原子核集団現象への平均場アプローチ
第３回 QRPA

日野原 伸生
筑波大学計算科学研究センター
原子核理論研究室

Aug. 8, 20182018年度原子核三者若手夏の学校

講義計画
p 1. イントロダクション

p 平均ポテンシャルと一粒子運動・集団運動p 対相関p 2. 平均場理論

p Hartree-Fock, Hartree-Fock-Bogoliubov
p 原子核密度汎関数理論

p 3. QRPA

p RPA, QRPA
p 線形応答理論
p 有限振幅法

p 4. 大振幅集団運動の理論

p ボソン展開法
p 生成座標法
p 時間依存Hartree-Fock-Bogoliubov
p 自己無撞着集団座標法

前回まで

l

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i=1<latexit sha1_base64="GSc6q1XEhVPiMjdMaJDYu6jAkx4=">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</latexit>dˆ†i (i > N )b̂i (1  i  N )HF状態は粒子・空孔に関する真空dˆi | HF i = 0b̂i | HF i = 0<latexit sha1_base64="PiV0WkM0Lv0mD82mfJA8dTaa4mc=">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</latexit><latexit sha1_base64="0I8JmlY0nRPoL8s+kxzLE0C2iec=">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</latexit>

一粒子軌道aのエネルギー

原子核内の核子：平均ポテンシャル中を運動
平均ポテンシャル：平均場理論(Hartree-Fock)で相互作用から自己無撞着に決定
A
X
Y
†
†
ak =
Dlk cl| HF i =a†i |0i
粒子空間
空孔空間

対相関はBCS、Hartree-Fock-Bogoliubovで記述。準粒子の導入
一粒子軌道の占有率(vi2)

Bogoliubov変換
✓

↵µ
↵µ†

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◆

=

X ✓U†
k

VT

V†
UT

◆

✓ ◆
ck
c†k
µk

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HFB i = 0λ励起状態の理論
原子核の変形は実験でわかるのか？
→基底状態だけみてもわからない。励起状態に違いが現れる

平均ポテンシャル(密度)の集団振動

一粒子軌道でみると
10

V(r)

~

X

0

1p1h励起の組み合わせ
p

-10

"F

-20
-30

ph -40

h

-50
-60
0

1

2

3

4
r

5

6

7

8

p 低エネルギーの集団振動励起
p 巨大共鳴など
p pairingがない場合は1p1h励起の重ね合わせ(RPA)p pairingがある場合は2準粒子励起の重ね合わせ(QRPA)
自由度：たとえば300個の状態に100粒子つまっている場合(100 200)=20000次元

一粒子励起HF基底状態平均場HamiltonianĥHF = EHF +p<latexit sha1_base64="l3jvWMbTrdrrDnK7WF9HjbYoMyE=">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</latexit>

一粒子軌道aのエネルギーĥHF |

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粒子空間
空孔空間

X

"p d†p dp
粒子

空孔

HF i

粒子励起(A+1系)

ĥHF d†p |†HF i = (EHF + "p )dp |HF i"h )b†h |HF i

空孔励起(A-1系)
ĥHF b†h | HF i = (EHF
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集団励起

X

HF i = (EHF + "p"h )d†p b†h |HF i

残留相互作用によるph励起の重ね合わせ

集団励起
Hamiltonianはもともと二体以上

X

Ĥ =

1
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ijkl

Hartree-Fockでは二体部分はSlater行列式を使って一体場に近似
ph励起状態間の相互作用も考慮する必要がある
残留相互作用(residual interaction)

Ĥ

ĥHF

励起状態はHF状態のph励起の重ね合わせX<latexit sha1_base64="8TpAlh9qaFLtbDktKeTwA/6LAA8=">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</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="cgEdB8bhV33+oobq0VQa/1TdU5g=">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</latexit>
sha1_base64="ob6nrdauCpDHQudnp+T2fbZ8Y1o=">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</latexit>

Tamm-Dancoff近似(TDA)|⌫i =<latexit sha1_base64="ofWf/4ZSgMC/Rzar4Gz1dMI7Kos=">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</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="Jsw4HFYnp8dPCTkM79dPBSN5P18=">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</latexit>
sha1_base64="/FY7oRzTRdZvcKcmp8aw2hl7gCE=">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</latexit>Ô⌫† =<latexit sha1_base64="Jv8YPi4khEOOX5PRVBtDmzvDdYo=">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</latexit>

X
ph

X

⌫ † †
Cph
d p bj |⌫ † †Cph
d p bhph†
HF i = Ô⌫ |HF i

Ĥ = ĥHF +X⌫<latexit sha1_base64="4Lde0TkmdJYaLtTWkXFRNyxafA8=">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</latexit>

phTamm-Dancoff方程式(固有値方程式)<latexit sha1_base64="qhwbmFlWx2XyxgBN5+Vp6Klfp7I=">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</latexit>

p0 h0

[("p0

h

のうちd+b+db型を取り込む<latexit sha1_base64="ClVhsx73iE/Dkk77muj3ZylUzEg=">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</latexit>

