$N$-electron Slater determinants from nonunitary canonical transformations of fermion operators

Mean-field methods such as Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) constitute the building blocks upon which more elaborate many-body theories are based. The HF and HFB wave functions are built out of independent quasiparticles resulting from a unitary linear canonical transformation of the elementary fermion operators. Here, we discuss the possibility of allowing the HF transformation to become nonunitary. The properties of such HF vacua are discussed, as well as the evaluation of matrix elements among such states. We use a simple ansatz to demonstrate that a nonunitary transformation brings additional flexibility that can be exploited in variational approximations to many-fermion wave functions. The action of projection operators on nonunitary-based HF states is also discussed and applied, in a variation-after-projection approach, to the one-dimensional Hubbard model with periodic boundary conditions.

Figure 1

Left: Total correlation energy (in units of $t$), predicted by several methods for 1D Hubbard model calculations as a function of the number of sites $N_s$. Right: Corresponding correlation energy per particle. Calculations were performed at half-filling, with $U=4t$. The correlation energy has been defined with respect to the broken-symmetry HF solution.

Figure 2

Structure parameters ${\Delta }_1\left(j\right)$ and ${\Delta }_2\left(j\right)$ [defined by Eqs. (89) and (90), respectively] for the broken-symmetry Slater determinants in unitary (left)- and nonunitary (right)-based methods. For the latter, we show the structure parameters for the two determinants $|\Phi \rangle$ and $|\overline{\Phi }\rangle$. The calculations have been performed for a 48-site periodic 1D Hubbard lattice at $U=4t$. LMS${}_z$HF denotes the simultaneous projection into good $S_z$ and LM quantum numbers.

Figure 3

Correlation energy per electron (in units of $t$), predicted by a variety of approximate methods for a 14-site periodic 1D Hubbard model as a function of $N/N_s$. The correlation energy has been defined with respect to the broken-symmetry HF solution. We were unable to converge KHF for $N=8$.