field theory

Within the framework of exact diagonalization (ED), we compute the ground state of Anderson impurity problem using the variational approach based on the configuration interaction (CI) expansion. We demonstrate that an accurate ground state can be obtained by iteratively diagonalizing a matrix with the dimension that is less than 10$\%$ of the full Hamiltonian. The efficiency of the CI expansion for different problems is analyzed. By way of example, we apply this method to the single-site dynamical mean-field theory using ED as the impurity solver. Specifically, to demonstrate the usefulness of this approach, we solve the attractive Hubbard model in the grand-canonical ensemble, where the $s$-wave superconducting solution is explicitly obtained.

Figure 1

(a) Basis states for four fermions occupying eight spin orbitals. Picking up one determinantal state as the reference, all basis states can be classified by the number of particle-hole pairs with respect to the reference state. (b) Six determinantal states constructed from CAS(2,4). Two secondary and active orbitals are always empty and filled, respectively. (c) Some allowed and truncated states in CAS(2,4)CIS or equivalently RAS(2,−1;2,1). The inactive orbitals can have at most one hole, while secondary orbitals can at most have one particle, therefore the last two states are not included.

Figure 2

(a) The spectral function of the spin-up electron for $U=-1,-3,-5$ (5 bath sites) at quarter-filling. A gap at the Fermi energy opens when $\left|U\right|$ increases. (Inset) The corresponding order parameter $\varphi$ as a function of $U$. (b) The spectral function of the spin-up electron for $U=-3$ for 5, 7, 9 bath sites. The gap structures around Fermi energy are almost identical.