Unitary Correlation Operator Method

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The
Unitary Correlation
Operator Method
or
”The Taming of the Core“
H. Feldmeier, T. Neff & R. Roth
Gesellschaft für Schwerionenforschung — Darmstadt/Germany
“Nuclear Structure for the 21 st Century” — INT Seattle
11/2000

Why Effective Interactions?
The Problem: Short-Range Correlations
Interaction
many realistic two-body
interactions show a strong
short-range repulsion
(e.g. nucleon-nucleon & van der
Waals interactions)
Effective Interaction
replace the full potential by a
tamed effective interaction
Correlations
core induces strong
short-range correlations
in many-body state
(e.g. correlation hole in
two-body density)
Unitary Correlation Operator Method
UCOM
Product States
short-range correlations
cannot be described by
product-type states
(e.g. mean-field, superposition
of few product states,...)
Correlated States
include correlations in many-body
model-space
R. Roth

G † = G
C † C = 1
Concept of the
Unitary Correlation Operator Method
Correlation Operator
Short-range correlations are represented by a
state-independent unitary correlation operator C that
describes a radial distance-dependent shift in the
relative coordinate of the two-body system.
C = exp[−i G] = exp − i
gij Correlated States
ψ = C ψ
g = 1 r
2 [ s(r) r
i

Correlated Wave Function
Malfliet-Tjon V potential
• correlator acts on the relative part of the
two-body wave function
r, X C ψ = R−(r) R−(r) r
r , X ψ
r, X C † ψ = R+(r) R+(r) r
r , X ψ
• norm-conserving coordinate transformation
[a.u.]
1
0.5
0
-0.5
r ↦→ R±(r) r
r
r φ r C φ
v(r)
0 1 2 3
r [fm]
Two-Body System
Correlated Wave Function
Correlation Function R±(r)
• metric factor and inverse transformation
R±(r) = R±(r)
r
• connection with s(r)
[fm]
0.15
0.1
R±(r) dξ
±1 =
s(ξ)
r
R+(r) − r
R±[R∓(r)] = r
[fm]
2
1
R ′ ± (r)
R±(r) ≈ r ± s(r)
R+(r)
R−(r)
0.05
0
0 1 2 3
0
r [fm]
0 1 2 3
r [fm]
R. Roth

Lennard-Jones potential
[a.u.]
1
0.5
0
-0.5
r φ
r C φ
v(r)
0 2 4 6 8 10
r [ ˚A]
[ ˚A]
1.5
1
R+(r) − r
[ ˚A]
8
6
4
2
R+(r)
R−(r)
0.5
0
0 2 4 6 8 10
0
r [ ˚A]
0 2 4 6 8 10
r [ ˚A]

Many-Body System
Correlated Operators & Cluster Expansion
Cluster Expansion
decompose the correlated operator into
a sum of irreducible k-body operators
H = C † H C = H [1] + H [2] + H [3] + · · ·
if the range of correlations is small compared
to the average distance between the particles
then higher cluster orders are negligible
Two-Body
Approximation
H C2 = H [1] + H [2]
Smallness Parameter
κ = ρ VC
κ ≪ 1 κ ≪/ 1
Effective Corrections
e.g. density-dependent
correlation functions
in H C2
Cluster
Decomposition
Principle
VC = d 3 r r C 1 − r 1 2
= d 3 r [R+(r) − 1] 2
Three-Body
Approximation
H C3 = H [1] + H [2] + H [3]
R. Roth

Local Potentials
Two-Body Approximation
Correlated Hamiltonian & Effective Interaction
v(r) = v[R+(r)]
u(r) =
2µR ′ +
′′
1 2R +(r)
2
(r)
H = T0 +
i

The Basics of
Fermionic Molecular Dynamics
One-Body States
x ψ =
n
ν=1
ξν : mean position
πν : mean momentum
αν : complex width
exp − (x − ξν) 2
2 αν
Many-Body State
m
− i πνx
s
ν ⊗ m t
• 8n variational parameters
per nucleon
Ψ = C A ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψA
• exactly antisymmetrized product states
• short-range correlations described by UCOM
Structure
• energy minimization by variation of the parameters
gives groundstate wave function
• diagonalization within a small subspace (spanned by
rotation & scaling of the variational groundstate)
Dynamics
• time-dependent variational principle yields equations
of motion for the parameters
R. Roth

[AMeV]
40
20
0
-20
-40
-60
15.09
-12.59
4 He
2.50
18.90
-7.62
-26.52
intrinsic kinetic energy
potential energy
total energy
20.79
Malfliet-Tjon-V Potential
Groundstate Energies
5 He 6 He 7 Li 8 Be
3.46
25.52
-34.01
26.13
4.07
32.26
33.58
3.81
-7.83
-8.49
-17.33
-8.6
-11.00
-22.06
-11.06
-14.91
(-11.91)
-43.26
-29.76
41.57
-56.48
39.03
3.52
-35.52
reference
SVM • Varga, Suzuki; Phys. Rev. C52(95)2885
VMC • Wiringa; priv. comm. in Phys. Rev. C52(95)2885; except 7 Li
49.08
-19.81
-68.89
R. Roth

