An Upgraded Algorithm for Shell Model Calculations and its Implementation

in Medium-Heavy Nuclei D. Bianco

Naples F. Andreozzi F. Knapp N. Lo IudiceA. Porrino

Vietri10

Currently adopted methods• Direct Diagonalization: Lanczos:Antoine E. Caurier et al. Rev.

Mod. Phys. 77, 427 (2005) for short review

• Stochastic methodology: Monte Carlo (C.W. Johnson et al.

PRL 92), suitable for ground state. Minus sign problem.• In between: Quantum MC (M. Honma et al. PRL 95).

MC to select the relevant basis states. Problems: Redundancy, symmetries broken by the stochastic procedure.

• Truncation method: Density Matrix Renormalization Group (DMRG) (J. Dukelsky and S. Pittel, Rep. Prog. Phys. 67, 513 (2004)

Iterative diagonalization algorithmA. Andreozzi, A. Porrino, and N. Lo Iudice J. Phys. A 02

• Let I = Σ |i><i| A (= A† ) ≡ { aij } = { < i | Â | j > }

A ≡

Goal: Determine the lowest eigenvalue and eigenvector

a11 a12 a13 a14 …….. a1N

a21 a22 a23 a24 …….. a2N

a31 a32 a33 a34 …….. a3N

a41 a42 a43 a44 …….. a4N

………………………..

aN1 ………………….. aNN

(λ2 , | φ2 > ) b23 = < φ2| Â | 3 >

(λ3 , | φ3 > ) ………………………………………….

b34 = < φ3| Â | 3 >

………………………………………………………………………..

b23

a33

λ2

b23

a12

a22

a11

a21

λN E(1) , |φN > = | (1)

> = ci(N) | i > i = 1, N

End first iteration loop

λ3

b34

bN-1 N

aNN

(λN-1 , | φN > )

(bN = <φ3|A|N> )

λN-1

bN-1 N (λN , | φN > )

b34

a44

first iteration loop

Def. λ1 = E(1) |φ1> = | ψ(1) >

Compute b1 = < φ1 | Â | 1 >

…… ith Iteration loop

Second iteration loop

E(1), (1) E(2), (2) …….. E(i), (i) …..

THEOREMIf the sequence E(i) converges , then

E(i) (i) E

From 1 to v eigensolutions • The algorithm has a variational foundation.• Because of its matrix formulation, it can be easily

generalized so as to generate at once an arbitrary number v of eigensolutions

The structure of the algorithm remains unchanged: We need just to make the correspondence

bj

ajj

λj-1

bj à BhT

Bh Λh

Implementation Start with a vxv matrix Av, after diagonalization (Av →Λv) and

redefinition of the basis (|i> →|φi>), compute bij and embed Λv into a larger matrix Av+r

Av+r =

• diagonalize Av+r and extract the new lowest v roots • redefine the basis (| φi > →|ψi>) and iterate as for the 2x2

matrix

λ1 0 …. 0 b11 ...…. b1j

0 λ2 …. 0 b21 ……. b2j

………. ……. 0 0 ….. λv bv1 ……. bvj

b11 ...…. bv1 a11 …….. a1j

b12 …… bv2 a21 …….. a2j

……. ……

b1j ……. bvj aj1 ..….. ajj

cFeatures of the algorithm

• Easy implementation • Variational foundation• Robust Convergence to the extremal eigenvalues Numerically stable and ghost-free solutions Orthogonality of the computed eigenvectors

• Fast : O( N2) operations• But not enough!

M-scheme • Virtue : calculation of H very easy.

• Shortcoming: the basis dimensions become huge.

• Remedy: H is sparse in m scheme

Procedure Sort {j1

n1… jmnm} (≡ partitions)

(according to increasing energies) Choose D Id = Σd |i><i| = Σd | j1

n1… jmnm > < j1

n1… jmnm |

Property Jk | j1

n1… jmnm > D

New Hamiltonian HJ = H + c [J2 – M(M+1)]Generate (by the algorithm) {(HJ)ij} {λk}, {k

(J) } [k=1,..v (<d)]

M0 MMM0

Λ0≡{λ1..λi.. λv}0

Λ1≡{λ1..λi.. λv}1

Λ0

ΛN≡{λ1..λi.. λv}N

SM space decomposition I= M0 + M

exact eigenvalues

M’MM0

Λ1 M’’

ΛN-1

M1

M2

MN

Analogy: real space (Wilson) e density matrix (White) renormalization group

…….........

…

Sampling procedureI. Identify all configurations that couple to {k

(J) }

|< k(J)

|H |j>|2 / ( ajj – λk) > ε j=d+1,…..,r (<N)II. Confine ourselves to the space IS = Id Ir

III. Apply in IS the algorithm iterative procedure and generate new v eigensolutions

{Ek(J) (1) ,k

(J) (1)} (k=1,..v)Repeat steps I to III for the new set {k

(J) (1)}

After several iteration loops we generate the sequence{Ek

(J) (1) ,k(J) (1)}, {Ek

(J) (2) ,k(J) (2) }, …….. {Ek

(J) (i) ,k(J) (i)} ….

