A solvable model with an extreme AGP ground state: Relationships among fermion pairs, pairons, and natural spin geminals

Abstract

A model many-fermion Hamiltonian is presented for which the ground state is asymptotically an Antisymmetrized Geminal Powers (AGP) wave function with largest possible greatest eigenvalue for its two-particle reduced density matrix. Closed analytical expressions and plane-wave expansions are presented for the generating geminal of the AGP ground state and for its one-particle reduced density matrix. The natural orbitals for this generating geminal are plane waves. The generating geminal shows intensely local character in its intracule and corresponds to the formation of a quasi-boson from two fermions. One may appropriately modify this generating geminal to introduce zero occupation numbers of its one-particle reduced density matrix and to make all the nonzero occupation numbers of its one-particle reduced density matrix equal, thus making this geminal a generator of an extreme AGP wave function, with an extreme large eigenvalue for its two-particle reduced density matrix. Closed analytical expressions are also given for this modified geminal and for its one-particle reduced density matrix. The modified generating geminal develops anomolously long “tails” in the intracule function. Nevertheless, both generating geminals introduced here reduce to a renormalized Dirac δ-function of the intracule coordinate, in the appropriate limit. In this limit, both of these generating geminals produce an absolute extreme AGP wave function with the absolute extreme value, N/2, for the Lowdin-normalized occupation number of the generating geminal, and vanishing occupation numbers for all other natural geminals, in the two-particle reduced density matrix of this wavefunction. A demonstration is given showing the formation of quasi-bosons in this ground state and showing their relationship to the generating geminal and to the other natural geminals of the two-particle reduced density matrix of the extreme AGP wave function. The similarities and differences of the features of this model and the accepted models of the superconducting ground state of electrons in metals, and the superfluid ground state of liquid He4 are mentioned.