An iterative approach for the exact solution of the pairing Hamiltonian

Liqiang Wei, Chia‐Chung Sun, Zong-Hao Zeng

A. Kamimura

Michelangelo Sambataro

A method is described to construct the ground state of a general pairing Hamiltonian for a finite nuclear systemas a product of collective, real, distinct pairs. These are determined sequentially via an iterative variational procedure that resorts to diagonalizations of the Hamiltonian in restricted model spaces. Different applications of themethod are provided that include comparisonswith exact and projected BCS results. The quantities that are examined are correlation energies, occupation numbers and pair transfer matrix elements. The method is seen to work well leading, under some circumstances, to solutions that are basically exact.

R. W. Richardson, Noah Sherman

Xiaoyu Liu, Chong Qi

We present a program for solving exactly the general pairing Hamiltonian based on diagonalization. The program generates the seniority-zero shell-model-like basis vectors via the ‘01’ inversion algorithm. The Hamiltonian matrix is constructed in this seniority-zero space. The program evaluates all non-zero elements of the Hamiltonian matrix “on the fly” using the scattering operator and a search algorithm. The matrix is diagonalized by using the iterative Lanczos algorithm. The OpenMP parallel program thus developed, PairDiag, can efficiently calculate the ground-state eigenvalue and eigenvector of the general pairing Hamiltonian for both the even-mass and the odd-mass system. The program is packaged in a Fortran module, which makes it easy to use the program to replace the BCS approximation in standard self-consistent mean field calculations. For systems with dimension around 108, the calculation can be done within hours on standard desktop computers. Program Title: PairDiag. Program Files doi: https://doi.org/10.17632/dzzspfszsh.1 Licensing provisions: CC by NC 3.0. Programming language: Fortran 95. Nature of problem: The numerically exact solution of general pairing Hamiltonian can be solved by diagonalization in configuration spaces of fixed seniority. Solution method: The program constructs the seniority-zero space by the ‘01’ inversion algorithm, and diagonalizes the general pairing Hamiltonian by the Lanczos plus QR algorithm. Restrictions: The total number of orbits involved must be less than 63, and the dimension that can be calculated is restricted by the local RAM condition.

Stanley Engelsberg

The equations for the Green's functions of the theory of superconductivity are developed in an iterative scheme in which the two-particle function is rewritten as the functional derivative of the single-particle function with respect to external-source terms. When a source coupled to electron pairs in addition to an external potential coupled to the charge density is included in the Hamiltonian, the possibility for a gap in the single-particle excitations is found. The first order in the iteration scheme for the solution to the Green's function equations is independent of the way in which the two-particle function is generated, as the final result must be. Equations for the vertex functions are obtained and used to find the linear response of the current to an applied electromagnetic field. An exact solution to the vertex equation of interest is used to show that no current will be induced by a static longitudinal vector potential. In addition, all the functions calculated are found to have translationally invariant solutions. Thus, the original invariances of the Hamiltonian are restored when the source terms are set equal to zero. Corrections to the self-energy are estimated by using another of the solutions to the vertex equations.