S. De Baerdemacker

Inspired by the wavefunction forms of exactly solvable algebraic Hamiltonians, we present several wavefunction ansatze. These wavefunction forms are exact for two-electron systems; they are size consistent; they include the (generalized) antisymmetrized geminal power, the antisymmetrized product of strongly orthogonal geminals, and a Slater determinant wavefunctions as special cases. The number of parameters in these wavefunctions grows only linearly with the size of the system. The parameters in the wavefunctions can be determined by projecting the Schrödinger equation against a test-set of Slater determinants; the resulting set of nonlinear equations is reminiscent of coupled-cluster theory, and can be solved with no greater than O (N⁵) scaling if all electrons are assumed to be paired, and with O (N⁶) scaling otherwise. Based on the analogy to coupled-cluster theory, methods for computing spectroscopic properties, molecular forces, and response properties are proposed.

A three-configuration mixing calculation is presented in the context of the Interacting Boson Model (IBM1), with the aim to describe recently observed collective bands built on low-lying 0(+) states in the neutron-deficient lead isotopes. Possible effects on the nuclear binding energy are addressed, caused by mixing of these low-lying 0(+) intruder states into the ground state, and a new method is described in order to provide a consistent description of both ground-state and excited-state properties.

The algebraic derivation of the matrix elements of the quadrupole collective variables within the canonical basis of SU(1, 1) x SO(5) is applied to a simple fermionic system with j = 1/2 to illustrate the method.

The present contribution discusses a connection between the exact Bethe Ansatz eigenstates of the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and the multi-phonon states of the Tamm-Dancoff Approximation (TDA). The connection is made on the algebraic level, by means of a deformed quasi-spin algebra with a bosonic Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. Each exact Bethe Ansatz eigenstate is mapped on a unique TDA multi-phonon state, shedding light on the physics behind the Bethe Ansatz structure of the exact wave function. The procedure is illustrated with a model describing neutron pairing in ⁵⁶Fe.

Background: The reduced, level-independent, Bardeen-Cooper-Schrieffer Hamiltonian is exactly diagonalizable by means of a Bethe ansatz wave function, provided the free variables in the ansatz are the solutions of the set of Richardson-Gaudin equations. On the one side, the Bethe ansatz is a simple product state of generalized pair operators. On the other hand, the Richardson-Gaudin equations are strongly coupled in a nonlinear way, making them prone to singularities. Unfortunately, it is nontrivial to give a clear physical interpretation to the Richardson-Gaudin variables because no physical operator is directly related to the individual variables.
Purpose: The purpose of this paper is to shed more light on the singular behavior of the Richardson-Gaudin equations, and how this is related to the product wave structure of the Bethe ansatz.
Method: A pseudodeformation of the quasispin algebra is introduced, leading towards a Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. This enables an adiabatic connection of the exact Bethe ansatz eigenstates with pure bosonic multiphonon states. The physical interpretation of this approach is an adiabatic suppression of the Pauli exclusion principle.
Results: The method is applied to a so-called “picket-fence” model for the BCS Hamiltonian, displaying a typical critical behavior in the Richardson-Gaudin variables. It was observed that the associated bosonic multiphonon states change collective nature at the critical interaction strengths of the Richardson-Gaudin equations.
Conclusions: The Pauli exclusion principle is the main responsible for the critical behavior of the Richardson-Gaudin equations, which can be suppressed by means of a pseudodeformation of the quasispin algebra.
©2012 American Physical Society