cnrs, saclay, 6 june 2005. the shell model and the dmrg approach stuart pittel bartol research...

CNRS, Saclay, 6 June 2005.

The Shell Model and the The Shell Model and the DMRG ApproachDMRG Approach

Stuart PittelStuart Pittel

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

DelawareDelaware


IntroductionIntroduction

Will discuss approach for hopefully obtaining accurate Will discuss approach for hopefully obtaining accurate solutions to nuclear shell-model problem, in cases where solutions to nuclear shell-model problem, in cases where exact diagonalization not feasible.exact diagonalization not feasible.

Method is based on use of Density Matrix Renormalization Method is based on use of Density Matrix Renormalization Group (DMRG).Group (DMRG).

The DMRG:The DMRG:- Introduced by Steven White in the early 90s to treat - Introduced by Steven White in the early 90s to treat

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


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

Subsequently applied with great success to other 1D lattices Subsequently applied with great success to other 1D lattices (spin chains, t-J, Hubbard models).(spin chains, t-J, Hubbard models).

Original formalism based on real space lattice sites, also Original formalism based on real space lattice sites, also applied - though with less success – to some 2D lattices.applied - though with less success – to some 2D lattices.

Subsequently reformulated so as not to work solely in terms Subsequently reformulated so as not to work solely in terms of real space lattice sites, replacing sites by energy (or of real space lattice sites, replacing sites by energy (or momentum) levels.momentum) levels.


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

Suggests possible usefulness of the method in the description Suggests possible usefulness of the method in the description of another finite Fermi system, of another finite Fermi system, the nucleusthe nucleus..

Recent review article on the subject:Recent review article on the subject:

- J. Dukelsky and SP, - J. Dukelsky and SP, The density matrix renormalization The density matrix renormalization group for finite fermi systemsgroup for finite fermi systems, J. Dukelsky and S. Pittel, Rep. , J. Dukelsky and S. Pittel, Rep. Prog. Phys. Prog. Phys. 6767 (2004) 513. (2004) 513.


OutlineOutline

Briefly review key steps of DMRG algorithm.Briefly review key steps of DMRG algorithm.

Discuss nuclear physics calculations of Papenbrock and Discuss nuclear physics calculations of Papenbrock and Dean.Dean.

Describe the angular-momentum-conserving JDMRG Describe the angular-momentum-conserving JDMRG approach we are developing and show first test results.approach we are developing and show first test results.


CollaboratorsCollaborators

Jorge Dukelsky (Madrid)Jorge Dukelsky (Madrid)

Nicu Sandulescu (Bucharest, Saclay)Nicu Sandulescu (Bucharest, Saclay)

Bhupender Thakur (University of Delaware graduate student)Bhupender Thakur (University of Delaware graduate student)


Brief Review of the DMRGBrief Review of the DMRG

DMRG a method for systematically taking into account DMRG a method for systematically taking into account all the degrees of freedom of a problem, without letting all the degrees of freedom of a problem, without letting problem get numerically out of hand.problem get numerically out of hand.

Method rooted in Wilson's original RG procedure, Method rooted in Wilson's original RG procedure, whereby we systematically add degrees of freedom whereby we systematically add degrees of freedom (sites or levels) until all have been treated.(sites or levels) until all have been treated.


Wilson’s RG ProcedureWilson’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×sp×s states. states.

• RG procedure implements truncation of these p×sp×s states to p states to p states, exactly as before states, exactly as before enlargement.enlargement.


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

This is done till all layers are treated.This is done till all layers are treated.

Calculation done as function of Calculation done as function of pp, the # of states kept, until , the # of states kept, until change with increasing change with increasing pp is acceptably small. is acceptably small.


Key construct for block enlargementKey construct for block enlargement

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

and store them.and store them.

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

sadjohermiteanaaaaaaaaa aaa lkjikjijijii int , , , , ,


How to do the truncationHow to do the truncation

Wilson:Wilson: Diagonalize Diagonalize hamiltonian in states of hamiltonian in states of enlarged block and truncate to enlarged block and truncate to the lowest the lowest pp eigenstates. eigenstates.