X

p

E⌫TDA Ô⌫† Ô⌫ + · · ·✓ †◆ ✓ †◆ ✓ ◆ ✓ ◆ddddc† c† cc ⇠ a† a† aa ⇠
b
b
b†
b†
<latexit sha1_base64="/37WXL2CQyPXAFXJjskizHGA5Do=">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</latexit>

⌫
Cph

⌫
"h0 ) pp0 hh0 + v̄p0 hh0 p ] Cp⌫0 h0 = E⌫TDA Cph

粒子空間

空孔空間

RPA
TDA: 励起状態にはph間の相互作用が取り込まれるが基底状態はHFのまま
乱雑位相近似(random-phase approximation; RPA) 基底状態にも相関を入れる

Ĥ| i = E | i<latexit sha1_base64="dJEgjmsPpULsa/aTEJ4HjR4HAeE=">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</latexit>

フォノン演算子の真空として基底状態を定義†| i = Ô |0i

Ô |0i = 0<latexit sha1_base64="rO2QoEK71N2ZB0GNfK6u5qeQqL0=">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</latexit>

TDAのときはÔ⌫† =<latexit sha1_base64="+v+329gsAu+9PzoArfS2xwXrOUI=">AAACw3ichVHNStxQGD2m1dppq9O6EdwEB4ur4YsILYUWqRTc+ddRwWi4yVxnwuSP5GbAhnmBvoCLrlrqQnwAH8CNdNt24SOIS4VuXPhNEpBWtF9I7rnnnvPdHD478txEEZ0OaA8eDg49Gn5cefL02cho9fmLtSRMY0c2nNAL4w1bJNJzA9lQrvLkRhRL4dueXLc78/3z9a6MEzcMPqrdSG75ohW4O64jFFNW9Z3ZFiozHeHpi71tsylalhmk+lvdTFLfyqJ2T5/fZqaAzUIR6XYB2la1RnXKS78NjBLUUNZSWD2CiSZCOEjhQyKAYuxBIOFnEwYIEXNbyJiLGbn5uUQPFfamrJKsEMx2+Nvi3WbJBrzv90xyt8O3ePzG7NQxRb/pgC7ohA7pjK7u7JXlPfr/ssurXXhlZI1+Hl/981+Xz6tC+8Z1j8Nm9f2ZFHbwOs/icrYoZ/opnaJ/99PexeqblansJX2jc873lU7pmBMG3Utnf1mufEGFB2T8O47bYG2mblDdWJ6tzb0vRzWMCUximufxCnNYwBIafO93/MBP/NI+aB0t1lQh1QZKzxj+Kq13Df3WqQ4=</latexit>

RPAでは

†

Ô =
<latexit sha1_base64="pgOss/X+q6IvRLdl31uO6C3rtKI=">AAAC5nichVFPa9RAHH2NVuta7aoXwcvQpdKLy6QIiiAUvXizf9x2pemGSTK7CZ0kQzK7UMN+AQ/epAcvKngQP4AfwIN+gR76EUQ8VezFg79NAmKLdUIy7/fmvd/k8fO0inLD+cGUdebs9LnzMxcaF2cvXZ5rXrm6kafDzJcdP1Vp1vVELlWUyI6JjJJdnUkRe0puejsPJ+ebI5nlUZo8MbtabsdikET9yBeGKLcpnFCYwvGFYo/HPScQA9dRZA8Eu8+cfBi7hQ7HrNur2aoMKqVmXgVCdos97RW1ZlyJPKIDV7vNFm/zcrGTwK5BC/VaSZsf4SBACh9DxJBIYAgrCOT0bMEGhyZuGwVxGaGoPJcYo0HeIakkKQSxO/QdULVVswnVk5556fbpFkVvRk6GBb7P3/ND/oV/4F/5r3/2Ksoek3/Zpd2rvFK7c8+vrx/91xXTbhD+cZ3i8Eh9eiaDPu6WWSLKpktmktKv+o+e7R2u31tbKG7yt/wb5XvDD/gnSpiMfvjvVuXaKzRoQPbxcZwEG0ttm7ft1dut5Qf1qGZwA/NYpHncwTIeYQUduvczvuMnjqzQemG9tPYqqTVVe67hr2W9/g04mrch</latexit>

X<latexit sha1_base64="GSC/SouxX3ElKqJkfNxMyAFoKHA=">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</latexit>

⌫ † †
Cph
d p bh

基底状態はHF状態<latexit sha1_base64="lBYxjMDxXy9vnvQ8xgVivxZTLF4=">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</latexit>

ph

X
ph

|0i = |Xph d†p b†hYph bh dp

前方振幅

後方振幅

✓ †◆ ✓ †◆ ✓ ◆ ✓ ◆ddddc† c† cc ⇠ a† a† aa ⇠
b
b
b†
b†

HF i

RPAでの基底状態(真空)は|HF>に相関がはいったもの<latexit sha1_base64="ClVhsx73iE/Dkk77muj3ZylUzEg=">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</latexit>のうち、d+b+db, d+d+b+b+, ddbbを取り込む

†

Ô =
<latexit sha1_base64="pgOss/X+q6IvRLdl31uO6C3rtKI=">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</latexit>