Convergence

{Ek(J) (i) ,k

(J) (i)} → {Ek(J) ,k

(J)}.

Eigenvalues: Convergence

Eigenvalues: Convergence

B(E2): Convergence

B(E2): Convergence

Spectra

Spectra

Th Exp

THANK YOU

Numerical applications: 48Cr

- Model space (pf) shells for both Hn and Hp

- Hamiltonian

H = HNils + GKB3

- Space Dimensions N= 1.9 x 106

48Cr

B(M1): Convergence

48Cr: Comparative analysis

48Cr: IS and convergence

Sn

Model space

1h11/2

3s1/2 2d3/2

1g7/2

2d5/2

Hamiltonian H = HNils + GCDBonn

Dimensions (116Sn)

N = 1.6x 107

110Sn

116Sn: Convergence and Extrapolation

116Sn: B(E2)

Saturation reached at

48Cr n~ 8x 105 with a ratio x=n/N ~ 0.4

116Sn n~ 6x 104 with a ratio x=n/N ~ 0.004 !!! Extrapolation law

λ (n) = a + b (N/n) + c exp(- N/n)

Extrapolation

Conclusions• The algorithm is simple, robust and has a

variational foundation• Once endowed with the importance sampling, a) it keeps the extent of space truncation under

strict control b) It reaches saturation very early c) it allows for extrapolation to exact eigensolutions

• It is very promising for heavy nuclei

THANK YOU

Sampling and truncation

Energy distribution of basis states

• Convergence• The eingenvalues reach

rapidly a plateau

• The convergence is even much faster in 133Xe

• The convergence is generally smooth

• Notice, however, the sharp turn in 48Cr. It corrsponds to energy crossing proving the stability of the solutions

• Convergence of wave functions

• The overlaps with the exact eigenvectors reach soon unity even if the starting value is small

• Notice the fluctuations at small n. They correspond to the energy crossing pointed out already as an indicaion of the stability of the solutions

• B(E2)• A very stringent test

since the states are non collective and the E2 operator M(E2) may pick components with very small amplitudes

• The very fast convergenge is to be noticed indicating the

• reliability and efficiency of the sampling

• Scaling with n (number of sampled states)

λ (n) = a + b (N/n) exp(-c N/n)

ε (n) = (d/n2) exp(-c N/n)

high precision extrapolation n N

Scaling laws(For an alternative scaling see Hori et al. PRL 82 (1999))

Heuristic argument• consistency ε(n) dλ / dn• from the sampling condition Δλ = Σj Δλj = Σ i = 1,v | λi’ - λi | = Σj bj

2 / ( ajj – λ- Δλj ) by expanding in Δλj, we obtain in lowest

order

Δλ = Σj Δλj = Σj bj2 / ( ajj - λ)

In the convergence region• ajj - ann - n

• bj2 = <j-1 |H| j>2 = (i ci <i| H|j>)2

( i << j) Hij small and random for j < n = 0 for j> n

bj2 exp (- j/n)

• Δλ b (N/n) exp(-c N/n)

• ε(n) dλ / dn (d/n2) exp(-c N/n)

Δλ = Σj Δλj = Σj bj2 / ( ajj - λ)

ajj - n bj2 exp (- j/n)

Conclusions• The algorithm is simple, robust and has a

variational foundation• Once endowed with the importance

sampling, a) it keeps the extent of space truncation

under strict control b) it allows for extrapolation to exact

eigensolutions• It is very promising for heavy nuclei

THANK YOU

• The solution of the eigenvalue problem requires the diagonalization of the many-body Hamiltonian H in a space of very large dimensions

• Standard diagonalization methods are very lengthy: The CPU time goes like N3 (N ≡ dimensions of the H matrix) (Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford 1965)

• In most cases, only a few (very often one) eigenstates of a given J and T are needed.

• The non zero matrix elements of H grow only linearly with N

• Adaptive diagonalization algorithms which efficiently identify the relevant pieces of H are more suitable

Relation to Renormalization Group (RG)

• RG methods were (are) adopted in solid state to study strongly correlated systems (where mean field breaks down)

• One adopts simplified model Hamiltonians incorporating the core ingredients to describe specific phenomena

• Hubbard model H = - Σtij + U Σninj • Heisenberg model H = ΣJij SiSj

• In many cases only exact diagonalization or MC are reliable

• RG aims at solving the exact problem in the most efficient way

Real space RGK.G. Wilson Reb. Mod. Phys. 47, 773 (1975)

• Describe interactions on a sublattice forming a block A of length l by a block Hamiltonian HA in a space of dimensions M

• Form a compound block AA of length 2l and the Hamiltonian HAA of dimensions MxM

• Diagonalize HAA to find the lowest eigenstates• Project HAA into the truncated space spanned by the lowest M

eigenstates• Restart the process• The method, successfull for the Kondo problem, gives poor

results in general• Reason: Boundary conditions• When solving in A, the BC impose vanishing of WF at the

extremes, but this zero corresponds to a maximum in AA!

