The%20Shell%20Model%20and%20the%20DMRG%20Approach

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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

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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).

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• 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.

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• 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.

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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.

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Collaborators

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

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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.

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Wilsons 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 L1st) admits s
  states. Thus, enlarged block has ps states.
• RG procedure implements truncation of these ps
  states to p states, exactly as before enlargement.

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• 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.

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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.

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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.

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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.

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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.

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• 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.

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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 (nljm 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.

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Their Results for 28Si
Also did calculations for 56Ni, but results not
as good.
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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.

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Key new construct for JDMRG

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

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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.

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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.

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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.

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• 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.

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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 p18
• - Warmup gives EGS-180 with all 158 states.
• - Any number of sweeps give the same results
  since full space always used.

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• Results for p10
• - After first sweep, obtain EGS-180 with a
  basis of 38 states

• Results for p8
• - After first sweep, get EGS-180 with a basis
  of 32 states

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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.