RPA/QRPAX

Xph d†p b†hphYph bh dp

Ĥ = EHF +†[Ĥ, Ô ] = ~⌦ Ô

†<latexit sha1_base64="Csx3YZPnbmog8hoIDOCCJHxaQDg=">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</latexit>[Ĥ, Ô ] =<latexit sha1_base64="tD+INuw0hB+KjlmQfSfX/OJgP6Q=">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</latexit>✓

RPA方程式<latexit sha1_base64="dmtcsx6b6PIVK8TnsbEjbYwjv/4=">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</latexit>

Aphp0 h0 = ("ph | 0i =<latexit sha1_base64="KQMlWxYa1lpce08FpLLm4vFvoOE=">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</latexit>

0=<latexit sha1_base64="VpqpTrY50edEPbOmrCaLB8Ig1+0=">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</latexit>

pp0 hh0 =

0

X⇣
ph
X

=h

RPA |[Ô0⇤
Xph Xph, Ô† ]|

0⇤Yph YphXph⇤ Xp0 h0⌘~⌦ Ô† Ô

~⌦ Ô

調和振動子と同じ方程式

◆✓ ◆
0
X
1
Y

"h ) pp0 hh0 + v̄ph0 hp0<latexit sha1_base64="8h2j6b5+mhpdyQDcYNp4OteCyYY=">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</latexit>

規格直交性<latexit sha1_base64="S2MpmkGrMgBb6Pf2+yqEVwQUx60=">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</latexit>

◆✓ ◆
✓
B
X
1
=
~⌦
A⇤
0
Y

A
B⇤

ERPA +X

RPA i ⇠ h

Bphp0 h0 = v̄pp0 hh0<latexit sha1_base64="Qv2roTel/5igI5hfglGasL0GKTs=">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</latexit>

HF |[Ô, Ô† ]|

HF i

Yph⇤ Yp0 h0

QRPA (quasiparticle RPA) pairingがあるとき<latexit sha1_base64="OJf64S8TWm0ffCnd0Xlzk4RJ96I=">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</latexit>†

O =<latexit sha1_base64="zyqZF/lDsyDoDQ2iOfCZ1XpzELs=">AAADCHichVE9S+RQFL3Jrl/jrjOujcs2wUHZxuFGBEUQZG3s/BydxWh4ybwZgy8fJHkDbpjKzj9gYaVgIdtsJ9vuNv4BC3+CWCrYWHgnExEV9YbknXvuPfflcK1AOFGMeKGoHz52dHZ19+R6P33uyxf6v6xGvgxtXrZ94YcVi0VcOB4vx04seCUIOXMtwdes7dlWfa3Bw8jxvZV4J+AbLqt7Ts2xWUyUWdhNDJsJbb65aVRZ3TQESatMm9aMSLpmYrjS8GRTq2xmlUfKYCLYYpnMlU9zT2qj2s/XRGn9AbrSLBSxhGloL4GegSJkseAXTsGAKvhggwQXOHgQExbAIKJnHXRACIjbgIS4kJCT1jk0IUdaSV2cOhix2/StU7aesR7lrZlRqrbpFkFvSEoNhvEcT/Aaz/A3XuLdq7OSdEbrX3botNpaHpj5vcHl23dVLp0xbD2q3lBY1P22pxhqMJl6cchbkDItl3Z7fuPX/vXy1NJwMoJHeEX+DvEC/5NDr3FjHy/ypQPI0YL05+t4CVbHSjqW9MXx4syPbFXd8A2G4DvtYwJmYA4WoEz3Xip5ZVD5qu6pf9RT9W+7VVUyzQA8CfXfPZH1wxw=</latexit>Ĥ<latexit sha1_base64="8ynmhSMXjOBfmuZeIRv7BLuqfTQ=">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</latexit>

XXµ⌫ ↵µ† ↵⌫†Yµ⌫ ↵⌫ ↵µµ⌫N̂ = EHFB +EQRPA +

X

二準粒子励起の重ね合わせ~⌦ Ô† Ô

遷移強度と巨大共鳴
基底状態|0>から励起状態|λ>への演算子Fによる遷移確率

集団励起：集団演算子(多重極演算子など)で強く励起される状態
集団低励起：ΔN=0の励起
巨大共鳴(高励起)
巨大共鳴の種類

ΔN=2励起(正パリティ)、ΔN=1(負パリティ)
古典的な形状振動
T=0(アイソスカラー)

T=1(アイソベクター)

λ=0(単極子共鳴)

