Two approximate approaches for Two approximate approaches for solving the large-scale shell solving the large-scale shell

model problemmodel problem

Sevdalina S. DimitrovaInstitute for Nuclear Research & Nuclear Energy

Bulgarian Academy of Sciences

CollaboratorsCollaborators

DMRGDMRGJorge DukelskyJorge DukelskyInstituto de Estructura de la Materia, Instituto de Estructura de la Materia,

Madrid,SpainMadrid,Spain

Stuart PittelStuart PittelBartol Research Institute, University of Bartol Research Institute, University of

Delaware, USADelaware, USA

Mario StoitsovMario StoitsovInstitute for Nuclear Research & Nuclear

Energy, Sofia

ISAISANicola Lo IudiceNicola Lo IudiceUniversity of Naples, ItalyUniversity of Naples, Italy

Antonio Porrino Antonio Porrino University of Naples, ItalyUniversity of Naples, Italy

Francesco AndreozziFrancesco AndreozziUniversity of Naples, ItalyUniversity of Naples, Italy

Davide BiancoDavide BiancoUniversity of Naples, ItalyUniversity of Naples, Italy

Contains Contains

Large - scale shell modelLarge - scale shell model Density matrix renormalization group methodDensity matrix renormalization group method Importance sampling algorithmImportance sampling algorithm Calculations for Calculations for 4848Cr in the fp-shellCr in the fp-shell

Hamiltonian:Hamiltonian:

Large-scaleLarge-scale shell model shell model

ConfigurationConfiguration space space fp -shell fp -shell

Goal of the project:Goal of the project: Develop the Density Matrix Renormalization Group (DMRG) Develop the Density Matrix Renormalization Group (DMRG)

method for use in nuclear structure;method for use in nuclear structure;

Background: Background: DMRG method introduced by Steven White in 1992 as an DMRG method introduced by Steven White in 1992 as an

improvement of Ken Wilson’s Renormalization Group;improvement of Ken Wilson’s Renormalization Group; Used extensively in condensed matter physics and quantum Used extensively in condensed matter physics and quantum

chemistrychemistry;;

DMRG principle:DMRG principle: Systematically take into account the physics ofSystematically take into account the physics of all all single-particle single-particle

levels:levels: Still the Still the orderingordering of the single-particle levels plays a crucial role for of the single-particle levels plays a crucial role for

the convergence of the calculations;the convergence of the calculations;

DMRGDMRG

Q: How to construct optimal approximation to the ground state Q: How to construct optimal approximation to the ground state wave function when we only retain certain number of particle and wave function when we only retain certain number of particle and hole states?hole states?

A:A: Choose the states that maximize the overlap between the truncated Choose the states that maximize the overlap between the truncated

state and the exact ground state.state and the exact ground state. Q: How to do this?Q: How to do this? A:A:

Diagonalize the HamiltonianDiagonalize the Hamiltonian

Define the reduced density matrices for particles and holesDefine the reduced density matrices for particles and holes

DMRGDMRG

Diagonalize these matrices:Diagonalize these matrices:

P,HP,H represent the probability of finding a particular represent the probability of finding a particular -state in the -state in the

full ground state wave function of the system;full ground state wave function of the system;

Optimal truncationOptimal truncation corresponds to retaining a fixed number of corresponds to retaining a fixed number of eigenvectors that have largest probability of being in ground state, i.e., eigenvectors that have largest probability of being in ground state, i.e., have largest eigenvalues;have largest eigenvalues;

Bottom lineBottom line:: DMRG is a method for systematically building in DMRG is a method for systematically building in correlations from all single-particle levels in problem. As long as correlations from all single-particle levels in problem. As long as convergence is sufficiently rapid as a function of number of states kept, convergence is sufficiently rapid as a function of number of states kept, it should give an accurate description of the ground state of the system, it should give an accurate description of the ground state of the system, without us ever having to diagonalize enormous Hamiltonian matrices;without us ever having to diagonalize enormous Hamiltonian matrices;

DMRGDMRG

DMRGDMRG

Subtleties:Subtleties: Must calculate matrix elements of all relevant operators Must calculate matrix elements of all relevant operators

at each step of the procedure.at each step of the procedure. This makes it possible to set up an iterative procedure This makes it possible to set up an iterative procedure

whereby each level can be added straightforwardly. whereby each level can be added straightforwardly. Must of course rotate set of stored matrix elements to Must of course rotate set of stored matrix elements to optimal (truncated) basis at each iteration.optimal (truncated) basis at each iteration.

