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Physical Review A

Excitation operators associated with antisymmetrized geminal-power states

Brian Weiner, Osvaldo Goscinski

Published Jan 1, 1983• 29 citations
Publisher

Abstract

Various expressions for antisymmetrized geminal-power (AGP) states are reviewed. With the use of a simple relationship for creation and annihilation operators of BCS (Bardeen, Cooper, and Schrieffer)-type states, such operators are characterized for AGP and GAGP (generalized AGP) schemes. The effect of degeneracy in the first-order reduced density matrix on these operators is displayed.

DOI
10.1103/physreva.27.57

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Various expressions for antisymmetrized geminal-power (AGP) states are reviewed. With the use of a simple relationship for creation and annihilation operators of BCS (Bardeen, Cooper, and Schrieffer)-type states, such operators are characterized for AGP and GAGP (generalized AGP) schemes. The effect of degeneracy in the first-order reduced density matrix on these operators is displayed.
The generalized antisymmetrized geminal-power (GAGP) function, shown to be the ground state for the oneparticle-hole propagator, can be computed using the rather simple properties of its reduced density matrices and the properties of unitary transformations. The practical procedure involves limited storage notwithstanding the rather large number of configurations with which the GAGP function can be associated. Using the stability properties previously derived for this function, we construct the self-consistent one-particle-hole propagator.
The antisymmetrized geminal power (AGP) wave function has a long history and considerable conceptual appeal, but in many situations its accuracy is wanting. Here, we consider a form of configuration interaction (CI) based upon the AGP wave function and take advantage of its killing operators to construct an excitation manifold. Our geminal CI reduces to standard single-determinant-based CI in the limit in which AGP reduces to a single determinant. It substantially improves upon AGP in the reduced BCS Hamiltonian, which serves as a prototype for the kinds of strong pairing correlations relevant in Bardeen-Cooper-Schrieffer-style superconductivity. Moreover, our geminal CI naturally generalizes to add correlation to more general geminal-based wave functions than AGP.
Functions which can be expressed as an antisymmetrized power of a single two-particle function (a geminal) occur in the BCS ansatz. They constitute a comparatively tractable generalization of a single Slater determinant. The second-order reduced density matrix and bounds on its eigenvalues are obtained for the most general such antisymmetrized geminal power (AGP). These eigenvalues are physically of great importance, being analogs, in an analysis of the wavefunction into pair states, of the one-particle occupation numbers of the independent-particle model. A necessary and sufficient condition is given that an alleged first-order reduced density matrix be derivable from an N-particle AGP function. It is shown that only for exceptional functions (called extreme) is the generating geminal of the AGP function an eigenfunction of the 2-matrix. The extreme 2-matrix is diagonalized explicitly and provides a vivid example of the fact that one-particle occupation numbers alone are unable to convey certain information of decisive significance about the wavefunction. The extreme 2-matrix satisfies the assumption basic to current theories of superconductivity.
The antisymmetrized geminal power (AGP), a wave function equivalent to number-projected Hartree-Fock-Bogoliubov (HFB), and number-projected Bardeen-Cooper-Schrieffer (BCS) when working in the paired (natural orbitals) basis, has proven to be an excellent reference for strong pairing interactions. Several correlation methods have also been applied on top of AGP. In this work, we show how AGP can also be applied to spin systems by simply basing its formulation on a spin su(2)su(2)su(2) algebra. We here implement spin AGP and spin AGP-based correlation techniques and benchmark them on the XXZ and ${\mathrm{J}}_{1}\text{\ensuremath{-}}{\mathrm{J}}_{2}$ Heisenberg models, both in one and two dimensions. Our results indicate that spin AGP is a promising starting point for modeling spin systems.
Some of the difficulties encountered in the approximation of Green's functions are reviewed with particular emphasis on the self-consistent polarization propagator. Some new basic theorems are presented, involving a general procedure, which lead to both a derivation and alternate prescriptions for the computations. As demonstrated by Linderberg and Öhrn the ground state is an AGP (antisymmetrized geminal power). An energy optimized AGP ( E AGP ) in a Brueckner orbital representation can be used to generate a manifold of excited states| K ⟩ = O K + |AGP⟩, O K |AGP⟩ = 0 with the excitation operators, O + K , and killers, O K , being combinations of particle-hole and hole-particle operators. The SCPP can be expressed then as ⟨⟨ X, Y ⟩⟩ E = - ⟨AGP| XT 0 (- E + E AGP ) Y + YT 0 ( E + E AGP ) X |AGP⟩ in terms of the resolvent T 0 ( ) = P ( - PHP ) -1 with P spanned by the states| K ⟩⟨ K |, in the sense of Löwdin. Our results establish a natural relationship between hermiticity, completeness, non-redundance of operator manifolds with stationarity and killer conditions.
The antisymmetrized geminal power (AGP) wave function has a long history and is known by different names in various chemical and physical problems. There has been recent interest in using AGP as a starting point for strongly correlated electrons. Here, we show that in a seniority-conserving regime, different AGP-based correlator representations based on generators of the algebra, killing operators, and geminal replacement operators are all equivalent. We implement one representation that uses number operators as correlators and has linearly independent curvilinear metrics to distinguish the regions of Hilbert space. This correlation method called J -CI provides excellent accuracy in energies when applied to the pairing Hamiltonian.