Properties of nuclei from chiral EFT interactions
Stefano Gandolfi
Los Alamos National Laboratory (LANL)
Progress in Ab Initio Techniques in Nuclear Physics,TRIUMF, Vancouver BC, Canada. Feb 26 - March 1, 2019.
www.computingnuclei.org
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 1 / 23
The big picture
protons
neutrons
The Grand Nuclear Landscape(finite nuclei + extended nucleonic matter)
82
50
28
28
50
82
2082
28
20stable nucleistable nuclei
known nucleiknown nuclei
126terra incognitaterra incognita
neutron stars
neutron
drip line
probably known only up to oxygen
known up to Z=91
superheavynucleiZ=118, A=294
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 2 / 23
Outline
The nuclear Hamiltonian and the method
Some “issue” of chiral Hamiltonians
Light nuclei and neutron matter
Medium nuclei
Conclusions
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 3 / 23
Nuclear Hamiltonian
Model: non-relativistic nucleons interacting with an effectivenucleon-nucleon force (NN) and three-nucleon interaction (TNI).
H = − ~2
2m
A∑
i=1
∇2i +
∑
i<j
vij +∑
i<j<k
Vijk
vij NN fitted on scattering data.
Vijk typically constrained to reproduce light systems (A=3,4).
“Phenomenological/traditional” interactions (Argonne/Illinois)
Local chiral forces up to N2LO (Gezerlis, et al. PRL 111, 032501(2013), PRC 90, 054323 (2014), Lynn, et al. PRL 116, 062501(2016)).
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 4 / 23
Quantum Monte Carlo
Propagation in imaginary time:
H ψ(~r1 . . . ~rN) = E ψ(~r1 . . . ~rN) ψ(t) = e−(H−ET )tψ(0)
Ground-state extracted in the limit of t →∞.
Propagation performed by
ψ(R, t) = 〈R|ψ(t)〉 =
∫dR ′G (R,R ′, t)ψ(R ′, 0)
Importance sampling: G (R,R ′, t)→ G (R,R ′, t) ΨI (R′)/ΨI (R)
Constrained-path approximation to control the sign problem.Unconstrained-path calculation possible in several cases (exact).
GFMC includes all spin-states of nucleons in the w.f., nuclei up to A=12AFDMC samples spin states, bigger systems, less accurate than GFMC
Ground–state obtained in a non-perturbative way. Systematicuncertainties within 2-3 %.
See Carlson, et al., Rev. Mod. Phys. 87, 1067 (2015)
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 5 / 23
Traditional approach (credit D. Furnsthal, T. Papenbrock)
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 6 / 23
Light nuclei spectrum computed with GFMC
-100
-90
-80
-70
-60
-50
-40
-30
-20
Ene
rgy
(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+
8He0+
2+2+
2+1+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+
2+0+
12C
Argonne v18with Illinois-7
GFMC Calculations
Carlson, et al., Rev. Mod. Phys. 87, 1067 (2015)
Also radii, densities, matrix elements, ...
Unfortunately phenomenological Hamiltonians are not useful to addresssystematical uncertainties.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 7 / 23
Neutron matter and the ”puzzle” of the three-body force
0.04 0.06 0.08 0.1 0.12 0.14 0.16
ρ [fm-3
]
6
8
10
12
14
16
18
20
En
erg
y p
er N
eutr
on
[M
eV]
AV8’+UIXAV8’+IL7AV8’
Note: AV8’+UIX and (almost) AV8’ are stiff enough to support observedneutron stars. → How to reconcile with nuclei???
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 8 / 23
Nuclear Hamiltonian
Expansion in powers of Q/Λ, Q∼100 MeV, Λ ∼1 GeV.
Long-range physics given by pion-exchanges (no free parameters).
Short-range physics: contact interactions (LECs) to fit.
Operators need to be regulated → cutoff dependency!
Order’s expansion provides a way to quantify uncertainties!
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 9 / 23
Truncation error estimate
Error quantification (one possible scheme). Define
Q = max
(p
Λb,mπ
Λb
),
where p is a typical nucleon’s momentum or kF for matter, Λb is thecutoff, and calculate:
∆(N2LO) = max(Q4|OLO |, Q2|OLO − ONLO |, Q|ONLO − ON2LO
)
Epelbaum, Krebs, Meissner (2014).
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 10 / 23
Chiral three-body forces, issue (I)?