ISGMR

IVGMR

λ=1(双極子共鳴)- (ISGDR)IVGDR

λ=2(四重極共鳴） ISGQR

IVGQR

ガモフ・テラー共鳴(F=στ)、スピン巨大共鳴など

the energy-weighted
sum ruleof themultipole operators
without
additional model
space
truncation
is not possiIV,ed out inconsidered here. This indicates that with the given tranble. The table also shows the convergence (of integrationaor.
generalsition operator, the gauge invarianceof the term ⇢⌧ j 2
QRPA和則nderlyingalongshould
A2 ) isenoughwhen
for linking
k = Thouless
4 sum rule
which,
notnotbe broken
theoremandhowgacommutaX
ever,QRPA
is notsum
as importantotherenergymomentsshown
rules. Other as
terms
in thetime-odd
functionalues
2
和則(sumrule)donotchangetheenergy-weighed
sum
rule.
The
local
m
(
F̂
)
=
(~⌦
)
|h
|
F̂
|0i|
there.
k⌫theoximation
gauge symmetry also connects the term with Ctrj to thetheunctional
spin-orbit term CtrJ , but the numerical results show thisteraction.
ven
term does not a↵ect the energy-weighted sum rule of mul和則比は巨大共鳴の平均励起エネルギーとも対応
ulomb
its ex-B. tipoler theorem for energy-weighted sum ruleThouless
operators.
m1
Hamiltomk
heE 1  E0  E1  Ē ⌘
 E2
Ek =
mation tomm
k 2 gives the relation between the 0snd
discrep- Thouless theorem
A1To theenergy-weighted sum rules m (F̂ ) for isoscalar operators
6].
1
P
PA
not beenhusA
いくつかの和則は平均場から直接計算できる
foroperators
inverse energy-weighted
F̂ = C. i=1Dielectricf (r̂i ) ortheorem
isovector
F̂ = i=1 f (r̂i )ˆ
⌧3masumofruleand the expectation value
the double commutator at
FB value”
er, Thoulessの定理(k=1, energy-weighted sum rule)
he energythe ground state [8]Thouless, Nucl. Phys. 22, 78 (1961)ery
re shown
The dielectric theorem connects the inverse-energycspace.
is In
weighted sum
energy curvature.
1 rule with the potential
1
igible difxis.
m1 (theoremF̂ ) = h[wasF̂ , [proposed
Ĥ, F̂ ]]i =h[F̂[9], [T̂for, F̂the
]]i(1
)
This
in Ref.
HF+case,2
2
FAM
and it has been provenin the HFB framework in Ref. [19].
! and
2 Z
ncreasing~From the dielectric
theorem, 2the inverse energy-weighted
his
=
|rf
(r)|the⇢(r)dr(1(19)
k=3,5,...にも同様の定理が存在
ce we obsum rule is computed
from
relation + ) ,2m
me,
ion of thederivative
誘電定理(k=-1, dielectric theorem)

med accuonal comto underults show
good apr Skyrme<latexit sha1_base64="KE3n+V8Hiju7UvB94OVeDUt0dz4=">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</latexit>

1 @2
m 1 (F̂ ) =E[R( )]
2 @ 2

=
=0

1 @ h ( )| F̂ | ( )i
2
@

,
=0

(21)

|φ(λ)>線形拘束-λFで計算したHFB状態

QRPAコードのチェック、近似の妥当性をみるのみも使われる
where
the generalized density R( ) is computed from theQRPA計算例
巨大共鳴

低励起集団状態

双極子巨大共鳴γ振動

Terasaki and Engel, Phys. Rev. C 84, 014332 (2011)

Yoshida and Nakatsukasa, Phys. Rev. C 83, 021304(R) (2011)

Skyrme SkM* functional

QRPA解は行列の固有値・固有ベクトルで離散的Lorentzian smearing

dB
=
d!
2⇡
<latexit sha1_base64="KXmHjChwy2O2YG7T9RxgiumYlR8=">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</latexit>

X

>0

†

2

|h |F̂ |0i|
(⌦
!)2 +

2

2

|h |F̂ |0i|(⌦ + !)2 +2!

QRPA解とHFB状態の安定性
QRPA方程式の性質 X,Y, Ωが解であれば、 Y*, X*, -Ωも解HFBが安定解であればすべてのΩは実数(正と負)HFBが不安定であれば虚数となるΩが存在
RPA解の詳細な構造：Nakada, Prog. Theor. Exp. Phys. 2016, 063D02, 2017, 023D03

安定解

不安定解

任意の変位(ph自由度、変形度など)ある変位(ph自由度、変形度など)

例えば対称性に制限のあるコードでHFB方程式を解くと不安定解が現れる
・球対称コードで計算したHFB解が変形の自由度に対して不安定
・軸対称変形コードで計算したHFB解が非軸対称変形に対して不安定
・パリティ対称性を課して計算したHFB解がY30変形に対して不安定
QRPA解の性質をみることでHFB解の安定性がわかる

QRPA方程式のPQ表示
QRPA方程式[Ĥ, Ô† ] = ~⌦ Ô†[Ĥ, Ô ] =<latexit sha1_base64="S2MpmkGrMgBb6Pf2+yqEVwQUx60=">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</latexit><latexit sha1_base64="tD+INuw0hB+KjlmQfSfX/OJgP6Q=">AAAC23ichVE9SxxRFD1OTDSbDzexCdgMWQwpwnInBAyBgCSNnV/ZVdjZDG9mn+vgmw9m3i6YYat0YisWqSJYiD/Axs7GP5DCNp1oZ8DGIndnB0JiYu7w5p137j33vcN1Y+WnmuhkyLg1fPvOyOjd0r37Dx6OlR89rqdRJ/FkzYtUlCy7IpXKD2VN+1rJ5TiRInCVXHLX3vfzS12ZpH4UftDrsWwGoh36K74nNFNOud6wV4XOZnovzBzYnlDmbO+j3RJtJ7MVd2qJXtN8y2lXJPZsINvCKfi/SIqMU65QlfIwrwOrABUUMReVD2CjhQgeOgggEUIzVhBI+WvAAiFmromMuYSRn+cleiixtsNVkisEs2v8b/OpUbAhn/s901zt8S2KV8JKE5P0jfbogo5pn07p6p+9srxH/y3rvLsDrYydsY0ni5f/VQW8a6z+Ut2gcLn6Zk8aK3ide/HZW5wzfZfeoH/30/bF4puFyewZ7dAZ+/tKJ3TEDsPuD293Xi58QYkHZP05juug/rJqUdWaf1WZfleMahQTeIrnPI8pTGMGc6jxvYf4jjOcG03js7FhbA5KjaFCM47fwtj6Cf3msuI=</latexit>~⌦ Ô

