Advances in time-dependent methods for quantum many-body system
2013.10.16 @ ECT*, Torento
Time-dependent mean field theory including pairing correlation,
and applications to linear response calculation
EBATA Shuichiro
Contents
Meme media Laboratory,
Hokkaido University
Time-dependent method
Cb-TDHFB
Nuclear Reaction Data Center, Faculty of Science,
Hokkaido University (JCPRG)
*JCPRG: Japan Charged-Particle nuclear
Reaction data Group
Linear response calculation
Comparison to Exp. & QRPA
Systematic study for E1
Application to other modes
NAKATSUKASA Takashi
RIKEN Nishina Center
Summary and Perspective
ECT* workshop ATDmQMB
2013.10.14-18
Subject: Nuclear many body system
Nucleus is a core of atom, which is composed nucleons.
Atom
Nucleon
Nucleus
Neutron
Finite quantum
many-body system
electron
~ 10
-10
[m]
~ few 10
-15
[m]
Proton
Nucleus has
magic number
Nucleus makes
some average potential.
→ Mean field
→ Shell structure
Neutron
∆B = Binding number
Energy(Exp.) – Empirical formula
ECT* workshop ATDmQMB
2013.10.14-18
In Nuclear system, many degree of freedom appear,
while depending on a number of nucleons and energy of the system.
Single-particle
excitation
Alpha-particle
excitation
Collective
excitation
Etc.
To understand the complexity of nuclear system,
we should know several excited modes of several nuclei.
ECT* workshop ATDmQMB
2013.10.14-18
Several mean filed theories for several applications
No Pairing
For static
For dynamics
Hartree-Fock(HF)
Time-Dependent HF
(TDHF, RPA)
With
BCS Pairing
HF+BCS
With Pairing
Hartree-FockBogoliubov
(HFB)
ECT* workshop ATDmQMB
2013.10.14-18
TDHF+BCS
Cb-TDHFB
TDHFB
(QRPA)
HF vs. HF+BCS vs. HFB
HF
Pairing correlation
HF+BCS
: Canonical basis
: BCS quasi-particle state
Generalize
HFB
: Generalized quasi-particle
state
*One body density matrix is diagonalized in Canonical basis.
N : nucleon #
N' : canonical basis #
M : basis #
Dimension
N = N'
N < N'
TDHFB
: Generalized
density matrix
ECT* workshop ATDmQMB
2013.10.14-18
: Generalized
Hamiltonian
: Pair potential
Recipe for the Canonical-basis TDHFB (Cb-TDHFB)
Ebata et al, Phys. Rev. C82, 034306
TDHFB
: Density matrix
: Pair tensor
: Arbitrary complete set
: Canonical basis
Canonical basis diagonalize Density matrix.
In this Canonical-basis,
the number of matrix elements compress to diagonal components.
The computational cost of TDHFB can be reduced
in Canonical-basis representation !?
: Time-dependent Canonical basis
: Time-dependent Canonical single-particle basis
This set is assumed
to be
orthonormal.
ECT* workshop
ATDmQMB
2013.10.14-18
Note: HFB state can be represented in Canonical basis at each time ...
HFB at t
HFB at t'
...
...
Can we know
the time dependence of unitary transformation
without the solution of TDHFB ?
?
BCS at t
Time
BCS at t'
...
...
ECT* workshop ATDmQMB
2013.10.14-18
Recipe for the Cb-TDHFB
1, Canonical-basis representation
Ebata et al, PRC82, 034306
TDHFB
: Pair of k-state
(no restriction of time-reversal)
: Occupation probability
: Pair probability
: Density matrix
: Pair tensor
Inversion
We can obtain the derivatives of ρk(t) and κk(t) with respect to time.
ECT* workshop ATDmQMB
2013.10.14-18
Recipe for the Cb-TDHFB
Ebata et al, PRC82, 034306
TDHFB
are identical to gap parameters of BCS approximations,
in the case where pair potential is computed as
We can get the time-dependent equation for ρk(t) and κk(t)
workshop ATDmQMB
withECT*
orthonormal
canonical basis
2013.10.14-18
?
Recipe for the Cb-TDHFB
Ebata et al, PRC82, 034306
Can we describe the inversion for this part with the orthonormal canonical basis ?
We can not invert this pairing potential,
because the two-particle state
do not span the whole space.
2, Assumption for Pairing potential
… Pair potential is diagonal.
We can invert
the pairing potential.
Cb-TDHFB equations
Properties of Cb-TDHFB
TDHF
HF+BCS
ECT* workshop ATDmQMB
2013.10.14-18
When we apply Cb-TDHFB to ...
Ebata et al, PRC82, 034306
3, If we adopt a schematic pairing functional:
This pairing potential violate the gauge invariance related to
the phase degree of freedom of canonical basis.
Cb-TDHFB equations are invariant with respect to the phase of canonical basis.
This schematic
pairing potential violate
We must choose the special gauge in this schematic pairing functional.