π π π
c1, c3, c4 cD cE
In the Fourier transformation of VD two possible operator structures arise:
VD1 =gAcDm
2π
96πΛχF 4π
∑i<j<k
∑cyc
τi · τk[Xik(rkj)δ(rij) + Xik(rij)δ(rkj) −
8π
m2π
σi · σkδ(rij)δ(rkj)
]
VD2 =gAcDm
2π
96πΛχF 4π
∑i<j<k
∑cyc
τi · τk[Xik(rik) − 4π
m2π
σi · σkδ(rik)
] [δ(rij) + δ(rkj)
]Xij(r) = T (r)Sij + Y (r)σi · σj
Navratil (2007), Tews et al PRC (2016), Lynn et al PRL (2016).
Equivalent only in the limit of an infinite cutoff. Implications in real life?
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 11 / 23
Chiral three-body forces, issue (II)?
π π π
c1, c3, c4 cD cE
Equivalent forms of operators entering in VE (Fierz-rearrangement):
1, σi ·σj , τi ·τj , σi ·σjτi ·τj , σi ·σjτi ·τk , [(σi×σj)·σk ][(τi×τj)·τk ]
Epelbaum et al (2002). We investigated the following choices:
VEτ =cE
ΛχF 4π
∑
i<j<k
∑
cyc
τi · τkδ(rkj)δ(rij)
VE1 =cE
ΛχF 4π
∑
i<j<k
∑
cyc
δ(rkj)δ(rij)
Qualitative differences expected, i.e. consider 4He vs neutron matter!
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 12 / 23
Chiral three-body forces
Coefficients cD and cE fit to reproduce the binding energy of 4He andneutron-4He scattering. → more information on T=3/2 part ofthree-body interaction. (vs A=3, 4)
0 1 2 3 4 50
30
60
90
120
150
180
Ec.m. (MeV)
δ JL (
degr
ees)
12
+
12
-
AV18AV18+UIXAV18+IL2R-Matrix
32
-
GFMC neutron-4He resultsusing Argonne Hamiltonians.
Nollett, Pieper, Wiringa,Carlson, Hale, PRL (2007).
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 13 / 23
4He binding energy and p-wave n-4He scattering
Regulator: δ(r) = 1πΓ(3/4)R3
0exp(−(r/R0)4)
Cutoff R0 taken consistently with the two-body interaction.
−1.0 0.0 1.0 2.0 3.0 4.0cD
−1.5
−1.0
−0.5
0.0
0.5
1.0
c E
R0 = 1.0 fm
R0 = 1.2 fm
R0 = 1.0 fm(a)
N2LO (D1, Eτ )
N2LO (D2, Eτ )
N2LO (D2, E1)
N2LO (D2, EP)
0 1 2 3 4Ecm (MeV)
0
20
40
60
80
100
120
140
δ(d
eg.)
32
−
12
−
(b) NLO
N2LO (D2, Eτ )
N2LO (D2, EP)
R−matrix
No fit can be obtained for R0 = 1.2 fm and VD1 - Issue (I)
Lynn, Tews, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk PRL (2016).
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 14 / 23
A=3, 4 nuclei at N2LO
−9
−8
−7
−6
−5
E(M
eV)
3H 3He 4He−31
−25
−19
−13
3H 3He 4He0.8
1.0
1.2
1.4
1.6
1.8
2.0
√〈r
2pt 〉
(fm)
NLO
N2LO (D2, Eτ )
Exp.
Lynn, Tews, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk PRL (2016).Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 15 / 23
Neutron matter at N2LO
EOS of pure neutron matter at N2LO, R0=1.0 fm
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
n (fm−3)
0
2
4
6
8
10
12
14
16
18
E/A
(MeV
)
N2LO (D2, E1)
N2LO (D2, EP)
N2LO (D2, Eτ )
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Neutron Density (fm-3
)
0
2
4
6
8
10
12
14
16
18
En
erg
y p
er
Ne
utr
on
(M
eV
)
AV8’+UIX
AV8’
Lynn, Tews, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk PRL (2016).
Significant dependence to the choice of VE (Issue (II)), but similar resultsto phenomenological Hamiltonians.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 16 / 23
Neutron matter at N2LO
EOS of pure neutron matter at N2LO, R0=1.0 fm
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
n (fm−3)
0
2
4
6
8
10
12
14
16
18
E/A
(MeV
)
N2LO (D2, E1)
N2LO (D2, EP)
N2LO (D2, Eτ )
0.00 0.05 0.10 0.15 0.20 0.25 0.30
n[fm−3
]0
5
10
15
20
25
30
35
40
45
E/A
[ MeV
]
AV8′ + UIX
N2LO (TPE + VE1)
N2LO (TPE− only)
N2LO (TPE + VEτ)
Tews, Carlson, Gandolfi, Reddy, APJ (2018).