調和振動子と同じ形→座標・運動量による表示が可能<latexit sha1_base64="xOa2A6LZH69isEPxbWqyG/sW39Y=">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</latexit>

1
P̂ =
i

r

M ~⌦
(Ô
2

†

[P̂ , P̂ 0 ] = [Q̂ , Q̂ 0 ] = 0<latexit sha1_base64="UfXUhQLmoJcn7CVDubEQ4zJp/rg=">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</latexit>

PQ表示のQRPA方程式

2<latexit sha1_base64="+KdrxT47/sah9xT3AcuQ6gjOFww=">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</latexit>N̂ = EHFBP, Qはエルミート演算子<latexit sha1_base64="O71Ksz6JW5Q9NhjRENKQySsiGYY=">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</latexit>P̂ =<latexit sha1_base64="FUTEVGMBXLUpfcgX3To34D2DUyQ=">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</latexit>Xh<latexit sha1_base64="iCRTgvIAaUqL/qSV+QXIgVerHy8=">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</latexit>

HFB |[Q̂r, P̂ 0 ]|HFB i = i~

0

i~
P̂
M

<latexit sha1_base64="uVa22vtvsIHKsntN0bbz0EoFF/Q=">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</latexit>

X 1
1
M 2 2
TrA +
P̂ 2 +⌦ Q̂
22M
2Pµ⌫ ↵µ† ↵⌫† + Pµ⌫⇤ ↵⌫ ↵µ

Q̂ =

µ<⌫

Pµ⌫ = i<latexit sha1_base64="kau8Btxei7g45ixeiul7spWx+nI=">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</latexit><latexit sha1_base64="xll1IuaUjPVwVLG1ED5lZSQGdY4=">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</latexit>~(Ô + Ô† )2M ⌦[Ĥ, Q̂ ] =[Ĥ, P̂ ] = i~⌦ M Q̂
Ĥ

Q̂ =

Ô )

r

M ~⌦
(Xµ⌫ + Yµ⌫⇤ )2<latexit sha1_base64="RGAEG4SHKIfXv4s6CyTZJF561nw=">AAADDnichVFNa9RQFL2JVtvxo6NSKAgaHCpCYbgRQRGFYjdddlpnptDXhpfM60zoywfJy0AN8wN05c6FKwtdlO7rSly46R9wMT9BXLiYQjdd9M6bFGmL9YbknXvuPfflcN1Y+qlC7Bvmlatj166PT5Ru3Lx1e7J8524jjbLEE3UvklGy4vJUSD8UdeUrKVbiRPDAlaLpbs4P682uSFI/Ct+qrVisBbwd+hu+xxVRTvk963CVM49Lq9ZzmCRli1uvLZZmgZOzIHvFwqxn1daLkuY0xbiMO3ydtXjbIe5sHmbWLInyQsV4qnrnpbrrFAaZU65gFXVYF4FdgAoUsRiVvwKDFkTgQQYBCAhBEZbAIaVnFWxAiIlbg5y4hJCv6wJ6UCJtRl2COjixm/RtU7ZasCHlw5mpVnt0i6Q3IaUFM/gTd3GAB7iHv/D4n7NyPWP4L1t0uiOtiJ3JD9PLR/9VBXQq6PxVXaJwqftyTwo24IX24pO3WDNDl95ofvfdp8Hyy6WZ/DFu42/y9wX7+IMcht1Db6cmlj5DiRZkn1/HRdB4WrWxateeVebeFKsah/vwCJ7QPp7DHCzAItTp3j/GlPHAeGh+NPfNb+b3UatpFJp7cCbMgxPxhcXU</latexit>Qµ⌫ =<latexit sha1_base64="2efiXfHJwfAxE6UwohZCHJzqzAI=">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</latexit>

XQµ⌫ ↵µ† ↵⌫† + Qµ⌫⇤ ↵⌫ ↵µ

µ<⌫

r

~
(Xµ⌫
2M ⌦

Yµ⌫⇤ )

RPA方程式のPQ表示
RPA方程式

✓

A
B⇤

B
A⇤

<latexit sha1_base64="i+Db5KAlMv8Qxsm0xtEOWpsGcaU=">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</latexit>

✓

A
B⇤

B
A⇤

◆✓

Q

✓

<latexit sha1_base64="ay1p/y1V666Lii+dDfWCQmIozII=">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</latexit>

