Sevdalina S. DimitrovaInstitute for Nuclear Research and Nuclear

Energy, Sofia, Bulgaria

The Density Matrix Renormalization Group Method applied to

Nuclear Shell Model Problems

Collaborators

Jorge DukelskyInstituto de Estructura de la Materia, MadridStuart PittelBartol Research Institute, University of Delaware,

USAMario StoitsovInstitute for Nuclear Research & Nuclear Energy,

Sofia

Contains

• Introduction– Wilson’s Renormalization Group Method– Density Matrix Renormalization Group Method

• p-h DMRG basics• Application to nuclear shell model problems• Outlook

Wilson’s Renormalization Group (1974)

• The goal: to solve the Kondo problem (describes the antiferromagnetic interaction of the conduction electrons with a single localized impurity) after mapping it onto a 1D lattice in energy space.

• The assumption: low-energy states most important for law-energy behavior of large quantum systems

Wilson’s Renormalization Group (1974)• The idea: numerically integrate out the irrelevant

degrees of freedom• The algorithm:

→isolate finite subspace of the full configuration space

→diagonalize numerically →keep m lowest energy eigenstates→add a site →iterate

Sampling the configuration space

m s s s

Infinite procedure

“the onion picture”

superblock environment

•the size of the superblock stays the same•while the environment shrinks

From WRG to DMRG

•The WRG was the first numerical implementation of the RG to a non-perturbative problem like the Kondo model, for which it had enormous success.•WRG cannot be applied to other lattice problems. For 1D Hubbard models it begins to deviate significantly from the exact results.

•The problem resides in the fact that the truncation strategy is based solely on energy arguments.•The solution to this problem was proposed by White who introducted the DMRG: PRL 69 (1992) 2863 and PR B 48 (1993) 10345.

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. Husse 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 chemical calculation. J. Chem. Phys. 110 (1999) 4127.

• Since then applications in Quantum Chemistry, small metallic grain, nuclei, quantum Hall systems, etc…

• review article: U. Schollwöck, Rev. Mod. Phys. 77(2005)259

The particle-hole DMRG

Introduced by J. Dukelsky and G.Sierra to study systems of utrasmall superconducting grains

PRL 83 (1999) 172 and PRB 61 (2000) 12302

Motivation:

BCS breaks particle number. PBCS improves the superconducting state. Fluctuation dominated phase?

Level ordering:

•In Fermi systems, the Fermi level defines hole and particle sp states.

•Most of the correlations take place close to the Fermo level

p-h DMRG basics

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Let's consider for simplicity axially-symmetric Nilsson-like levels, which admit four states (s=4):

When we add the next level:

• number of particle states goes from m to s×m• number of hole states goes from m to s×m• number of states involving particles coupled to

holes also goes up.

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Basic idea of DMRG method:

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truncate from the s×m states for particles to the optimum m of them, and likewise from the s×m states for holes to the optimum m of them.

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Finite procedure

medium environment superblock m x s x m

•starting point: infinite procedure•size of superblock and medium stay the same•while environment block shrinks•medium block stored from previous iteration•“zipping” back and forth → iterative convergence

warm up1st sweep2nd sweep

Sampling criterion: FAQ

Q: How to construct optimal approximation to the Q: How to construct optimal approximation to the ground state wave function when we only retain ground state wave function when we only retain certain number of particle and hole states?certain number of particle and hole states?A: Choose the states that maximize the overlap A: Choose the states that maximize the overlap between the truncated state and the exact between the truncated state and the exact ground state.ground state.Q: How to do this?Q: How to do this?A: A:

•Diagonalize the HamiltonianDiagonalize the Hamiltonian

…

•Define the reduced density matrices for particles Define the reduced density matrices for particles and holesand holes

•Diagonalize these matrices:Diagonalize these matrices:

represent the probability of finding a represent the probability of finding a particular particular -state in the full ground state wave -state in the full ground state wave function of the system;function of the system;