Errors grow quickly with the density.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 17 / 23
Heavier nuclei
What about heavier nuclei?
Many chiral Hamiltonians cannot predict both energies and radii.
Strategy: include medium nuclei properties in the fit (but sacrificenucleon-nucleon data)?
Ekstrom, Hagen, et al., Phys. Rev. C 91, 051301(R) (2015)
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 18 / 23
AFDMC calculations
Energies and charge radii, cutoff 1.0 fm:
-16
-14
-12
-10
-8
-6
-4
-2
3H 3He 4He 6He 6Li 12C 16O
(a) R0 = 1.0 fm
E/A
(MeV
)
Exp
LO
NLO
N2LO Eτ
N2LO E1
-16
-14
-12
-10
-8
-6
-4
-2
3H 3He 4He 6He 6Li 12C 16O
1.0
1.5
2.0
2.5
3.0
3.5
3H 3He 4He 6He 6Li 12C 16O
(a) R0 = 1.0 fm
r ch (f
m)
Exp
LO
NLO
N2LO Eτ
N2LO E1
1.0
1.5
2.0
2.5
3.0
3.5
3H 3He 4He 6He 6Li 12C 16O
Lonardoni, et al., PRL (2018), PRC (2018).
Qualitative good description of both energies and radii.
Good convergence (although uncertainties still large if LO included).
Different VE operators give similar results.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 19 / 23
AFDMC calculations
Energies and charge radii, cutoff 1.2 fm:
-16
-14
-12
-10
-8
-6
-4
-2
3H 3He 4He 6He 6Li 12C 16O
(b) R0 = 1.2 fm
E/A
(MeV
)
Exp
LO
NLO
N2LO Eτ
N2LO E1
-16
-14
-12
-10
-8
-6
-4
-2
3H 3He 4He 6He 6Li 12C 16O
1.0
1.5
2.0
2.5
3.0
3.5
3H 3He 4He 6He 6Li 12C 16O
(b) R0 = 1.2 fm
r ch (f
m)
Exp
LO
NLO
N2LO Eτ
N2LO E1
1.0
1.5
2.0
2.5
3.0
3.5
3H 3He 4He 6He 6Li 12C 16O
Lonardoni, et al., PRL (2018), PRC (2018).
Qualitative good description up to A=6.
Different VE operators give very different results for 16O.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 20 / 23
Energy contribution
Expectation value of the N2LO energy contributions 16O:
Potential Ekin + vij Vijk V 2π,P V 2π,S VD VE
2b, 1.0 −134(2)Eτ , 1.0 −130(2) −44(1) −55(1) 0.85(1) 0 8.50(4)E1, 1.0 −131(2) −41(1) −54(1) 0.72(1) −4.03(5) 15.7(1)
2b, 1.2 −151(3)Eτ , 1.2 −156(7) −202(3) −101(2) −0.72(9) −94(2) −5.43(3)E1, 1.2 −152(2) −26(1) −34(1) 0.94(1) 4.53(8) 1.90(1)
LECs cD and cE for different cutoffs and parametrizations of thethree-body force (other strengths are the same):
Vijk R0 (fm) cD cEEτ 1.0 0.0 −0.63E1 1.0 0.5 0.62Eτ 1.2 3.5 0.09E1 1.2 −0.75 0.025
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 21 / 23
Charge form factor
10-5
10-4
10-3
10-2
10-1
100
101
102
0.0 1.0 2.0 3.0 4.0
16O|FL(q)
|
q (fm-1)
exp
N2LO (1.0 fm)
N2LO (1.2 fm)
10-5
10-4
10-3
10-2
10-1
100
101
102
0.0 1.0 2.0 3.0 4.0
10-4
10-3
10-2
10-1
100
0.0 1.0 2.0 3.0 4.0
12C
AV18+IL7
10-4
10-3
10-2
10-1
100
0.0 1.0 2.0 3.0 4.0
Lonardoni, et al., PRL (2018), PRC (2018).
Hard interaction reproduces exp.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 22 / 23
Conclusions
Quantum Monte Carlo calculations for larger nuclei is now possible(at least up to A=16, work in progress...)
Chiral EFT provides a way to constrain nuclear interactions andestimate systematic uncertainties
But...