規格化

Q ⇤
<latexit sha1_base64="5+TPi0rSNAnhMgl3I5TeRaa+CQE=">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</latexit>

◆✓

P
⇤
P
Q
⇤
Q
0

P
0⇤
P

◆

◆

◆

= i~⌦2 M

~ 1
=
iM

= i~

✓

✓

P
P ⇤

Q
Q ⇤

◆

◆

0

A,B行列が実数のときはQRPAで対角化する行列の次元を半分にできる(A + B)(A<latexit sha1_base64="f4mGDqUvalsqLowwg878g0Og5BQ=">AAACv3ichVFNSxtRFD2Z1qrRNmm7KXQzNCiR0nAnFFqEQhI33dWo0UASw5vxmQyZL2YmgTT0D/gHXHSjggvxB/QHFEp3uunCn1C6tNCNC28mg6JBe4c3775zz7nvHa7uWWYQEp0llAcPJx5NTk0nZ2YfP0mlnz5bD9yub8iK4VquX9VFIC3TkZXQDC1Z9XwpbN2SG3pnaVjf6Ek/MF1nLex7smGLlmNum4YIGWqmF7Nq8XVpIVt8U1pQy5t1i6VbQv2g1tu68Dfzav2TLVuiGRcYuCI10xnKURTqeKLFSQZxLLvpb6hjCy4MdGFDwkHIuQWBgL8aNBA8xhoYMOZzZkZ1iS9IsrbLLMkMwWiH/y0+1WLU4fOwZxCpDb7F4uWzUsUc/aIjOqefdEy/6eLOXoOox/Atfd71kVZ6zdTOi9V//1XZvIdoX6vuUejMvt9TiG28j7yY7M2LkKFLY9S/93n3fHVxZW4wTwf0h/3t0xl9Z4dO769xWJYrX5HkAWm3xzGerOdzGuW08ttMoRSPagov8QpZnsc7FPARy6jwvXv4gROcKkWlpTiKN6IqiVjzHDdC6V8CfWOkMQ==</latexit>B)Q = ~2 ⌦2 Q

QRPAと対称性
HFB解での連続対称性の破れ：平均ポテンシャル(一体場で対称性が破れている)・並進対称性
・回転対称性(変形)・ゲージ対称性(対相関)
もともとのHamiltonianは対称性を持っているが一体平均場のHamiltonian
では自発的に対称性が破れる(P: 破れた対称性の演算子)[Ĥ, P̂ ] = 0[ĥHFB , P̂ ] 6= 0
<latexit sha1_base64="M4j9PC+A/f2Y+NEtSQbwoBcsXc4=">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</latexit><latexit sha1_base64="OIHWmO6Bn/WXROQgh7mG8rJQWOA=">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</latexit>

PQ表示のQRPA方程式
2[Ĥ, P̂ ] = i~⌦ M Q̂<latexit sha1_base64="+KdrxT47/sah9xT3AcuQ6gjOFww=">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</latexit>[Ĥ, P̂ ] = 0<latexit sha1_base64="OIHWmO6Bn/WXROQgh7mG8rJQWOA=">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</latexit>[Ĥ, Q̂ ] =

i~
P̂
M

◆✓

◆

<latexit sha1_base64="uVa22vtvsIHKsntN0bbz0EoFF/Q=">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</latexit>✓<latexit sha1_base64="duhdJHEMDDcu6w/MUtxHdhsffqU=">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</latexit>

A
B⇤

B
A⇤

P
P⇤

=0

対称性が破れた系でのPはRPAのゼロ・エネルギー解(spurious解)HFBで自発的に破れた対称性はQRPAで回復(南部・ゴールドストーンモード)

対称性の自発的破れと南部・ゴールドストーンモード
系が自発的に対称性が破れると状態が無限に縮退する対称性を回復するゼロ・エネルギー集団モード(南部・ゴールドストーン)が発生

破れた対称性

平均場

並進対称性

重心が原点に固定

回転対称性
SU(2)
粒子数保存
(U(1)ゲージ対称性)

アイソスピン対称性

南部・ゴールドストーン解(QRPA)

Kπ
(量子数)

変形

集団回転
z軸対称：x軸 or y軸回転
非軸対称：３次元回転

1 +, 2 +対凝縮・超伝導対回転

0+

中性子ー陽子混合

アイソスピン回転

0+

重心運動

1-

QRPAのゼロ・エネルギー解

XY表示では計算できない(規格化できない)

✓

A
B⇤

<latexit sha1_base64="dmtcsx6b6PIVK8TnsbEjbYwjv/4=">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</latexit>

B
A⇤

◆✓ ◆
✓
X
1
= ~⌦
0
Y

◆✓ ◆
0
X
1
Y

もともとのHamiltonianは対称性を持っているが一体平均場のHamiltonian
では自発的に対称性が破れる(P: 破れた対称性の演算子)i~
P: 破れた対称性の演算子(既知)[Ĥ, P̂ ] = 0[Ĥ, Q̂ ] =P̂
M
Q: 対称性演算子と正準共役な座標演算子(未知)h HFB |[Q̂ , P̂ ]| HFB i = i~
M: 慣性質量(QPRA mass, 未知)<latexit sha1_base64="w+b869C1z3mrm8LpAHsaHF6grmo=">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</latexit><latexit sha1_base64="uVa22vtvsIHKsntN0bbz0EoFF/Q=">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</latexit><latexit sha1_base64="78fu+vkF2jZ6zXhL31YcFytCHfc=">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</latexit>