,P H

…

Optimal truncationOptimal truncation corresponds to retaining a corresponds to retaining a fixed number of eigenvectors that have largest fixed number of eigenvectors that have largest probability of being in ground state, i.e., have probability of being in ground state, i.e., have largest eigenvalues;largest eigenvalues;

Parameter of the procedure:Parameter of the procedure: number of states number of states retained after each interaction;retained after each interaction;

Bottom line:Bottom line: DMRG is a method for DMRG is a method for systematically building in correlations from all systematically building in correlations from all single-particle levels in problem. As long as single-particle levels in problem. As long as convergence is sufficiently rapid as a function convergence is sufficiently rapid as a function of number of states kept, it should give an of number of states kept, it should give an accurate description of the ground state of the accurate description of the ground state of the system, without us ever having to diagonalize system, without us ever having to diagonalize enormous Hamiltonian matrices;enormous Hamiltonian matrices;

•Must calculate matrix elements of all relevant Must calculate matrix elements of all relevant operators at each step of the procedureoperators at each step of the procedure

•The highest memory consuming operators The highest memory consuming operators within a block arewithin a block are

•They can be contracted with the interaction They can be contracted with the interaction and be reduced to O(1) and O(L) and be reduced to O(1) and O(L)

Subtleties:

kjiijk

ijklllkjiijkl

ijkl cccVOccccVV and

kjilkji ccccccc and

Subtleties:

•This makes it possible to set up an This makes it possible to set up an iterative iterative procedureprocedure whereby each level can be added whereby each level can be added straightforwardly. Must of course straightforwardly. Must of course rotate rotate set of stored set of stored matrix elements to optimal (truncated) basis at each matrix elements to optimal (truncated) basis at each iteration.iteration.•Procedure as described guarantees optimization of Procedure as described guarantees optimization of ground stateground state. To get optimal description of many . To get optimal description of many states, we may need to construct density matrices that states, we may need to construct density matrices that simultaneously include info on several states of the simultaneously include info on several states of the system. system. •Legeza and Solyom used quantum information Legeza and Solyom used quantum information concepts like block entropy and entanglement to concepts like block entropy and entanglement to conclude that the DMRG is extremely conclude that the DMRG is extremely sensitive to the sensitive to the level ordering and the initialization procedure.level ordering and the initialization procedure.

ph-DMRG: model calculations

– Hamiltonian

– 40 particles in j=99/2 shell

– size of the superblock ndim~10 26

– parameters: 2.0;1.0;1 g

ph-DMRG: realistic nuclear structure calculations

• Hamiltonian

1 1 2 2 1 1 2 2

1 1 2 2

1 1 2 2 4 4 3 3

1 1 2 2

3 3 4 4

1 1 2 2 4 4 3 3

1 1 2 2

3 3 4 4

, ,,

,

,112344

,,

1234,,

m m m mm m

m m m mm mm m

m m m mm mm m

H a a

V a a a a

H H H V a a a a

ph-DMRG: realistic nuclear structure calculations

• configuration space

ph-DMRG: 24Mg in m - scheme

sd-shell 4 valent protons 4 valent neutrons USD interaction

SphHF

ph-DMRG: Infinite vs. finite procedure

ph-DMRG: 48Cr in the j-scheme

ph-DMRG: 48Cr in the m-scheme

The Oak Ridge DMRG program

Thomas Papenbrock from ORNL developed an alternative program for doing nuclear structure calculations with the DMRG:

•DMRG with sweeping in the m-scheme

•Axial HF basis.

•The 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.

The Oak Ridge DMRG program

T. Papenbrock and D. J. Dean, J.Phys. G 31 (2005) S1377

56Ni

Comments about the results & Outlook

• By working in symmetry broken basis we did not preserve angular momentum.•Conservation of angular momentum would require:

Work in spherical single particle basisInclude all states from a given orbit in a single shot.Avoid truncations within a set of degenerate density matrix eigenvalues.

Comments about the results & Outlook

Include sweeping (results do not show expected improvement) •Apply an effective interaction theory that renormalizes the interaction within the superblock space.