Effect of the cutoff important to explore
Effect of using different (“equivalent”) operators important toexplore
Similar issues with electroweak currents?
Acknowledgments:
J. Carlson (LANL), D. Lonardoni (LANL and FRIB)
J. Lynn, A. Schwenk (Darmstadt)
K.E. Schmidt (ASU)
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 23 / 23
Extra slides
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 1 / 18
Scattering data and neutron matter
Two neutrons have
k ≈√Elab m/2 , → kF
that correspond to
kF → ρ ≈ (Elab m/2)3/2/2π2 .
Elab=150 MeV corresponds to about 0.12 fm−3.
Elab=350 MeV to 0.44 fm−3.
Argonne potentials useful to study dense matter above ρ0=0.16 fm−3
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 2 / 18
Phase shifts, AV8’
0 100 200 300 400 500 600E
lab (MeV)
-40
-20
0
20
40
60δ (
deg
)Argonne V8’
SAID
1S
0
0 100 200 300 400 500 600E
lab (MeV)
-20
0
20
40
60
80
100
120
δ (
deg
)
Argonne V8’
SAID
3S
1
0 100 200 300 400 500 600E
lab (MeV)
-40
-20
0
δ (
deg
)
Argonne V8’
SAID
1P
1
0 100 200 300 400 500 600E
lab (MeV)
-40
-20
0
20
δ (
deg
)
Argonne V8’
SAID
3P
0
0 100 200 300 400 500 600E
lab (MeV)
-50
-40
-30
-20
-10
0
δ (
deg
)
Argonne V8’
SAID
3P
1
0 100 200 300 400E
lab (MeV)
-10
0
10
20
30
δ (
deg
)
Argonne V8’
SAID
3P
2
0 100 200 300 400 500 600E
lab (MeV)
-40
-30
-20
-10
0
δ (
deg
)
Argonne V8’
SAID
3D
1
0 100 200 300 400 500 600E
lab (MeV)
0
10
20
30
40
50
60
δ (
deg
)
Argonne V8’
SAID
3D
2
0 100 200 300 400 500 600E
lab (MeV)
0
2
4
6
8
δ (
deg
)
Argonne V8’
SAID
ε1
Difference AV8′-AV18 less than 0.2 MeV per nucleon up to A=12.Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 3 / 18
Nuclear Hamiltonian
Phase shifts, LO, NLO and N2LO with R0=1.0 and 1.2 fm:
0 50 100 150 200 250
Lab. Energy [MeV]
0
10
20
30
40
50
60
70
Ph
ase
Sh
ift
[deg
]
LONLO
N2LO
0 50 100 150 200 250
Lab. Energy [MeV]
0
50
100
150
200
0 50 100 150 200 250
Lab. Energy [MeV]
-25
-20
-15
-10
-5
0
0 50 100 150 200 250
Lab. Energy [MeV]
0
1
2
3
4
1S03S1
3D1 ε1
0 50 100 150 200 250
Lab. Energy [MeV]
-30
-25
-20
-15
-10
-5
0
Ph
ase
Sh
ift
[deg
]
0 50 100 150 200 250
Lab. Energy [MeV]
-10
-5
0
5
10
15
20
25
30
35
40
0 50 100 150 200 250
Lab. Energy [MeV]
-30
-25
-20
-15
-10
-5
0
1P1
3P03P1
Gezerlis, et al. PRC 90, 054323 (2014)Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 4 / 18
Three-body forces
Urbana–Illinois Vijk models processes like
π
π
∆
π
π
π
π∆
π
π
π
∆
π
∆
+ short-range correlations (spin/isospin independent).
Chiral forces at N2LO:
π π π
c1, c3, c4 cD cE
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 5 / 18
Quantum Monte Carlo
H ψ(~r1 . . . ~rN) = E ψ(~r1 . . . ~rN) ψ(t) = e−(H−ET )tψ(0)
Ground-state extracted in the limit of t →∞.
Propagation performed by
ψ(R, t) = 〈R|ψ(t)〉 =
∫dR ′G (R,R ′, t)ψ(R ′, 0)
Importance sampling: G (R,R ′, t)→ G (R,R ′, t) ΨI (R′)/ΨI (R)
Constrained-path approximation to control the sign problem.Unconstrained calculation possible in several cases (exact).
