Transcript The Density Matrix Renormalization Group Method for

The Density Matrix Renormalization
Group Method applied to
Nuclear Shell Model Problems
Sevdalina S. Dimitrova
Institute for Nuclear Research and Nuclear Energy,
Sofia, Bulgaria
Collaborators
Jorge Dukelsky
Instituto de Estructura de la Materia, Madrid
Stuart Pittel
Bartol Research Institute, University of Delaware, USA
Mario Stoitsov
Institute for Nuclear Research & Nuclear Energy, Sofia
Contains
• Introduction
– Wilson’s Renormalization Group Method
– Density Matrix Renormalization Group Method
• p-h DMRG basics
• Application to nuclear shell model problems
• Outlook
Wilson’s Renormalization Group (1974)
• The goal: to solve the Kondo problem (describes the
antiferromagnetic interaction of the conduction
electrons with a single localized impurity) after
mapping it onto a 1D lattice in energy space.
• The assumption: low-energy states most important
for law-energy behavior of large quantum systems
Wilson’s Renormalization Group (1974)
• The idea: numerically integrate out the irrelevant
degrees of freedom
• The algorithm:
→isolate finite subspace of the full configuration
space
→diagonalize numerically
→keep m lowest energy eigenstates
→add a site
→iterate
Sampling the configuration space
superblock environment
m
s
s
s
Infinite procedure
•the size of the superblock stays the same
•while the environment shrinks
“the onion picture”
From WRG to DMRG
•The WRG was the first numerical implementation of
the RG to a non-perturbative problem like the Kondo
model, for which it had enormous success.
•WRG cannot be applied to other lattice problems. For
1D Hubbard models it begins to deviate significantly
from the exact results.
•The problem resides in the fact that the truncation
strategy is based solely on energy arguments.
•The solution to this problem was proposed by White
who introducted the DMRG:
PRL 69 (1992) 2863 and PR B 48 (1993) 10345.
From 1D lattices to finite Fermi systems
• S. White introduced the DMRG to treat 1D lattice
•
•
•
•
•
models with high accuracy. PRL 69 (1992) 2863 and
PR B 48 (1993) 10345.
S. White and D. Husse studied S=1 Heisenberg chain
giving the GS energy with 12 significant figures. PR B
48 (1993) 3844.
T. Xiang proposed the k-DMRG for electrons in 2D
lattices. PR B 53 (1996) R10445.
S. White and R. L. Martin used the k-DMRG for
quantum chemical calculation. J. Chem. Phys. 110
(1999) 4127.
Since then applications in Quantum Chemistry, small
metallic grain, nuclei, quantum Hall systems, etc…
review article: U. Schollwöck, Rev. Mod. Phys.
77(2005)259
The particle-hole DMRG
Introduced by J. Dukelsky and G.Sierra to study
systems of utrasmall superconducting grains
PRL 83 (1999) 172 and PRB 61 (2000) 12302
Motivation:
BCS breaks particle number. PBCS improves the
superconducting state. Fluctuation dominated phase?
Level ordering:
•In Fermi systems, the Fermi level defines hole and
particle sp states.
•Most of the correlations take place close to the
Fermo level
p-h DMRG basics
Let's consider for simplicity axially-symmetric
Nilsson-like levels, which admit four states (s=4):
ip | a | j p 
F
ih | a | jh 
When we add the next level:
• number of particle states goes from m to s×m
• number of hole states goes from m to s×m
• number of states involving particles coupled to
holes also goes up.
ikp p| a| a| j| plp  i 'p |  k p | a | l p  | j 'p  
i 'p | a | j 'p  kl  (1)nl  k p | a | l p  ij
i'p | a | j 'p 
F
ih' | a | jh' 
ih | a | jh   ih' |  kh | a | lh  | jh'  
kih' | a | ljhh'  kl  (1) nl  kh | a | lh  ij
…
i p | a | j p   i 'p |  k p | a | l p  | j 'p  
k' p | a | l' p 
i p | a | j p  kl  (1)n  k p | a | l p  ij
l
F


i'p | a | j 'p 
ih' | a | jh' 
ih | a | jh   ih' |  kh | a | lh  | jh'  
ikh' h| a| a| |jh'lh kl  (1) nl  kh | a | lh  ij
…
F


i p | a | j p   i 'p |  k p | a | l p  | j 'p  
k p | a | l p 
i 'p | a | j 'p  kl  (1)nl  k p | a | l p  ij
i'p | a | j 'p 
ih' | a | jh' 

Basic idea of DMRG method:
F

truncate from the s×m
states for particles to
the optimum m of them,
i'p | a | j 'p  and likewise from the
s×m states for holes to
the optimum m of them.
ih' | a | jh' 
Finite procedure
st
up
2nd1warm
sweep
sweep
•starting
medium point:
environment
infinite procedure
superblock
•sizemof superblock
x
s andx medium
m stay the same
•while environment block shrinks
•medium block stored from previous iteration
•“zipping” back and forth → iterative convergence
Sampling criterion: FAQ
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
•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;
Parameter of the procedure: number of states
retained after each interaction;
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;
Subtleties:
•Must calculate matrix elements of all relevant
operators at each step of the procedure
•The highest memory consuming operators
within a block are
cicj ck cl and cicj ck
•They can be contracted with the interaction
and be reduced to O(1) and O(L)
V  Vijklci c j ck cl and Ol  Vijklci c j ck
ijkl
ijk
Subtleties:
•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 density matrices that simultaneously
include info on several states of the system.
•Legeza and Solyom used quantum information concepts like
block entropy and entanglement to conclude that the
DMRG is extremely sensitive to the level ordering and the
initialization procedure.
ph-DMRG:
model calculations
– Hamiltonian
– 40 particles in j=99/2 shell
– size of the superblock
ndim~10 26
– parameters:
  1; g  0.1 ;   0.2
ph-DMRG:
realistic nuclear structure calculations
• Hamiltonian
H  , 

1m1 , 2 m2
1
4

1 m1 , 2 m2
 3 m3 , 4 m4
 ,m , m a m a m 
1
1
2
2
1
1
2
2
 , 
V1234
a1m1 a2 m2 a4 m4 a3m3
H  H  H 

1 m1 , 2 m2
 3 m3 , 4 m4
  
V1234
a1m1 a2m2 a 4 m4 a3m3
ph-DMRG:
realistic nuclear structure calculations
• configuration space
ph-DMRG:
24Mg
in m - scheme
sd-shell
 4 valent protons
 4 valent neutrons
 USD interaction

Sph
HF
ph-DMRG:
Infinite vs. finite procedure
ph-DMRG:
48Cr
in the j-scheme
ph-DMRG:
48Cr
in the m-scheme
The Oak Ridge DMRG program
Thomas Papenbrock from ORNL developed an
alternative program for doing nuclear structure
calculations with the DMRG:
•DMRG with sweeping in the m-scheme
•Axial HF basis.
•The levels from the Fermi energy.
•In the warm up, protons are the medium for neutrons
and vice versa.
•In the sweeping, protons are to the left of the chain
and neutrons to the right.
The Oak Ridge DMRG program
56Ni
T. Papenbrock and D. J. Dean, J.Phys. G 31 (2005) S1377
Comments about the results & Outlook
• By working in symmetry broken basis we did not
preserve angular momentum.
Conservation of angular momentum would require:
Work in spherical single particle basis
Include all states from a given orbit in a single
shot.
Avoid truncations within a set of degenerate
density matrix eigenvalues.
•
Comments about the results & Outlook
Include sweeping (results do not show expected
improvement)
Apply an effective interaction theory that
renormalizes the interaction within the superblock
space.
•