White's DMRG approach: White's DMRG approach: Consider the enlarged block B’ Consider the enlarged block B’ in the presence of a medium M in the presence of a medium M that approximates the rest of that approximates the rest of the system. Carry out the the system. Carry out the truncation based on the truncation based on the importance of the block states importance of the block states in states of full in states of full superblocksuperblock. .


Implementation of DMRG Truncation StrategyImplementation of DMRG Truncation Strategy

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

MB,1sp1,i

j| i| | tj

ij

where where tt denotes the number of states in the medium. 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.

jitj

ijBii '

,1

*'


The finite vs the infinite algorithmThe finite vs the infinite algorithm

So far, have described infinite DMRG algorithm, in which we So far, have described infinite DMRG algorithm, in which we go thru set of sites (degrees of freedom) once.go thru set of sites (degrees of freedom) once.

Will work well if correlations between layers fall off sufficiently Will work well if correlations between layers fall off sufficiently fast.fast.

Usually won't work well, since truncation in early layers has no Usually won't work well, since truncation in early layers has no way of knowing about coupling to subsequent layersway of knowing about coupling to subsequent layers ..


Can avoid this limitation by using a Can avoid this limitation by using a sweepingsweeping algorithm. algorithm.

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

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

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


Work of Papenbrock and DeanWork of Papenbrock and Dean

The best calculations to date using DMRG in nuclei reported The best calculations to date using DMRG in nuclei reported recently by Dean and Papenbrock. [lanl preprint # recently by Dean and Papenbrock. [lanl preprint # nucl-th/0412112.]nucl-th/0412112.]

The approach they follow is based on the finite-algorithm The approach they follow is based on the finite-algorithm approach.approach.- Partition neutron versus proton orbitals. Neutron orbits on - Partition neutron versus proton orbitals. Neutron orbits on one side of the “chain” and proton orbits symmetrically on the one side of the “chain” and proton orbits symmetrically on the other.other.- Use orbits that admit two particles (nlj+m and nlj-m).- Use orbits that admit two particles (nlj+m and nlj-m).- Such an m-scheme approach violates angular-momentum - Such an m-scheme approach violates angular-momentum conservation, which may be severe if truncation is significant.conservation, which may be severe if truncation is significant.- Order the orbits so that most active (i.e., those nearest the - 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 Fermi surface) are at the center of the chain. This is based on work of Legeza and collaborators. work of Legeza and collaborators. - Use closed shell plus 1p-1h states to define output from - Use closed shell plus 1p-1h states to define output from warmup phase.warmup phase.


Their Results – for Their Results – for 2828SiSi

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


Our approachOur approach

We are developing a DMRG strategy that works directly in a We are developing a DMRG strategy that works directly in a J-scheme or angular momentum conserving basis. We call it J-scheme or angular momentum conserving basis. We call it the J-DMRG. An example of a non-Abelian DMRG [I. P. the J-DMRG. An example of a non-Abelian DMRG [I. P. McCulloch and M. Gulacsi, Europhys. Lett. 57 (2002) 852.]McCulloch and M. Gulacsi, Europhys. Lett. 57 (2002) 852.]

It is our hope that by not violating angular momentum It is our hope that by not violating angular momentum conservation in the truncation steps, we can get more conservation in the truncation steps, we can get more accurate results, with smaller matrices. This is experience accurate results, with smaller matrices. This is experience from other non-Abelian DMRG work.from other non-Abelian DMRG work.

Code being developed by Nicu, Jorge and I is in absolutely Code being developed by Nicu, Jorge and I is in absolutely final throes of testing. Very first preliminary test results final throes of testing. Very first preliminary test results obtained Friday. More general code being developed by my obtained Friday. More general code being developed by my graduate student, Bhupender Thakur.graduate student, Bhupender Thakur.