Ground–state obtained in a non-perturbative way. Systematicuncertainties within 1-2 %.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 6 / 18
Overview
Recall: propagation in imaginary-time
e−(T+V )∆τψ ≈ e−T∆τe−V∆τψ
Kinetic energy is sampled as a diffusion of particles:
e−∇2∆τψ(R) = e−(R−R′)2/2∆τψ(R) = ψ(R ′)
The (scalar) potential gives the weight of the configuration:
e−V (R)∆τψ(R) = wψ(R)
Algorithm for each time-step:
do the diffusion: R ′ = R + ξ
compute the weight w
compute observables using the configuration R ′ weighted using wover a trial wave function ψT .
For spin-dependent potentials things are much worse!
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 7 / 18
Branching
The configuration weight w is efficiently sampled using the branchingtechnique:
Configurations are replicated or destroyed with probability
int[w + ξ]
Note: the re-balancing is the bottleneck limiting the parallel efficiency.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 8 / 18
GFMC and AFDMC
Because the Hamiltonian is state dependent, all spin/isospin states ofnucleons must be included in the wave-function.
Example: spin for 3 neutrons (radial parts also needed in real life):
GFMC wave-function:
ψ =
a↑↑↑a↑↑↓a↑↓↑a↑↓↓a↓↑↑a↓↑↓a↓↓↑a↓↓↓
A correlation like
1 + f (r)σ1 · σ2
can be used, and the variational wave
function can be very good. Any operator
accurately computed.
AFDMC wave-function:
ψ = A[ξs1
(a1
b1
)ξs2
(a2
b2
)ξs3
(a3
b3
)]We must change the propagator by usingthe Hubbard-Stratonovich transformation:
e12
∆tO2=
1√
2π
∫dxe−
x2
2+x√
∆tO
Auxiliary fields x must also be sampled.
The wave-function is pretty bad, but we
can simulate larger systems (up to
A ≈ 100). Operators (except the energy)
are very hard to be computed, but in some
case there is some trick!
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 9 / 18
Propagator
We first rewrite the potential as:
V =∑
i<j
[vσ(rij)~σi · ~σj + vt(rij)(3~σi · rij~σj · rij − ~σi · ~σj)] =
=∑
i,j
σiαAiα;jβσjβ =1
2
3N∑
n=1
O2nλn
where the new operators are
On =∑
jβ
σjβψn,jβ
Now we can use the HS transformation to do the propagation:
e−∆τ 12
∑n λO
2nψ =
∏
n
1√2π
∫dxe−
x2
2 +√−λ∆τxOnψ
Computational cost ≈ (3N)3.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 10 / 18
Three-body forces
Three-body forces, Urbana, Illinois, and local chiral N2LO can be exactlyincluded in the case of neutrons.
For example:
O2π =∑
cyc
[{Xij ,Xjk} {τi · τj , τj · τk}+
1
4[Xij ,Xjk ] [τi · τj , τj · τk ]
]
= 2∑
cyc
{Xij ,Xjk} = σiσk f (ri , rj , rk)
The above form can be included in the AFDMC propagator.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 11 / 18
Three-body forces
V 2π,PWa = A2π,PW
a
∑i<j<k
∑cyc{~τi · ~τk , ~τj · ~τk}{σ
αi σ
γk, σµkσβj}XiαkγXkµjβ
= 4A2π,PWa
∑i<j
~τi · ~τjσαi σ
βj
∑k 6=i,j
XiαkγXkγjβ , (1)
V 2π,PWc = A2π,PW
c
∑i<j<k
∑cyc
[~τi · ~τk , ~τj · ~τk ][σαi σγk, σµkσβj
]XiαkγXkµjβ
= −4A2π,PWc
∑i<j<k
∑cycτηiτξjτφkεηξφσ
αi σ
βjσνk ενγµXiαkγXkµjβ (2)
= A2π,PWc
∑i<j<k
∑cyc
[~τi · ~τk , ~τj · ~τk ][σαi σγk, σµkσβj
]
Xiαkγ − δαγ4π
m3π
∆(rik )
Xkµjβ − δµβ4π
m3π
∆(rkj )
(3)
= V∆∆c + V∆δ
c + Vδδc (4)
V 2π,SW = A2π,SW ∑i<j<k
∑cycZikαZjkασ
αi σ
βj~τi · ~τj
= A2π,SW ∑i<j
σαi σ
βj~τi · ~τj
