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.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.

Computer Physics CommunicationsApr 28, 202012 citations

Exact diagonalization: the Bose–Hubbard model as an example

J. M. Zhang, R X Dong

We take the Bose-Hubbard model to illustrate exact diagonalization techniques in a pedagogical way. We follow the road of first generating all the basis vectors, then setting up the Hamiltonian matrix with respect to this basis, and finally using the Lanczos algorithm to solve low lying eigenstates and eigenvalues. Emphasis is placed on how to enumerate all the basis vectors and how to use the hashing trick to set up the Hamiltonian matrix or matrices corresponding to other quantities. Although our route is not necessarily the most efficient one in practice, the techniques and ideas introduced are quite general and may find use in many other problems.We take the Bose-Hubbard model to illustrate exact diagonalization techniques in a pedagogical way. We follow the road of first generating all the basis vectors, then setting up the Hamiltonian matrix with respect to this basis, and finally using the Lanczos algorithm to solve low lying eigenstates and eigenvalues. Emphasis is placed on how to enumerate all the basis vectors and how to use the hashing trick to set up the Hamiltonian matrix or matrices corresponding to other quantities. Although our route is not necessarily the most efficient one in practice, the techniques and ideas introduced are quite general and may find use in many other problems.

arXiv:1102.4006European Journal of PhysicsApr 7, 2010144 citations

On-the-fly exact diagonalization solver for quantum electronic models

Кашин Илья Владимирович, Vladimir V. Mazurenko

We propose a distinct numerical approach to effectively solve the problem of partial diagonalization of the super-large-scale quantum electronic Hamiltonian matrices. The key ingredients of our scheme are the new method for arranging the basis vectors in the computer's RAM and the algorithm allowing not to store a matrix in RAM, but to regenerate it on-the-fly during diagonalization procedure. This scheme was implemented in the program, solving the Anderson impurity model in the framework of dynamical mean-field theory (DMFT). The DMFT equations for electronic Hamiltonian with 18 effective orbitals that corresponds to the matrix with the dimension of 2.4 * 10^9 were solved on the distributed memory computational cluster.We propose a distinct numerical approach to effectively solve the problem of partial diagonalization of the super-large-scale quantum electronic Hamiltonian matrices. The key ingredients of our scheme are the new method for arranging the basis vectors in the computer's RAM and the algorithm allowing not to store a matrix in RAM, but to regenerate it on-the-fly during diagonalization procedure. This scheme was implemented in the program, solving the Anderson impurity model in the framework of dynamical mean-field theory (DMFT). The DMFT equations for electronic Hamiltonian with 18 effective orbitals that corresponds to the matrix with the dimension of 2.4 * 10^9 were solved on the distributed memory computational cluster.

arXiv:1508.04895Aug 20, 20150 citations

Numerical Study of the 8-32-Particle Eigenstates of the Pairing Hamiltonian

R. W. Richardson

Exact eigenstates of the pairing-force Hamiltonian have been calculated for $2N$ particles in $2N$ equispaced, doubly degenerate single-particle levels for $N=4, 8, 12, \mathrm{and} 16$ . The states calculated are the ground state and the first two excited seniority-zero and seniority-two states. These states were calculated for values of the interaction strength that give pairing energies that are typical of nuclear systems. Numerical results are given for the energies and occupation probabilities of these states and are compared with the corresponding results of the Bardeen-Cooper-Schrieffer (BCS) theory. It is shown that the BCS theory is a worse approximation for the larger values of $N$ than it is for the smaller values, with errors as high as 140% (20%) for the excitation energies of the seniority-zero (-two) states. The methods used in the calculation of these states are outlined.Exact eigenstates of the pairing-force Hamiltonian have been calculated for $2N$ particles in $2N$ equispaced, doubly degenerate single-particle levels for $N=4, 8, 12, \mathrm{and} 16$. The states calculated are the ground state and the first two excited seniority-zero and seniority-two states. These states were calculated for values of the interaction strength that give pairing energies that are typical of nuclear systems. Numerical results are given for the energies and occupation probabilities of these states and are compared with the corresponding results of the Bardeen-Cooper-Schrieffer (BCS) theory. It is shown that the BCS theory is a worse approximation for the larger values of $N$ than it is for the smaller values, with errors as high as 140% (20%) for the excitation energies of the seniority-zero (-two) states. The methods used in the calculation of these states are outlined.

Physical ReviewJan 21, 1966200 citations

Treatment of like-particle pairing with quartets

Michelangelo Sambataro, Nicolae Sandulescu

The ground state correlations induced by a general pairing Hamiltonian in a finite system of like fermions are described in terms of four-body correlated structures (quartets). These are real superpositions of products of two pairs of particles in time-reversed states. Quartets are determined variationally through an iterative sequence of diagonalizations of the Hamiltonian in restricted model spaces and are, in principle, all distinct from one another. The ground state is represented as a product of quartets to which, depending on the number of particles (supposed to be even, in any case), an extra collective pair is added. The extra pair is also determined variationally. In case of pairing in a spherically symmetric mean field, both the quartets and the extra pair (if any) are characterized by a total angular momentum J=0. Realistic applications of the quartet formalism are carried out for the Sn isotopes with the valence neutrons in the 50-82 neutron shell. Exact ground state correlation energies, occupation numbers and pair transfer matrix elements are reproduced to a very high degree of precision. The formalism also lends itself to a straightforward and accurate description of the lowest seniority 0 and 2 excited states of the pairing Hamiltonian. A simplified representation of the ground state as a product of identical quartets is eventually discussed and found to improve considerably upon the more traditional particle-number projected-BCS approach.The ground state correlations induced by a general pairing Hamiltonian in a finite system of like fermions are described in terms of four-body correlated structures (quartets). These are real superpositions of products of two pairs of particles in time-reversed states. Quartets are determined variationally through an iterative sequence of diagonalizations of the Hamiltonian in restricted model spaces and are, in principle, all distinct from one another. The ground state is represented as a product of quartets to which, depending on the number of particles (supposed to be even, in any case), an extra collective pair is added. The extra pair is also determined variationally. In case of pairing in a spherically symmetric mean field, both the quartets and the extra pair (if any) are characterized by a total angular momentum J=0. Realistic applications of the quartet formalism are carried out for the Sn isotopes with the valence neutrons in the 50-82 neutron shell. Exact ground state correlation energies, occupation numbers and pair transfer matrix elements are reproduced to a very high degree of precision. The formalism also lends itself to a straightforward and accurate description of the lowest seniority 0 and 2 excited states of the pairing Hamiltonian. A simplified representation of the ground state as a product of identical quartets is eventually discussed and found to improve considerably upon the more traditional particle-number projected-BCS approach.

Aug 16, 201615 citations

Multipair approach to pairing in finite nuclear systems

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.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.

AIP conference proceedingsJan 1, 20120 citations

^{8-9}

, the calculation can be done within a day on standard desktop computers.

Exact eigenstates of the pairing-force Hamiltonian have been calculated for

2N

particles in

2N

equispaced, doubly degenerate single-particle levels for

N=4, 8, 12, \mathrm{and} 16

. The states calculated are the ground state and the first two excited seniority-zero and seniority-two states. These states were calculated for values of the interaction strength that give pairing energies that are typical of nuclear systems. Numerical results are given for the energies and occupation probabilities of these states and are compared with the corresponding results of the Bardeen-Cooper-Schrieffer (BCS) theory. It is shown that the BCS theory is a worse approximation for the larger values of

N

than it is for the smaller values, with errors as high as 140% (20%) for the excitation energies of the seniority-zero (-two) states. The methods used in the calculation of these states are outlined.