ECT* workshop ATDmQMB
2013.10.14-18
Recipe for the Cb-TDHFB
Ebata et al, PRC82, 034306
TDHFB
1, Canonical-basis representation
Essential points
of derivation
2, Assumption for Pairing potential
Cb-TDHFB
3, We adopt a schematic pairing functional.
We choose the special gauge.
For applications of Cb-TDHFB
ECT* workshop ATDmQMB
2013.10.14-18
Essential points
of application
Contents
Time-dependent method
Cb-TDHFB
Linear response calculation
Comparison to Exp. & QRPA
Systematic study for E1
Application to other modes
Summary and Perspective
ECT* workshop ATDmQMB
2013.10.14-18
Contents
Time-dependent method
Cb-TDHFB
Linear response calculation
Comparison to Exp. & QRPA
Systematic study for E1
Application to other modes
Summary and Perspective
ECT* workshop ATDmQMB
2013.10.14-18
Linear response calculation with TD scheme (procedure)
Calculate HF or HF+BCS ground state
Adding
a instantaneous external field
to ground state
one-body operator
Calculate the time-evolution with TDHF or Cb-TDHFB
Strength function S(E;F) is gotten as Fourier transformed TD-
: Smoothing parameter
ECT* workshop ATDmQMB
2013.10.14-18
.
Linear response cal. with TD scheme (ex. 20Ne w/ BKN int.)
We need to confirm the dependence on
the amplitude k, and the operator.
ECT* workshop ATDmQMB
2013.10.14-18
Example : Photo-absorption cross section of 172Yb (W/ SkM*)
3D Cb-TDHFB
Cb-TDHFB can reproduce
the photo-absorption cross section of 172Yb.
β =0.32
∆n = 0.76 [MeV]
∆p = 0.55 [MeV]
Heavy nucleus
Deformed nucleus
Including pairing
Γ= 1.0 [MeV]
K=0-
Total cal. cost : 300 CPU hours
(with a Single processor; Intel Core i7 3.0 GHz)
Box size : R=15[fm], mesh=1[fm] (3D-Spherical)
Canonical-basis space (HF+BCS g.s.) :
146 states for neutron,
98 states for proton
Experimental data:
A.M.Goryachev and G.N.Zalesnyy Vopr. Teor. Yad. Fiz. 5, 42 (1976).
ECT* workshop ATDmQMB
2013.10.14-18
K=1-
Proton
Neutron
Example of Cb-TDHFB (for 172Yb ; numerical cost)
Cb-TDHFB can reproduce the similar results of QRPA
with very small numerical cost.
S. Ebata using Cb-TDHFB
( based on PRC82, 034306 )
172
J. Terasaki and J. Engel
PRC82, 034326
Yb
172
Yb
β =0.34
β =0.32
∆n = 0.76 [MeV]
∆p = 0.55 [MeV]
<∆n>= 0.77 [MeV]
<∆p>= 1.25 [MeV]
Total cal. cost : 300 CPU hours
(with a Single processor; Intel Core i7 3.0 GHz)
Total cal. cost : 100,000 CPU hours
(with Kraken; Super computer of ORNL)
Box size : R=15[fm], mesh=1[fm] (3D-Spherical)
Box Size : ρ = z±=20[fm], b-spline (Cylindrical)
Canonical-basis space (HF+BCS g.s.) :
146 states for neutron,
98 states for proton
Single-quasiparticle space (HFB g.s.) :
5348 states for neutron,
4648 states for proton
ECT* workshop ATDmQMB
2013.10.14-18
Example of Cb-TDHFB (for 188Os ; triaxial deformed nucleus )
Cb-TDHFB in 3D-coordinate representation
can describe the cross section of deformed nuclei.
Cb-TDHFB
188
TDHFB
I.Stetcu, et al.
PRC84 051309(2011)
Os
Skyrme : SkM*
Skyrme : SLy4
Deformation : β =0.202 γ =21.2°
Gap energy : ∆n = 0.87 [MeV] ∆p = 0.00 [MeV]
Deformation : β = ??? γ = ???°
Gap energy : ∆n = ??? [MeV] ∆p = ??? [MeV]
ECT* workshop ATDmQMB
2013.10.14-18
Contents
Time-dependent method
Cb-TDHFB
Linear response calculation
Comparison to Exp. & QRPA
Systematic study for E1
Application to other modes
Summary and Perspective
ECT* workshop ATDmQMB
2013.10.14-18
Systematic Study of Pygmy Dipole Resonance (PDR)
GDR
Soft dipole mode ?
Experiments
26
Strength
Ne: J. Gibelin et al., PRL101, 212503.
68
Ni: O. Wieland et al., PRL102, 092502.
130, 132
Sn: P. Adrich et al., PRL95, 132501.
140
Ce: R.-D. Herzberg et al., PLB 390, 49.
138
Ba: R.-D. Herzberg et al., PRC60, 051307.
138
Ba,140Ce,144Sm: A. Zilgas et al., PLB542, 43.
208
Pb: N. Ryezayeva et al., PRL89, 272502.
204-208
Pb: J. Enders et al., NPA724, 243.
208
Pb: A. Tamii et al., PRL107, 062502.
?