Relation to the real space RGK.G. Wilson Reb. Mod. Phys. 47, 773 (1975)

• Describe a block A of length l by a block Hamiltonian HA in a space of dimensions M

• Form a compound block AA of length 2l and the Hamiltonian HAA of dimensions MxM

• Diagonalize HAA to find the lowest eigenstates

• Project HAA into the truncated space spanned by the lowest M eigenstates

• Start again• The method gives poor results

because of inconsistent boundary conditions in A ersus AA

• In our case the block A is the vxv matrix

• We embed A into a larger matrix of sizes (v+k)x(v+k)

• We diagonize the larger matrix and pick the lowest v eigenvalues

• We redefine a new basis in a v-dimensional space

• We restart the iteration with the new basis

• In our case the boundary conditions are the same in any space

Density matrices RG (DMRG)S. R. White PRL 69, 2863 (1992)

• Assume a block S of length l in a Ms dimensional space and a similar block E

• Add a site to each block and form new enlarged blocks S’and E’

• Form the superblock of dimension 2l+2 from S’ and E’• Diagonalize H in the superspace• Diagonalize the density matrix ρ = TrEΨΨ obtaining the

eigenvalues wα (weights)• Form a new reduced basis for S’ out of the MS eigenstates

with the largest weights wα• By diagonalizing only in the superspace, the inconsistency

coming from the boundary conditions is avoided• The method is quite successful in solid state

DMRG in Nuclear PhysicsJ. Dukelsky and S. Pittel, Rep. Prog. Phys. 67, 513 (2004)

• In the original version the blocks were composed of sets of particle and hole states respectively

• Results: not very encouraging• In the new version (T. Papenbrock and D.J. Dean, J. Phys. G

31, S1377 (2005), S. Pittel and N. Sandulescu PRC 73, 014301 (2006)),

• they partition the SM space according to the phylosophy adopted in our approach. The only difference is that they still use the diagonalization of the density matrix to truncate the space

• As pointed out, however, there are no compelling reasons for doing so

I. Iterate up to N-1II. Update the basis { | φN-1 >, | N > }

III. Compute bN = < φN-1| Â | N >

IV. Diagonalize the matrix

Det( ) = 0  

V. Pick up the lowest eigenvalue and eigenvector

λN E(1) , |φN > = | (1) > = ci

(N) | i > i = 1, N

End first iteration loop

bN

aNN -λ

λN-1 -λ bN

We, then, start a new iteration obtaining the sequence

E(1), (1) E(2), (2) …… E(i), (i) …..

THEOREM

If the sequence E(i) converges , then

E(i) E (eigenvalue of the matrix A)

(i) (eigenvector of the matrix A)

 

Iterate up to N

Pick up the lowest eigenvalue and eigenvector

λN E(1) ,

|φN > = | (1) > = ci

(N) | i > i = 1, N

End first iteration loop

Def. |φ1> = | ψ(1) > λ1 = E(1)

Compute b1 = < φ1 | Â | 1 >

the states { | φ1 >, |1> } are not linearly independent

(|φ1> = | ψ(1) > = ci(N) | i > ) Generalized eigenvalue problem

Det( -λ )= 0

Iterate again up to N

Second iteration loop

b1

a11

λ1

b1

< φ1 |1>

1 1 < φ1 |1>

• In practice (Sampling condition) to avoid the diagonalization we search for all j states for which

Δλ = Σ i = 1,v | λi’ - λi | Σj bj2 / ( ajj – λi) > ε

and keep all of them for the diagonalization procedure

M-scheme and J-projection- New Hamiltonian HJ = H + c [J2 – M(M+1)]- Basis states characterized by the partition | j1

n1… jmnm > sorted

according to increasing energies - Crucial property<j1

n1… jmnm |J2 | j1

n’1… jmn’m > δ n1n’1

..δ nmn’m

Starting with a subspace {| j1n1… jm

nm >} large enoughit is possible to single out a set of pivotal {|J

(piv) >}

with good J

IMPORTANCE SAMPLING(F. Andreozzi, N. Lo Iudice, A. Porrino, J. Phys. G 03)

• | Ψ > = Σ ci | i >only m ( « N ) ci important (localization property )

Sampling procedure• {aij} Λ v = diag (λi) (i,j = 1, …, v)• Diagonalize Aj = for j = v+1 , N

• bj = {b1j , … , bvj}

Λv bj

bjT ajj

IMPORTANCE SAMPLING(F. Andreozzi, N. Lo Iudice, A. Porrino, J. Phys. G 03)

| Ψ > = Σ ci | i >only m ( « N ) ci important (localization property )Sampling procedure {aij} Λ v = diag (λi) (i,j = 1, …, v)Diagonalize Aj = for j = v+1 , N bj = {b1j , … , bvj} select the new v lowest eigenvalues λ1, … , λv λ1’ , … , λv’

if Σ i = 1,v | λi’ - λi | > ε accept the j –th state

Λv bj

bjT ajj