ĤN̂ = EQRPA +<latexit sha1_base64="NnUbbhopeWyB2/tWdMxE7uopPeQ=">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</latexit>

X

>0

†

~⌦ Ô Ô +X P̂µ2µNG解からのエネルギーの寄与

2Mµ

R
R
I
MNG = 2[PNG(A + B) 1 PNG+ PNG(A<latexit sha1_base64="lNH15384LrRmi7fa5Zya2jX2b0s=">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</latexit>IB) 1 PNG]

RI
PNG = PNG
+ iPNG<latexit sha1_base64="lFveod1S50RyxgehnG5NJvShTgQ=">AAACsHichVFNLyRRFD0Kg8ZobCQ2lekQiaRzSyREIhEW2Ehrmk4wlaqa1zzqK1XVndDxB+ysLKxILMQPsCWzmT8wCz9BLE0yGwu3qivECHMrr965591z3zu5pm/LMCK6a1KaW1q/tLV3ZDq7ur/2ZHv71kKvGliiZHm2F5RNIxS2dEUpkpEtyn4gDMe0xbq5Nxefr9dEEErPXY32fbHlGNuurEjLiJjSs6MFvb4ZOOrS/KE6rRa+F1/TUVUysfhCZPRsjvKUhPoeaCnIIY2Cl73GJn7Ag4UqHAi4iBjbMBDytwENBJ+5LdSZCxjJ5FzgEBnWVrlKcIXB7B7/tznbSFmX87hnmKgtvsXmFbBSxRD9pkt6pF90Rff09GGvetIjfss+72ZDK3y952hg5e9/VQ7vEXZeVZ8oTK7+3FOECiYTL5K9+QkTu7Qa/WsHJ48rU8Wh+jCd0wP7O6M7+skO3dof62JZFE8RD0j7dxzvwdpYXqO8tjyem5lNR9WOQXzDCM9jAjNYQAElvvcY17jBrTKmlBVdMRqlSlOq6cebUHafAYfWnuM=</latexit>

平均場

P

Q

M

軸対称変形核(対称軸z)角運動量演算子Jx,
Jy

回転角演算子

慣性モーメント
(MOI)

重心を固定

重心運動量演算子

重心座標演算子

全質量(mA)

超伝導状態

粒子数演算子

対回転角演算子

対回転のMOI

陽子・中性子混合

アイソスピン演算子

アイソスピン回転角演算子

アイソスピン回
転のMOI

回転対称性の破れの場合対称性を破った変形状態(物体固定系)

変形核の計算：主軸の固定が必須(固定しないと反復が収束しない)

xʼ

zʼ

yʼ

h

HFB |x̂ŷ|

HFB i =0h

HFB |ẑ x̂|HFB i =0h<latexit sha1_base64="Ef7ct/5tazCAqonl3X6hg+fDot4=">AAADV3icpVHLSiNBFL2d+IgZH9HZCLNpDMqswm0ZGBEEURhcxkdUsDV0t2VSWP2guxJM2vyAP+DClYIL8QP8gNnMesBFPkFcOjAbwbn9AFExs/A21ffWqXtO1eGanuCBROwqmWxf/8Bgbij/aXhkdKwwPrEZuA3fYhXLFa6/bRoBE9xhFcmlYNuezwzbFGzLPFyOzreazA+462zIlsd2baPm8ANuGZKgauG3LgynJpiql+u8Guq+ra78WOqox6peN2R41IlTi4BXDbqf8BZmVNT1fG+ZViLT/qBMu5M+qocM5quFIpYwDvVtoaVFEdIou4Ub0GEfXLCgATYwcEBSLcCAgL4d0ADBI2wXQsJ8qnh8zqADeeI2qItRh0HoIf1rtNtJUYf2kWYQsy26RdDyianCNN7iFT7gL7zGO3x8VyuMNaK3tCibCZd51bGTyfW//2XZlCXUn1k9GCZ19/Yk4QDmYi+cvHkxErm0Ev1m+/RhfX5tOpzBC7wnf+fYxZ/k0Gn+sS5X2doZRAPSXo/jbbE5W9KwpK1+Ky4upaPKwReYgq80j++wCCtQhgpYypyyp9SUeqabecoOZHNJa0ZJOZ/hRWTH/wEoUNqq</latexit>

HFB |ŷẑ|<latexit sha1_base64="KSS6XVlQSZn5suOi2Bv8acMeSgs=">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</latexit>0i✓ Jˆx
(✓)i
=
e
|
HFB

x

z

HFB i =0y

主軸がずれた状態：同一エネルギーのHFB状態(縮退)|

対称性の回復した状態(実験室系)

HFB i

回転対称性が破れたHFB状態は無限に縮退
NGモードの励起
(回転)

J=8
J=6
J=4
J=2
J=0

(J mixed)
軸対称変形状態(平均場)

