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### The Shell Model and the DMRG Approach

Stuart Pittel

Bartol Research Institute and Department of Physics and Astronomy, University of Delaware

Introduction

- Will discuss approach for hopefully obtaining accurate solutions to nuclear shell-model problem, in cases where exact diagonalization not feasible.
- Method is based on use of Density Matrix Renormalization Group (DMRG).
- The DMRG:

- Introduced by Steven White in the early 90s to treat quantum lattices.

[S. R. White, PRL 69, 2863 (1992); S. R. White, PRB48, 10345 (1993); S. R. White and D. A. Huse, PRB48, 3844 (1993).]

Enormously successful, producing for g.s energy of the spin-one Heisenberg chain results accurate to 12 significant figures.

- Subsequently applied with great success to other 1D lattices (spin chains, t-J, Hubbard models).
- Original formalism based on real space lattice sites, also applied - though with less success – to some 2D lattices.
- Subsequently reformulated so as not to work solely in terms of real space lattice sites, replacing sites by energy (or momentum) levels.

Reformulated versions have proven useful in describing several finite Fermi systems (e.g., in quantum chemistry, in small metallic grains, and in 2D electron systems).

- Suggests possible usefulness of the method in the description of another finite Fermi system, the nucleus.
- Recent review article on the subject:

- J. Dukelsky and SP, The density matrix renormalization group for finite fermi systems, J. Dukelsky and S. Pittel, Rep. Prog. Phys. 67 (2004) 513.

Outline

- Briefly review key steps of DMRG algorithm.
- Discuss nuclear physics calculations of Papenbrock and Dean.
- Describe the angular-momentum-conserving JDMRG approach we are developing and show first test results.

Collaborators

- Jorge Dukelsky (Madrid)
- Nicu Sandulescu (Bucharest, Saclay)
- Bhupender Thakur (University of Delaware graduate student)

Brief Review of the DMRG

- DMRG a method for systematically taking into account all the degrees of freedom of a problem, without letting problem get numerically out of hand.
- Method rooted in Wilson\'s original RG procedure, whereby we systematically add degrees of freedom (sites or levels) until all have been treated.

Wilson’s RG Procedure

- Assume we\'ve already treated a given number of sites (L) and that the total number of states we have kept to describe them is p. Refer to that portion of the system as the block.
- Assume that the next layer (the L+1st) admits s states. Thus, enlarged block has p×s states.
- RG procedure implements truncation of these p×s states to p states, exactly as before enlargement.

Process continues by adding the next layer and implementing again a truncation to p states.

- This is done till all layers are treated.
- Calculation done as function of p, the # of states kept, until change with increasing p is acceptably small.

Key construct for block enlargement

- At each step of process, evaluate matrix elements of all hamiltonian sub-operators

and store them.

- Having this info for the block plus the additional level/site enables us to calculate them for the enlarged block.

How to do the truncation

- Wilson: Diagonalize hamiltonian in states of enlarged block and truncate to the lowest p eigenstates.
- White\'s DMRG approach: Consider the enlarged block B’ in the presence of a medium M that approximates the rest of the system. Carry out the truncation based on the importance of the block states in states of full superblock.

Implementation of DMRG Truncation Strategy

- Hamiltonian is diagonalized in superblock, yielding a ground state wave function

where t denotes the number of states in the medium.

Ground state density matrix for the enlarged block is then constructed and diagonalized.

Truncate to the p eigenstates with largest eigenvalues. By definition, they are the most important states of the enlarged block in the ground state of the superblock, i.e. the system.

The finite vs the infinite algorithm

- So far, have described infinite DMRG algorithm, in which we go thru set of sites (degrees of freedom) once.
- Will work well if correlations between layers fall off sufficiently fast.
- Usually won\'t work well, since truncation in early layers has no way of knowing about coupling to subsequent layers.

Can avoid this limitation by using a sweeping algorithm.

- After going thru all layers, reverse direction and update the blocks based on results stored in previous sweep. Done iteratively until acceptably small change from one sweep to the next.

