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In collaboration with Stuart Pittel and German Sierra
1) r-DMRG versus k-DMRG.
2) Single-particle basis, level ordering and symmetries.
3) Grains and nuclei.
Rep. Prog. Phys. 67 (2004) 513-552.
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. Huse 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 chemistry
calculations. J. Chem. Phys. 110 (1999) 4127.
• Since then applications in Quantum Chemistry, small metallic grains,
nuclei, quantum Hall systems, etc…
r-DMRG
•Hubbard Model


H  t  ci c j  c j ci  U  ni ni
ij 
Nearest neighbors
i
On site
1D
Two-body interaction is diagonal. Nearest-neighbor onebody interaction.
2D
Two-body interaction still diagonal. Nearest-neighbor
plus long range one-body interaction.
k-DMRG
U


H    k nk 
c
c
c c

k1  k 2  k 3  k1  k 2  k 3 
L k1k 2k 3
k
Single-particle
energies (on site)
Infinite range
interaction
1D: Doubly- degenerate single-particle energies
2D: Multiply-degenerate single-particle energies
Xiang’s crucial observation :
The highest memory consuming operators within a block are
ci c j ck cl and ci c j ck
There are order O(L4) and O(L3) different operators.
They can be contracted with the interaction and be reduced to O(1)
and O(L), respectively
V  Vijklci c j ck cl and Ol  Vijklci c j ck
ijkl
ijk
Level Ordering:
1) According to k (Xiang 96)
2) According to the distance to the Fermi level (Nishimoto et
al. PR B 65 (2002) 165114) for short-range hopping.
In 2D, the accuracy in k-DMG and r-DMRG decreases
with the system size.
k-DMRG is better for weak U ( U/W<1) while r-DMRG
is better for strong U.
What would come out from a DMRG calculation in a
HF basis (e.g. SDW)?
What would be the impact of broken symmetries?
SDW breaks spin and translational symmetry.
What would be the result in k-space with a
renormalized U dictated by the SDW?
Ultrasmall superconducting grains
In collaboration with G. Sierra, PRL 83 (1999) 172 and PRB 61 (2000) 12302
•A fundamental question posed by P.W. Anderson in J. Phys. Chem.
Solids 11 (1959) 26 :
•“at what size of particles will superconductivity actually cease?”
•Anderson argued that for a sufficiently small metallic particle, since
d~Vol-1, there will be a critical size d ~bulk at which superconductivity
must disappear.
•This condition indeed arises for grains at the nanometer scale.
•Main motivation for the revival of this old question came from the
experimental works of:
•D.C. Ralph, C. T. Black y M. Tinkham
•PRL’s 74 (1995) 3421 ; 76 (1996) 688 ; 78 (1997) 4087.
The model used to study metallic grains is the reduced BCS
Hamiltonian in a discrete basis
.
H    j    c j c j  d  c j  c j c j 'c j '
j
j j'
Single-particle energies are assumed equally spaced
 j  jd ,
j  1,, 
where  is the total number of levels given by the Debye frequency D
and the level spacing d is
  2 D / d
The BCS gap equation is

1
1

d j 1 2  j   2  2
whose solution in the limit  >>1 gives the bulk gap
 Bulk
1
 d / 2sinh  
 
From the condition =0 we get an equation for the critical value of
the mean level spacing d

1


d c j 1 2  j   F
Anderson criterion: dc=bulk
Phase diagram of metallic grains
PBCS study of ultrasmall grains:
•Braun y J. von Delft. PRL 81 (1998)47
Conclusions of F. Braun y J. von Delft :
 “The crossover from bulk superconductivity to the
fluctuation-dominated regime can be captured in full using
fixed-N projected BCS ansatz.”
 The condensation energy changes rather abruptly at some
“critical level spacing” dc, which depends on the parity of
the grain ( even or odd).
 There are qualitative differences between the
superconducting regime (SC) d<dc and the fluctuationdominated regime (FD) d>dc.
The particle-hole DMRG
Motivation:
BCS breaks particle number. Fluctuations  N1/2 .
PBCS improves the superconducting state. Fluctuation dominated
phase?
Level ordering:
In Fermi systems, the Fermi energy EF separates the single-particle
space into a primarily occupied space (hole space) and a primarily
empty space (particle space).
Most of the correlations take place close to EF
EF
Key point: Capture most of the correlations at the beginning of
the DMRG procedure.
h
h4
h3
h2
h1
p 1 p2
p3
p4
p5
5
Superblock
We grow the particle and the hole block simultaneously. The particle
block is the medium for the hole block and vice versa.
Schematically the superblock is constructed as • Bh Bp • .
The phDMRG implemented as an infinite algorithm will work if
the correlations fall off rapidly as we progress away from the
Fermi energy.
At the beginning of the iteration procedure, the optimal states
are selected without information of the levels outside the
superblock.
This limitation of the infinite algorithm phDMRG can be
partially overcome by an effective interaction theory that
renormalizes the interaction within the superblock space.
General effective interactions theories would require at each
iteration a new Vijkl , precluding the use of the Xiang trick.
Instead, we used a phenomenological renormalization.
The bulk gap for  levels at half filling is:
d
 
2 sinh 1 /  
An effective way of taking into account the levels outside the
superblock in the nth iteration is by requiring n =  .
2n  1
sinh 1 / n 
sinh 1 / 