Procedure as described guarantees optimization of Procedure as described guarantees optimization of ground state. To get optimal description of many states, ground state. To get optimal description of many states, we may need to construct mixed density matrices, we may need to construct mixed density matrices, namely density matrices that simultaneously include namely density matrices that simultaneously include info on several states of the system. info on several states of the system.

2424Mg in m - schemeMg in m - scheme

SphHF

sd-shell 4 valent protons 4 valent neutrons USD interaction

Importance SamplingImportance Sampling

F. Andreozzi, A. Porrino and N. Lo Iudice,

“A simple iterative algorithm for generating selected eigenspaces of large matrices” J. Phys. A: 35 (2002) L61–L66

an iterative algorithm for determining a selected set of an iterative algorithm for determining a selected set of eigenvectors of a large matrix, eigenvectors of a large matrix, robustrobust and yielding always to and yielding always to ghost-free ghost-free stable solutions;stable solutions;

algorithm with an importance sampling for reducing the sizes of algorithm with an importance sampling for reducing the sizes of the matrix, in full control of the matrix, in full control of the accuracythe accuracy of the eigensolutions; of the eigensolutions;

Importance SamplingImportance Sampling

the iterative dialgonalization algorithmthe iterative dialgonalization algorithm ::

zero approximation loopzero approximation loop:diagonalize the two-dimensional matrixselect the lowest eigenvalue and the corresponding eigenvector:

diagonalize the two-dimensional matrix

select the lowest eigenvalue and the corresponding eigenvector……….

¸2 ; j Á2i = c(2)1 j 1i + c(2)

2 j 2i ;

E(1) ´ ¸N ; j Ã(1) i ´ j ÁN i =P N

i=1 c(N )i j ii

eige

nval

ue p

robl

em o

f ge

nera

l for

mei

genv

alue

pro

blem

of

gene

ral f

orm

H=fHi j g= fhi j H j j ig j ii ; j j i = f j 1i ; j 2i ; : : : ; j N ig

fHi j g (i; j = 1;2)

µ¸ j ¡ 1 bjbj H j j

¶where bj = hÁj ¡ 1 j H j j i f or j = 3;:: :;N

approximate eigenvalue and eigenvector

Importance SamplingImportance Sampling

the importance sampling algorithmthe importance sampling algorithm

start with m basis (m >v) vectors and diagonalize the m-dimensional principal submatrix for j = v+1, …N diagonalize the v+1-dimensional matrix :

select the lowest eigenvalues and accept the new state only if

¸0i ;(i = 1;v)

Pi=1;v j ¸0

i ¡ ¸ i j> ²

When the truncated configuration space is determined, apply the iterative diagonalization algorithm

fHi j g(i; j = 1;m)

H =

äv

~bj~bT

j H j j

!

; where ~bj = fb1j ;b2j ; :::;bvj g

Importance SamplingImportance Sampling

The algorithm has been shown to be completely equivalent to the method of optimal relaxation of I. Shavitt and has therefore a variational foundation;

It can be proven that the approximate solution of the eigenvalue problem converges to the exact one;

A generalization to calculate several eigenvalues and eigenvectors is straightforward

4848Cr in the fp shell: j-schemeCr in the fp shell: j-scheme

4848Cr in the fp shell: m-schemeCr in the fp shell: m-scheme

E = E0 +A exp·¡

Nc

¸

4848Cr in the fp shell: m-schemeCr in the fp shell: m-scheme

4848Cr in the fp shell: m-schemeCr in the fp shell: m-scheme

4848Cr: m-scheme in the fp shellCr: m-scheme in the fp shell

J ¼ "D M R G "H F +D M RG "I SA "exact

0+ -32.249 -32.840 -32.913 -32.9532+ -31.650 -32.016 -32.098 -32.1484+ -31.149 -31.668 -31.111 -31.128

Conclusions:Conclusions:

•The first calculations within the ISA in the m-scheme prove the applicability of the method to large-scale shell-model problems.

•The DMRG is also a practical approach which needs more tuning.

•At it is, the ISA code requires a lot of disk space for considering 56Ni for example.

•At it is, the DMRG code requires large RAM memory to describe heavier nuclei.

•Both methods are applicable for odd mass nuclei as well.