Afnan-Tang S3M Potential
Spin-Isospin Dependent NN-Potentials
Spin-Isospin Dependent Central Interaction
• central two-body potential expressed by projection
operators ΠST on spin S and isospin T
v =
S,T
• e.g. Afnan-Tang S3M potential
vST (r) ΠST
Spin-Isospin Dependent Correlator
• unitary transformation depending on the (S, T ) quantum
numbers of the particle pair (two-body space)
c = exp −i
=
S,T
g ST ΠST
S,T
exp(−i gST ) ΠST =
cST ΠST
S,T
• different correlation function R ST
+ (r) for each (S, T )component
of the potential
[MeV]
100
50
0
-50
-100
[fm]
0.2
0.15
0.1
0.05
0
vST (r)
ST =
01
10
00 & 11
R ST
+ (r) − r
0 1 2 3
r [fm]
[Phys. Rev. 175 (1968) 1337], [Nucl. Phys. A371 (1981) 79]
R. Roth

Afnan-Tang S3M Potential
Groundstate Energies
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R. Roth

y [fm]
y [fm]
6
4
2
0
-2
-4
-6
1.0
1.5
0.5
0.1
0.01
4 He
y [fm]
6
4
2
0
-2
-4
-6
Afnan-Tang S3M Potential
Groundstate Densities
ρ(r) [ρ [ρ0] 0]
10 -1
10 -1
10 -2
10 -2
0.1
0.5
0.01
1
1.0
10 -3
10 -3
10 -4
10 -4
0.01
total
neutron
proton
y [fm]
7 Li
6
4
2
0
-2
-4
-6
ρ(r) [ρ [ρ0] 0]
4 He
6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 0 1 2 2 4 3 6 -6 4 -4 5 -2 0 0 1 2 2 4 3 6 -6 4 -4 5 -2 0 0 1 2 2 4 3 6 4 5 0
20 24 20 28 24 40 28
x [fm] Ne
1 x [fm] r Mg [fm] Ne 1 x [fm] r [fm] Si Mg 1 x [fm] r [fm] Ca Si 1
4
4
4
4
2
0
-2
-4
-6
1.0
0.01
0.01
0.5
0.1
0.1
y [fm]
-2
-4
-6
-6 -4 -2 0
x [fm]
2 4 6
2
0
ρ(r) [ρ [ρ0] 0]
0.1
0.01
0.01
10 -1
10 -1
10 -2
10 -2
1.0
0.5
10 -3
10 -3
10 -4
10 -4
• one gaussian per nucleon, fixed spin
0.1
0.5
total
neutron
proton
y [fm]
2
0
-2
-4
-6
ρ(r) [ρ [ρ0] 0]
0.1
0.01
10 -1
10 -1
1.0
1
10 -2
10 -2
10 -3
10 -3
0.5
0.1
0.01
10 -4
10 -4
10 -1
10 -1
10 -2
10 -2
1.0
0.5
10 -3
10 -3
10 -4
10 -4
0.01
1.0
0.01
0.1
1.0
total
neutron
proton
0.1
total
neutron
proton
12 C
y [fm]
6
4
2
0
-2
-4
-6
ρ(r) [ρ [ρ0] 0]
7 Li
1
10 -1
10 -1
10 -2
10 -2
1.0
0.5
10 -3
10 -3
0.1
0.01
10 -4
10 -4
0.01
0.1
total
neutron
proton
-6 -4 -2 0 0 1 2 2 4 3 6 -6 4 -4 5 -2 0 0 1 2 2 4 3 6 -6 4 -4 5 -2 0 0 1 2 2 4 3 6 4 5
x [fm] r [fm]
x [fm] r [fm]
x [fm] r [fm]
y [fm]
2
0
-2
-4
-6
ρ(r) [ρ [ρ0] 0]
0.01
10 -1
10 -1
10 -2
10 -2
0.5
1.0
10 -3
10 -3
0.1
0.01
10 -4
10 -4
0.1
1.5
0.5
total
neutron
proton
16 O
0.01
R. Roth
ρ(r) [ρ [ρ0] 0]
ρ(r) [ρ [ρ0] 0]
12 C
1
10 -1
10 -1
10 -2
10 -2
10 -3
10 -3
10 -4
10 -4
10 -1
10 -1
10 -2
10 -2
10 -3
10 -3
10 -4
10 -4
0

Outlook
Summary
UCOM & FMD
Summary & Outlook
• Unitary Correlation Operator Method describes short-range
correlations by a distance-dependent transformation in the relative
two-body coordinate
• correlated Hamiltonian / effective interaction in two-body
approximation is given in closed form; higher cluster orders are
negligible in case of finite nuclei
• variational model based on a Slater determinant of Gaussian trial
states (Fermionic Molecular Dynamics) to describe groundstate
structure of light and intermediate mass nuclei
• for central potentials (MTV/ATS3M) the results are comparable
with quasi-exact and expensive many-body techniques
• need realistic NN-interaction, e.g. Bonn or Argonne potential,
including a proper treatment of tensor correlations
• ready for systematic and nearly ab initio nuclear structure studies
• can be used for the description of low-energy nucleus-nucleus
collisions right away
R. Roth