∑k 6=i,j
ZikαZjkα (5)
VD = AD
∑i<j
σαi σ
βj~τi · ~τj
∑k 6=i,j
Xiαjβ [∆(rik ) + ∆(rjk )] (6)
VE = AE
∑i<j
~τi · ~τj∑k 6=i,j
∆(rik )∆(rjk ) (7)
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 12 / 18
Three-body forces
H ′ = H − V 2π,PWc + α1V
2π,PWa + α2VD + α3VE . (8)
The Hamiltonian H ′ can be exactly included in the AFDMC propagation.The three constants αi are adjusted in order to have:
〈V∆∆c 〉 ≈ 〈α1V
2π,PWa 〉
〈V∆δc 〉 ≈ 〈α2VD〉
〈V δδc 〉 ≈ 〈α3VE 〉 (9)
Once the ground state Ψ of H ′ is calculated with AFDMC as explainedabove, the expectation value of the Hamiltonian H is given by
〈H〉 = 〈Ψ|H ′|Ψ〉+ 〈Ψ|H − H ′|Ψ〉= 〈Ψ|H ′|Ψ〉+ 〈Ψ|V 2π,PW
c − α1V2π,PWa − α2VD − α3VE |Ψ〉 (10)
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 13 / 18
Variational wave function
E0 ≤ E =〈ψ|H|ψ〉〈ψ|ψ〉 =
∫dr1 . . . drN ψ
∗(r1 . . . rN)Hψ∗(r1 . . . rN)∫dr1 . . . drN ψ∗(r1 . . . rN)ψ∗(r1 . . . rN)
→ Monte Carlo integration. Variational wave function:
|ΨT 〉 =
∏
i<j
fc(rij)
∏
i<j<k
fc(rijk)
1 +
∑
i<j,p
∏
k
uijk fp(rij)Opij
|Φ〉
where Op are spin/isospin operators, fc , uijk and fp are obtained byminimizing the energy. About 30 parameters to optimize.
|Φ〉 is a mean-field component, usually HF. Sum of many Slaterdeterminants needed for open-shell configurations.
BCS correlations can be included using a Pfaffian.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 14 / 18
Variational wave function
〈RS |ΨV 〉 = 〈RS |[∏
i<j
f c(rij)][
1 +∑
i<j
Fij +∑
i<j<k
Fijk
]|ΦJM〉 ,
〈RS |ΦJM〉 =∑
n
kn[∑
D{φα(ri , si )}]JM,
φα(ri , si ) = Φnlj(ri ) [Ylml(ri )ξsms (si )]jmj
,
In particular, we included orbitals in 1S1/2, 1P3/2, 1P1/2, 1D5/2, 2S1/2,and 1D3/2.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 15 / 18
The Sign problem in one slide
Evolution in imaginary-time:
ψI (R′)Ψ(R ′, t + dt) =
∫dR G (R,R ′, dt)
ψI (R′)
ψI (R)ψI (R)Ψ(R, t)
note: Ψ(R, t) must be positive to be ”Monte Carlo” meaningful.
Fixed-node approximation: solve the problem in a restricted space whereΨ > 0 (Bosonic problem) ⇒ upperbound.
If Ψ is complex:
|ψI (R′)||Ψ(R ′, t + dt)| =
∫dR G (R,R ′, dt)
∣∣∣∣ψI (R
′)
ψI (R)
∣∣∣∣ |ψI (R)||Ψ(R, t)|
Constrained-path approximation: project the wave-function to the real
axis. Extra weight given by cos ∆θ (phase of Ψ(R′)Ψ(R) ), Re{Ψ} > 0 ⇒ not
necessarily an upperbound.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 16 / 18
Unconstrained-path
GFMC unconstrained-path propagation:
0 10 20 30 40
-93
-92
-91
nu
⟨H⟩
(M
eV)
12C(0+) − AV18 & AV18+IL7 with various corrs. − ⟨H⟩ − 27 Feb 2010
g.s., IL7, 6 state 18
Changing the trial wave function gives same results.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 17 / 18
Unconstrained-path
AFDMC unconstrained-path propagation:
0 0.0005 0.001 0.0015 0.002 0.0025
τ (MeV-1
)
-110
-108
-106
-104
-102
-100
-98
-96
-94
-92
-90
E (
MeV
)
16O, AV6’+Coulomb
0 0.00025 0.0005 0.00075 0.001
τ (MeV-1
)
-100
-95
-90
-85
-80
E (
MeV
)
16O, AV7’+Coulomb
The difference between CP and UP results is mainly due to the presenceof LS terms in the Hamiltonian. Same for heavier systems.
Work in progress to improve Ψ to improve the constrained-path.
Stefano Gandolfi (LANL) Properties of nuclei from chiral interactions 18 / 18