diﬀusionmontecarloinmomentumspacebasedon ... · directdiagonalization...

Diffusion Monte Carlo in momentum space based onCoupled Cluster wave functions

A.Roggero, A.Mukherjee (ECT*), F.Pederiva

Otranto - 31 May, 2013

Outline of the talk

Ab–initio methods for nuclear structureDirect DiagonalizationMonte Carlo

Configuration Interaction Monte CarloQuantum Monte Carlo in Fock spaceImportance Sampling and Sign-ProblemCoupled Cluster solutions as trial wave–functions

Benchmark case: 3D Electron Gas

Conclusions

A.Roggero, A.Mukherjee (ECT*), F.Pederiva (University of Trento)Diffusion Monte Carlo in momentum space based on Coupled Cluster wave functionsOtranto - 31 May, 2013 2 / 1

Ab-initio methods for nuclear structure

A = 3,4 Several exact methods:Faddev-Yakubovsky,EIHH,NCSM,..(see eg. Kamada [PRC.64.044001])

A >4 Very few ab–initio methodsHamiltonian matrix diagonalization in basis–functionspace: NCSM (Navrátil [J.Phys.G.36.083101])Stochastic approach in R–space: GFMC


Direct Diagonalization

choose model single–particle space Swrite the Hamiltonian in the Hilbert space spanned by the N–particleSlater Determinants |n〉, where n = {na}, na = 0, 1 and a ∈ S.

H =∑a∈S

εaa†aaa +12

∑abcd∈S

Vabcd a†aa†bac ad + . . .

write interaction in the model–spacediagonalize the resulting matrix

Problemcomputational cost ∝ to dimension of the A-particle Hilbert space!




1960’s → 102 dim. (p-shell)1980’s → 105 dim. (sd -shell)1990’s → 109 dim. (fp-shell)2000’s → 1010 dim.· · ·

Monte Carlo methods

Use a projection operator to filter the ground–state

P[H]|Ψn〉 = |Ψn+1〉 ⇒ limn→∞

P[H]n|ΦT 〉 = |0〉

eg. Pa[H] = 1−∆τ H or Pb[H] = e−∆τ H

the projection is then performed stochastically.

can handle much larger systemsvirtually exact resultsformulation in R–space limits the choice of interactions to a limitedclass (ie. Argonne-Urbana family)


Monte Carlo methods

Use a projection operator to filter the ground–state

P[H]|Ψn〉 = |Ψn+1〉 ⇒ limn→∞

P[H]n|ΦT 〉 = |0〉

eg. Pa[H] = 1−∆τ H or Pb[H] = e−∆τ H

the projection is then performed stochastically.

can handle much larger systemsvirtually exact resultsformulation in R–space limits the choice of interactions to a limitedclass (ie. Argonne-Urbana family)


-100

-90

-80

-70

-60

-50

-40

-30

-20En

ergy

(MeV

)

AV18AV18+IL7 Expt.

0+

4He0+2+

6He 1+3+2+1+

6Li3/2−1/2−7/2−5/2−5/2−7/2−

7Li

0+2+

8He 2+2+

2+1+

0+

3+1+

4+

8Li

1+

0+2+

4+2+1+3+4+

0+

8Be

3/2−1/2−5/2−

9Li

3/2−1/2+5/2−1/2−5/2+3/2+

7/2−3/2−

7/2−5/2+7/2+

9Be

1+

0+2+2+0+3,2+

10Be 3+1+

2+

4+

1+

3+2+

3+

10B

3+

1+

2+

4+

1+

3+2+

0+

0+

12C

Argonne v18with Illinois-7

GFMC Calculations24 November 2012

S. Pieper, R. Wiringa et. al (ANL)

What is missing?

We need an ab–initio method that has both:a level of accuracy comparable to few-body techniques (DD methods)favorable scaling with system–size (MC methods)ability to treat general interactions

In general R–space MC are not efficient when used with non–local (ie.momentum dependent) interactions, which unfortunately are veryimportant in many fields:

χ–EFT potentials in Nuclear Physicspseudopotentials in Solid–state Physics


What has been done so far?

keep R–space as the stage and express EFT potentials in local form(eg. Gezerlis [arXiv:1303.6243] )

can account only for a limited number of terms

perform MC in Fock–spaceuse HS–transform to linearize interaction(eg. SMMC: Alhassid [Int.J.Mod.Phys.B15.1447])

limited only to certain types of interactions (sign problem)use approximate solution to guide the random walk