Key new construct for JDMRGKey new construct for JDMRG

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

)]~~[][ ( ,)~][ ( ,]~[ ,][ , 0Klk

Kji

Lk

Kji

Kki

Kjii aaaaaaaaaaaa


Our implementation of J-DMRGOur implementation of J-DMRG

Input:Input:

(1) Model space; (1) Model space;

(2) number of active (2) number of active neutrons and protons; neutrons and protons;

(3) shell-model H; (3) shell-model H;

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


Warm-up phaseWarm-up phase

Calculate and store initial Calculate and store initial reduced matrix elements for all reduced matrix elements for all possible sets of orbits, e.g. jpossible sets of orbits, e.g. j11 →→ jj22, j, j1 1 → → jj33, …, j, …, j11 → → jj5 5 , for neutrons , for neutrons and correspondingly for protons.and correspondingly for protons.

Here, we have treated first two Here, we have treated first two orbits, both for neutrons and orbits, both for neutrons and protons. protons.

Now add third neutron level 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 phaseThe sweep phaseSweep down and then up through Sweep down and then up through neutron and proton orbits separately. In neutron and proton orbits separately. In each case, use remainder of orbits each case, use remainder of orbits (from warmup or previous sweep stage) (from warmup or previous sweep stage) plus the full set of orbits of the other plus the full set of orbits of the other type as the medium for density matrix type as the medium for density matrix truncation.truncation.

Here we have just treated proton orbits Here we have just treated proton orbits 9 and 10 forming a block. We add 9 and 10 forming a block. We add proton orbit 8, creating enlarged proton proton orbit 8, creating enlarged proton block consisting of 8 block consisting of 8 →→ 10. 10.

We use neutron orbits 7 and 6 to define We use neutron orbits 7 and 6 to define neutron medium and entire proton block neutron medium and entire proton block to define the proton medium.to define the proton medium.

Superblock obtained by coupling Superblock obtained by coupling enlarged proton block to the two parts enlarged proton block to the two parts of medium.of medium.


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

Sweep down and up through one type of particle, then thru Sweep down and up through one type of particle, then thru the other. This updates information on the optimal truncation the other. This updates information on the optimal truncation within blocks, taking into account information about the within blocks, taking into account information about the medium from the previous sweep.medium from the previous sweep.

Sweep as many times as needed till change from one sweep Sweep as many times as needed till change from one sweep to another is acceptably small.to another is acceptably small.

Program has been written, checked and preliminary tests Program has been written, checked and preliminary tests have been carried out. Will report first test results.have been carried out. Will report first test results.


Test resultsTest results

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

Exact result:Exact result:

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

Results for p=18:Results for p=18:

- - WarmupWarmup gives E gives EGSGS=-180 with all 158 states.=-180 with all 158 states.- Any number of - Any number of sweepssweeps give the same results since full give the same results since full space always used.space always used.



- - After first sweep, getAfter first sweep, get EEGSGS=-180 with a basis of 32 =-180 with a basis of 32 statesstates


- After first sweep, obtain E- After first sweep, obtain EGSGS=-180 with a basis of 38 =-180 with a basis of 38 statesstates


SummarySummary

First reviewed basic ingredients and ideas behind the DMRG First reviewed basic ingredients and ideas behind the DMRG method, with nuclei specifically in mind.method, with nuclei specifically in mind.

Then described calculations of Papenbrock and Dean, which Then described calculations of Papenbrock and Dean, which work in m-scheme. Showed reasonably promising results for work in m-scheme. Showed reasonably promising results for 2828Si, albeit less so for Si, albeit less so for 5656Ni.Ni.

Then discussed how to implement an angular momentum Then discussed how to implement an angular momentum conserving variant of the DMRG method, including sweeping. conserving variant of the DMRG method, including sweeping. Preliminary test results seem promising and we will now Preliminary test results seem promising and we will now continue to do more tests and then hopefully some serious continue to do more tests and then hopefully some serious calculations.calculations.

cnrs, saclay, 6 june 2005. the shell model and the dmrg approach stuart pittel bartol research...

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