PDR
Excitation Energy
What is a nature of PDR ?
EXP.
Collective mode or not
Characteristic mode of unstable nucleus
Light and Heavy mass region
For deformed nuclei
…
→ UNCLEAR !!
P. Adrich et al.,
PRL95, 132501.
ECT* workshop ATDmQMB
2013.10.14-18
Calculation Setup
Isovector Dipole
External filed :
Isovector dipole mode (for E1 strength)
Proton
Effective Interaction : Skyrme force (SkM*),
Smoothed Pairing strength G (ref. N. Tajima et al. NPA603(1996)23)
: cutoff function
Neutron
Neutron and Proton
vibrate in anti-phase.
Nucleus : 14-28O, 18-32Ne, 18-40Mg, 24-46Si, 28-50S, 32-58Ar, 34-64Ca,
56-84
Ni, 60-88Zn, 64-98Ge, 68-104Se, 72-118Kr, 76-118Sr, 80-122Zr, 84-124Mo, 88-130Ru,
92-134
Pd, 96-138Cd, 100-140Sn, 128-142Te, 130-142Xe, etc. ( about 350 kinds of Nucleus )
Calculation space (3D-Spherical meshed box):
For light nuclei (8 < Z < 20),
we use the box has radius 12 [fm] and meshed by 0.8 [fm].
For heavy nuclei (Z > 28),
we use the box has radius 15 [fm] and meshed by 1.0 [fm].
ECT*
workshop ATDmQMB
Lattice
points
2013.10.14-18
1.0 [fm]
15 [fm]
15 [fm]
To quantify the low-lying E1 strength systematically ...
The ratio of low-lying E1 strength
in Total E1 strength (sum rule).
ex.)
We use the ratio to analyze
the low-lying E1 strength
for all calculated nuclei.
26
“Low-energy”
Ne on a 208Pb target
−λn=7.17
J. Gibelin, et al., Phys. Rev Lett. 101, 212503 (2008)
ECT* workshop ATDmQMB
2013.10.14-18
N-# dependence of PDR (light isotopes (82 items): Z < 20)
O, Ne, Mg
S, Ar, Ca
Solid line: Cb-TDHFB
Dashed line: HF+RPA
Solid line: Cb-TDHFB
Dashed line: HF+RPA
14
28
36
For O, Ne, Mg
The low-lying E1 strength appear from when s1/2 (N=16) orbital is occupied.
For S, Ar, Ca
The low-lying E1 strength appear from when p3/2 (N=30) orbital is occupied.
The ratio stays almost constant from when f5/2 (N=36) orbital is occupied.
Pairing effects
The low-lying E1 strength increases smoothly around N=28 in S and Ar isotopes,
due to the fractional occupation of orbitals caused by the Pairing correlation.
ECT* workshop ATDmQMB
2013.10.14-18
N-# dependence of PDR (heavy isotopes (272 items): Z > 28)
N=82
N=50
deformed
ECT* workshop ATDmQMB
2013.10.14-18
Contents
Time-dependent method
Cb-TDHFB
Linear response calculation
Comparison to Exp. & QRPA
Systematic study for E1
Application to other modes
Summary and Perspective
ECT* workshop ATDmQMB
2013.10.14-18
Isoscalar Quadrupole mode for deformed nuclei
3D Cb-TDHFB
(schematic-pairing)
β =0.37
34
Mg
∆n = 1.45 [MeV]
∆p = 0.00 [MeV]
Ext. Operator
34
Mg
2D TDHFB(QRPA)
(delta-pairing)
C. Losa et al.
Phys. Rev. C81, 064307 (2010)
β =0.36
<∆n>= 1.60 [MeV]
<∆p>= 0.00 [MeV]
ECT* workshop ATDmQMB
2013.10.14-18
Isovector-, p-, n-Quadrupole modes
and Absorbing Boundary Condition(ABC)
Cal. Box: 24-fm cube w/o ABC
Cal. Box: 36-fm cube w/ ABC (6 fm)
46
46
Ar
Ar
ABC region
PRC71, 024301 et al.
T. N & K.Y
24 fm
B(E2)
can be evaluated.
ECT* workshop ATDmQMB
2013.10.14-18
24 fm
36 fm
Contents
Time-dependent method
Cb-TDHFB
Linear response calculation
Comparison to Exp. & QRPA
Systematic study for E1
Application to other modes
Summary and Perspective
ECT* workshop ATDmQMB
2013.10.14-18
Summary
To study the properties of nucleus without restrictions of mass region,
we suggest the Cb-TDHFB method in 3D-coordinate representation.
Linear response calculation using Cb-TDHFB, can be compared with
TDHFB results for IVD, ISQ modes.
Due to the small computational cost, we can apply the method to
systematic study of the excited states.
Perspective
Apply to other modes (scissors, M1, E0, E3, etc.)
Apply to Large amplitude collective motion (Fusion, Fission, etc.)
Coupling between modes through the mean field dynamics
(ex. IVD ↔ ISD, ISQ ↔ pairing vibration in deformed system, etc.)
ECT* workshop ATDmQMB
2013.10.14-18
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