Configuration Interaction Monte Carlo ([arXiv:1304.1645])

yet another advantage in working on a Fock–spaceaccess to quantities that are extremely difficult to estimate in R–space

in k–space → Momentum Distributions


Easy-access to Momentum Distribution

Preliminary results for A = 66 particles with Yukawa interaction

0 1 2K / K

F

0.25

0.5

0.75

1n(K

)

major improvement on current R–space MC methods(eg. see Holzmann et al [PRL.107.110402])




0 1 2K / K

F

0.0001

0.01

1n(K

)





0 1 2K / K

F

1e-06

0.0001

0.01

1n(K

)



Configuration Interaction Monte Carlo

H =∑a∈S

εaa†aaa +12

∑abcd∈S

Vabcd a†aa†bac ad

Our model–space will be the set of SD n = {na} and na = 0, 1 for a ∈ S.

P = 1−∆τ(H − ET

)→ |Ψτ+∆τ 〉 = P|Ψτ 〉

Ψτ+∆τ (m) =∑

n

〈m|P|n〉Ψτ (n) =∑

n

(〈m|P|n〉∑m〈m|P|n〉

)(∑m

〈m|P|n〉

)Ψτ (n)

=∑

n

p(m,n)w(n)Ψτ (n)

MC sampling possible if p(m,n) > 0 → 〈m|H|n〉 > 0 are problematic !


Configuration Interaction Monte Carlo

In R–space the sign problem is circumvented using the so–called"fixed–node approximation" which employs importance sampling

P(m,n) → PFN(m,n) = ΨT (m)P(m,n)Ψ−1T (n)

In SD space we can do something similar (Ceperley et al.[PRB.51.13039])

〈m|P|n〉 = 1−∆τΨT (m)(H(m,n)− ET

)Ψ−1T (n)

H(m,n) =

{H(m,n) if s(n,n) < 0

0 if s(n,n) > 0

H(n,n) = H(n,n) +∑

s(m,n)>0

s(m,n)

where s(n,n) = ΨT (m)H(m,n)Ψ−1T (n).


Wave–functions for Importance Sampling

A good ΨT should beflexible enough to incorporate relevant correlations in the systemquick to evaluate

In R–spacealmost 50 years of experience in optimizing WFextremely good forms are available (Slater/BCS-Jastrow,..)

In K–spacealmost no experienceFT of R–space WF doesn’t work (high–k tails)


Wave–functions for Importance Sampling

A very accurate way to account for correlations in a generic Fock–space isthe Coupled Cluster ansatz:

|ΨT 〉 = e−T |ΦHF 〉 with T = T1 + T2 + . . .

Here we will restrict to CCD case: T = T2 = 12∑

ij ,ab tabij a†aa

†baj ai .

Is the CCD wave–function even quick to evaluate in SD space?

We need to calculate

ΦmCCD

( p1p2···pmh1h2···hm

)= ΦCCD(n) for |n〉 = a†p1 . . . a

†pmah1 . . . ahm |ΦHF〉

It turns out that one can write a recursive formula ([arXiv:1304.1549])

ΦmCCD ( ······ ) =

m∑γ=2

m∑µ<ν

(−)γ+µ+νtpµpνh1hγ Φm−2

CCD ( ······ )


Benchmark: 3DEG in momentum–space [arXiv:1304.1549]

weakly and strongly correlated regimes accessible tuning a singledensity–parameter: rs (Wigner–Seitz radius)single–particle space S =

{plane waves | k2 <= K 2

MAX}

Basis size-1

-0.58

-0.57

-0.56

-0.55

-0.51

-0.50

-0.49

-0.48

Co

rrel

atio

n e

ner

gy

(a.

u.)

162-1

342-1

2378-1

-0.41

-0.40

-0.39

2378-1

342-1

162-1

-0.33

-0.32

rs = 0.5

rs = 2.0

rs= 1.0

rs = 3.0

Results compare well with R–space MC with state of the art WF.


Conclusions

Summary:we have extended MC methods to work in general CI–space providingrigourus upper–bounds on energythe use of Coupled Cluster Wave–functions serves a dual pourpose:

extremely good guiding wave–functionprovides variational energies for CC solutions

access to accurate momentum distributions

Perspectives:including higher–order correlations in CC wfconsider spin–isospin dependent interaction (→ χ-EFT)response functions


Thanks for your attention


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