two approximate approaches for solving the large-scale shell model problem
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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.
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