The total dimension for a system of  levels at half filling is
  

Dim    
  / 2
For =24, dim(24)= 2.704.516, the condensation energy computed with
the ph-DMRG and m=60 states per particle or hole blocks agrees with
the Lanczos result up to 9 digits.
The largest superblock dimension in this computation is 3066.
For =100 (=0.4), dim(100)  1029
m
Ebare/d Edress/d
1-Pm
dim
40
-40.32502
-40.49884
4.3x10-9
1246
50
-40.44623
-40.50014
2.0x10-9
2108
60
-40.46887
-40.50061
1.6x10-10
3032
70
-40.48878
-40.50068
7.1x10-11
3622
80
-40.49588
-40.50072
4.2x10-11
4820
90
-40.49815
-40.50074
1.1x10-11
6306
100
-40.49919
-40.50075
4.8x10-12
7778
110
-40.49983
-40.50075
1.5x10-12
9720
=400 was the largest size studied. With =0.224 and m =60,
dim(400)  10119 , largest dim(DMRG)=3066.
Varying m the estimated error for the condensation energie was 
10-4.
After publication we were informed by R.W. Richardson that he
solved exactly the reduced BCS Hamiltonian back in 1964!!
Comparison between the Richardson’s exact solution and the
phDMRG:
=100
EDMRG=-40.50075 ,Eexact= -40.5007557623
=400
EDMRG=-22.5168, Eexact= -22.5183141,  relative error 10-4
Condensation energy for even and odd grains
PBCS versus DMRG
Gobert, Schollwök and von Delft, EPJ B 38 (2004) 501 studied the
Josephson coupling between two grains using phDMRG.
There are two scales in the problem, the Josephson coupling and the
size of the grains.
The phDMRG breaks down for weak Josephson couplings and large
grains.
The reason is that at weak coupling there are not enough correlations
between the blocks and the medium for selecting the optimal states.
A way out might be to implement a finite algorithm phDMRG.
Finite algorithm phDMRG
EF
h
h4
h3
h2
h4
h5
h1
p1 p2
p3
p4
p5
p5
p4
p3
5
Warm up
h
h2
h3
1
Sweeping
Block
Medium
p2
p1
Legeza and Solyom (LS) have done an exhaustive study of the level
ordering and the initialization procedure in molecules and in the 1D
Hubbard model.
They used quantum information concepts like block entropy,
entanglement and separability.
PR B 69 (2003) 195116.
They conclude that the DMRG is extremely sensitive to the level ordering
and the initialization procedure.
The optimal ordering corresponds to arranging the most active orbits at
the center of the chain (opposite to the phDMRG).
LS proposed a protocol called dynamically extended active space
(DEAS).
The initialization procedure in phDMRG is well defined (infinite phDMRG
algorithm) while it seems still open in k-DMRG.
The Nuclear Shell Model
•Two kinds of fermions, protons and neutrons.
•The nuclear interaction is strong and complex.
•The Pauli principle makes possible the existence of a mean field.
2
1
2
U r   r  Dl  C l  s
2
• There is a residual nuclear interaction
H  U c c j  V c c c c

ij i
ij
 
ijkl i j k l
ijkl
Shell Model Program
1. Good valence space (usually one major shell for each particle).
2. An effective nuclear interaction adapted to the valence space.
3. A shell model diagonalization code.
Nuclei in the p-f shell
Nucleus (,)
48Ti (2,6)
50Ti (2,8)
50Cr (4,6)
54Fe (6,8)
56Ni (8,8)
m scheme
634.744
1.967.848
14.625.240
345.400.174
1.087.455.288
J=0
14.177
39.899
267.054
5.220.621
15.443.684
Application of the phDMRG to 24Mg
with S. Pittel, M, Stoitsov and S. Dimitrova PR C65 (2002) 054319
4 protons and 4 neutrons outside doubly-magic 16O
We assume 16O as an inert core and distribute the 8 nucleons
over the 2s-1d orbits. The residual (USD) interaction can be
diagonalized exactly. Dimension in m-scheme is 28503.
Spherical SM
Axial HF
Solid triangles – HF spe’s
Open triangles – spherical spe’s
Comments about the results:
•
Include sweeping. (Preliminary results do not show
much improvement)
•
By working in symmetry broken basis we did not
preserve angular momentum.
•
Conservation of angular momentum would require:
1. Work in spherical single-particle basis
2. Include all states from a given orbit in a single shot.
3. Avoid truncations within a set of degenerate
density matrix eigenvalues.
PROSPECTS: the J-DMRG
Our current program for applying DMRG to nuclear structure
calculations is based on mapping full j-orbits into sites. A site thus has
a degeneracy of 2j+1.
The J-DMRG is rooted in the IRF method of Sierra and Nishino and in
the non-Abelian DMRG method of McCulloch and Gulácsi for
incorporating symmetries in the DMRG.
We will have to deal with large and non-equal orbitals. Techniques for
calculating the reduced matrix elements using the Wigner-Eckert
theorem of of all needed operators are well known in nuclear physics.
All matrix operations are performed in an angular-momentum coupled
representation making use of the Racah algebra.
Discussion of the J-DMRG scheme can be found in Rep. Prog. Phys.
67 (2004) 513. We expect to have the first results next fall.
The Oak Ridge DMRG program
Thomas Papenbrock and David Dean from ORNL are
developing an alternative program for doing nuclear structure
calculations with the DMRG. Preliminary results in the p-f
shell.
• DMRG with sweeping.
•Axial HF basis.
•Ordered 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.
SUMMARY
•There have been a lot of efforts to apply the DMRG to several
strongly correlated finite Fermi systems (2D Hubbard and t-J
lattices, molecules, nuclei, metallic grains, quantum Hall
system systems, etc…)
•In each case one has to define a single-particle basis, a level
ordering and an initialization procedure.
• These are still open issues:
.- Level ordering: how does phDMRG with sweeping
compare with QCDMRG?
.-Is it possible to use iterative renormalization of the
couplings during warm-up in other systems?
.- Symmetry conserving procedures, non-Abelian
DMRG, J-DMRG.
.- Interplay between symmetry broken basis and
restoration of symmetries.