Transcript Document 7322571

Two approximate approaches for
solving the large-scale shell
model problem
Sevdalina S. Dimitrova
Institute for Nuclear Research & Nuclear Energy
Bulgarian Academy of Sciences
Collaborators

DMRG

ISA
Jorge Dukelsky
Nicola Lo Iudice
Instituto de Estructura de la Materia,
Madrid,Spain
University of Naples, Italy
Stuart Pittel
Antonio Porrino
Bartol Research Institute, University of
Delaware, USA
University of Naples, Italy
Mario Stoitsov
Francesco Andreozzi
Institute for Nuclear Research & Nuclear
Energy, Sofia
University of Naples, Italy
Davide Bianco
University of Naples, Italy
Contains




Large - scale shell model
Density matrix renormalization group method
Importance sampling algorithm
Calculations for 48Cr in the fp-shell
Large-scale shell model

Hamiltonian:
Configuration space

fp -shell
DMRG

Goal of the project:


Background:



Develop the Density Matrix Renormalization Group (DMRG) method
for use in nuclear structure;
DMRG method introduced by Steven White in 1992 as an
improvement of Ken Wilson’s Renormalization Group;
Used extensively in condensed matter physics and quantum
chemistry;
DMRG principle:


Systematically take into account the physics of all single-particle
levels:
Still the ordering of the single-particle levels plays a crucial role for
the convergence of the calculations;
DMRG




Q: How to construct optimal approximation to the ground state
wave function when we only retain certain number of particle and
hole states?
A:
 Choose the states that maximize the overlap between the truncated
state and the exact ground state.
Q: How to do this?
A:
 Diagonalize the Hamiltonian

Define the reduced density matrices for particles and holes
DMRG

Diagonalize these matrices:
P,H represent the probability of finding a particular -state in the
full ground state wave function of the system;
Optimal truncation corresponds to retaining a fixed number of
eigenvectors that have largest probability of being in ground state, i.e.,
have largest eigenvalues;

Bottom line: DMRG is a method for systematically building in
correlations from all single-particle levels in problem. As long as
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,
without us ever having to diagonalize enormous Hamiltonian matrices;
DMRG

Subtleties:
 Must calculate matrix elements of all relevant operators
at each step of the procedure.
 This makes it possible to set up an iterative procedure
whereby each level can be added straightforwardly.
Must of course rotate set of stored matrix elements to
optimal (truncated) basis at each iteration.
 Procedure as described guarantees optimization of
ground state. To get optimal description of many states,
we may need to construct mixed density matrices,
namely density matrices that simultaneously include
info on several states of the system.
24Mg
in m - scheme
 sd-shell
 4 valent protons
 4 valent neutrons
 USD interaction
Sph
HF
Importance 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
eigenvectors of a large matrix, robust and yielding
always to ghost-free stable solutions;
algorithm with an importance sampling for reducing the
sizes of the matrix, in full control of the accuracy of the
eigensolutions;
Importance Sampling
eigenvalue problem of general form

the iterative dialgonalization
algorithm :
^
H= f Hi j g = f hi j H j j i g j i i ; j j i =
 zero approximation loop:
f j 1i ; j 2i ; : : : ; j N i g
f Hi j g (i ; j = 1; 2)
diagonalize the two-dimensional matrix
select the lowest
eigenvalue
and the corresponding eigenvector:
(
2)
(
2)
¸ 2 ; j Á2 i = c j 1i + c j 2i ;
1
2
µ
diagonalize
the ¶two-dimensional matrix
¸ j ¡ 1 bj
wher e bj = hÁj ¡ 1 j H^ j j i f or j = 3; : : : ; N
bj
Hj j
select the lowest eigenvalue and the corresponding eigenvector
……….
approximate eigenvalue and eigenvector
P
N c( N ) j i i
(
1)
(
1)
E ´ ¸ N ; j à i ´ j ÁN i =
i= 1 i
Importance Sampling

the importance sampling algorithm
 start with m basis (m >v) vectorsfand
the mHi j diagonalize
g (i ; j = 1; m)
dimensional principal submatrix
 for j = v+1, …N diagonalize the v+1-dimensional matrix :
Ã
!
¤ v ~bj
H=
; wher e ~bj = f b1j ; b2j ; :::; bv j g
~bT H
jj
j
¸ 0; (i = 1; v)
select the lowest eigenvalues i
P
state only if
j ¸ 0 ¡ ¸ i j> ²
i = 1;v
and accept the new
i
When the truncated configuration space is determined,
apply the iterative diagonalization algorithm
Importance 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
48Cr
in the fp shell: j-scheme
48Cr
in the fp shell: m-scheme
·
E = E 0 + A exp ¡
N
c
¸
48Cr
in the fp shell: m-scheme
48Cr
in the fp shell: m-scheme
48Cr:
m-scheme in the fp shell
J¼
"D M RG
"H F + D M RG
" I SA
" ex act
0+
2+
4+
-32.249
-31.650
-31.149
-32.840
-32.016
-31.668
-32.913
-32.098
-31.111
-32.953
-32.148
-31.128
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.