- Requires a first pass, called the warmup stage. Here we could, e.g., use the Wilson RG method to get a first approximation to the optimum states in each block. Since they will be improved in subsequent sweeps, not crucial that it be very accurate approximation.

- Called the finite algorithm. Usually needed when dealing with finite fermi systems such as nuclei.

Work of Papenbrock and Dean

- The best calculations to date using DMRG in nuclei reported recently by Dean and Papenbrock. [lanl preprint # nucl-th/0412112.]
- The approach they follow is based on the finite-algorithm approach.

- Partition neutron versus proton orbitals. Neutron orbits on one side of the “chain” and proton orbits symmetrically on the other.

- Use orbits that admit two particles (nlj+m and nlj-m).

- Such an m-scheme approach violates angular-momentum conservation, which may be severe if truncation is significant.

- Order the orbits so that most active (i.e., those nearest the Fermi surface) are at the center of the chain. This is based on work of Legeza and collaborators.

- Use closed shell plus 1p-1h states to define output from warmup phase.

Their Results – for 28Si

Also did calculations for 56Ni, but results not as good.

Our approach

- We are developing a DMRG strategy that works directly in a J-scheme or angular momentum conserving basis. We call it the J-DMRG. An example of a non-Abelian DMRG [I. P. McCulloch and M. Gulacsi, Europhys. Lett. 57 (2002) 852.]
- It is our hope that by not violating angular momentum conservation in the truncation steps, we can get more accurate results, with smaller matrices. This is experience from other non-Abelian DMRG work.
- Code being developed by Nicu, Jorge and I is in absolutely final throes of testing. Very first preliminary test results obtained Friday. More general code being developed by my graduate student, Bhupender Thakur.

Key new construct for JDMRG

- Now we must calculate reduced matrix elements of all coupled sub-operators of H:

Our implementation of J-DMRG

- Input:

(1) Model space;

(2) number of active neutrons and protons;

(3) shell-model H;

(4) single-shell reduced matrix elements for all active orbits and all sub-operators of H.

Warm-up phase

- Calculate and store initial reduced matrix elements for all possible sets of orbits, e.g. j1→ j2, j1 → j3, …, j1→ j5 , for neutrons and correspondingly for protons.
- Here, we have treated first two orbits, both for neutrons and protons.
- Now add third neutron level j3, using proton block to define medium for enlargement of neutron block and truncating based on resulting g.s density matrix.
- Continue till all neutron and proton blocks included.

The sweep phase

- Sweep down and then up through neutron and proton orbits separately. In each case, use remainder of orbits (from warmup or previous sweep stage) plus the full set of orbits of the other type as the medium for density matrix truncation.
- Here we have just treated proton orbits 9 and 10 forming a block. We add proton orbit 8, creating enlarged proton block consisting of 8 → 10.
- We use neutron orbits 7 and 6 to define neutron medium and entire proton block to define the proton medium.
- Superblock obtained by coupling enlarged proton block to the two parts of medium.

As always, truncation is to same number of states as before enlargement.

- Sweep down and up through one type of particle, then thru the other. This updates information on the optimal truncation within blocks, taking into account information about the medium from the previous sweep.
- Sweep as many times as needed till change from one sweep to another is acceptably small.
- Program has been written, checked and preliminary tests have been carried out. Will report first test results.

Test results

- Tests carried out for 2 neutrons and two protons in f-p shell subject to an SU(3) hamiltonian.
- Exact result:

- EGS=-180. Complete basis of 0+ states has 158 states.

- Results for p=18:

- Warmup gives EGS=-180 with all 158 states.

- Any number of sweeps give the same results since full space always used.

- After first sweep, obtain EGS=-180 with a basis of 38 states

- Results for p=8:

- After first sweep, getEGS=-180 with a basis of 32 states

Summary

- First reviewed basic ingredients and ideas behind the DMRG method, with nuclei specifically in mind.
- Then described calculations of Papenbrock and Dean, which work in m-scheme. Showed reasonably promising results for 28Si, albeit less so for 56Ni.
- Then discussed how to implement an angular momentum conserving variant of the DMRG method, including sweeping. Preliminary test results seem promising and we will now continue to do more tests and then hopefully some serious calculations.

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