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

iii

ijijji nnUcccctH

1D

Nearest neighbors On site

Two-body interaction is diagonal. Nearest-neighbor one-body interaction.

2D

Two-body interaction still diagonal. Nearest-neighbor plus long range one-body interaction.

k-DMRG

321

321321kkk

kkkkkkk

kk ccccL

UnH

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

kjilkji ccccccc and

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

kjiijk

ijklllkjiijkl

ijkl cccVOccccVV and

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 grainsIn 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 .

'

'' jj

jjjjjj

jj ccccdccH

Single-particle energies are assumed equally spaced

,,1, jdjj

where is the total number of levels given by the Debye frequency D and the level spacing d is

2D d/

The BCS gap equation is

221 2

11

j

jd

whose solution in the limit >>1 gives the bulk gap

1

sinh2/dBulk

From the condition =0 we get an equation for the critical value of the mean level spacing d

Fjjcd

2

1

1

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 fluctuation-dominated regime (FD) d>dc.

The particle-hole DMRGMotivation:

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.

p1 p2 p3 p4 p5h1h2h3h4h5

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 • .

Superblock

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:

/1sinh2

d

An effective way of taking into account the levels outside the superblock in the nth iteration is by requiring n = .

/1sinh12

/1sinh

n

n

The total dimension for a system of levels at half filling is

2/

Dim

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 dim40 -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

p1 p2 p3 p4 p5h1h2h3h4h5

h5h4h3h2h1 p4 p3 p2 p1p5

EF

Warm up

Sweeping

Block Medium

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.

• There is a residual nuclear interaction

slClDrrU 2

2

2

1

ij ijkl

lkjiijkljiij ccccVccUH

Shell Model Program1. 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 (,) m scheme J=048Ti (2,6) 634.744 14.17750Ti (2,8) 1.967.848 39.89950Cr (4,6) 14.625.240 267.05454Fe (6,8) 345.400.174 5.220.62156Ni (8,8) 1.087.455.288 15.443.684

Application of the phDMRG to 24Mg

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

with S. Pittel, M, Stoitsov and S. Dimitrova PR C65 (2002) 054319

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.