回転エネルギー~
J(J + 1)
2Jrot
2

E(J) =

角運動量演算子の固有状態

回転バンド

NGモードは慣性系の変換(物体固定系 実験室系)

回転の慣性モーメント
軸対称変形核：回転対称性の破れ
南部・ゴールドストーンモード：集団回転運動(対称軸(z)と直交する軸周り(x,y))
QRPAによる慣性モーメント：Thouless-Valatinの慣性モーメントJTV = 2Jx⇤ (A + B) 1 Jx
<latexit sha1_base64="jxAVtXzFT+XElkG7gitrPw/SjUE=">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</latexit>

Belyaevの慣性モーメント：QRPAで残留相互作用vをゼロにしたもの
X |h HFB |Jˆx |µ⌫i|2
JBelyaev = 2
Eµ + E⌫µ⌫3
実験値
Jexp =
E(2+1)<latexit sha1_base64="61STRUyBcsSgYj6ohvRrfG2nO4s=">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</latexit><latexit sha1_base64="KzO9tbW1ssfS3GAXczzNRQnxdyQ=">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</latexit>

Petrík and Kortelainen, Phys. Rev. C 97, 034321 (2018)

対相関：慣性モーメントを小さくする(0+のエネルギーを相対的に下げる)
残留相互作用の効果はBelayevと比べて30%ほど

超伝導状態とゲージ対称性
BCS状態: Bogoliubov準粒子の真空|

BCS i =Yi

↵i ↵ī |0i =Y↵µ |<latexit sha1_base64="jkEHIHUi7zL5a1wi4JCrN0MohHk=">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</latexit>

BCS i = 0

(ui + vi a†i a†ī )|0ii

粒子数保存を破る(軌道i,ibarに粒子対がある状態とない状態の重ね合わせ)
ui(非占有率)とvi(占有率)の２つの量：相対位相(新しい自由度)
通常のBCS/HFB計算ではuiとviは実数に選ぶ(位相の固定)<latexit sha1_base64="xYYg2OIHz9KPSXfMUpecGPAWVXE=">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</latexit>

|

0
BCS (

)i = ei N̂

|BCS i =

Y(ui + vi e2i a†i a†ī )|0i

i
X †
N̂ =
ai ai + a†ī aī 粒子数演算子
ゲージ回転の角運動量演算子i>0<latexit sha1_base64="C+OHOIwpYLxP8rlFVcn95/uxwf4=">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</latexit>

Bogoliubov変換

φ：ゲージ角<latexit sha1_base64="ItqEvTsw9PmBbnINOt92POsDShY=">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</latexit>↵i† =ui a†ivi aī

で (ui , vi ) ! (ui ei<latexit sha1_base64="Yj//VS3JmbNYtbjqGqxYoe7lCqE=">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</latexit>↵ī† =ui a†ī + vi ai(a†i , ai ) ! (a†i ei , ai e i )

⇤
⇤
(Viµ
Vjµ + Viµ̄
Vj µ̄ )

密度行列はゲージ回転不変µ>0
Hartree-Fockポテンシャルも不変X
⇤
⇤
ij =
(Viµ
Ujµ + Viµ̄
Uj µ̄ )
対密度行列は回転µ>0
対ポテンシャルも回転
1X⇢ij =<latexit sha1_base64="YaPmvbLkQ735JoKP9y5tasdO33o=">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</latexit>

反交換関係は不変<latexit sha1_base64="wHS+Bf5P1ZrZ2pqB+C+4CUcIjxc=">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</latexit><latexit sha1_base64="OrpRz+/sRwO/qAwhyXyNn5Cijao=">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</latexit>X, vi ei ) とするのと同じ

ij ( ) = h<latexit sha1_base64="Yug930mbI0qvInHeNecwFAzLqP8=">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</latexit>

0
HFB (

)|cj ci |0
HFB ()i = e 2i ij (0)<latexit sha1_base64="jJHxKrbj6/bdYx3eyXtBms99otM=">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</latexit>ij [] =2v̄ijkl klkl<latexit sha1_base64="Qp8yVIjV4bZDT/eLytrnp6VHVas=">AAACs3ichVFNSxtRFD0Ztcb0w1Q3QjdDg0UXDXek0lIoiHXhMn5EhUTDzPjU18wXMy+BdMgfcC104UrBRfEHCG7TRf9AF/4E6dJCN130zmTAL2rvMPPOPfee++ZwrcCRkSK6yGkDg0OPhvMjhcdPnj4bLT4fW4v8VmiLqu07frhhmZFwpCeqSipHbAShMF3LEetW82NSX2+LMJK+t6o6gdh0zV1P7kjbVEw1iuX6gnCU2Yjlp+5UPdiT0/oHXWzFr2dkknX1m3WabhRLVKY09PvAyEAJWVT84hnq2IYPGy24EPCgGDswEfFTgwFCwNwmYuZCRjKtC3RRYG2LuwR3mMw2+bvLWS1jPc6TmVGqtvkWh9+QlTom6Qd9pSv6Tqd0SX/+OStOZyT/0uHT6mtF0Bjdn1j5/V+Vy6fC3rXqAYXF3Q97UtjBu9SLZG9ByiQu7f789ucvVyvvlyfjV3RMP9nfEV1Qjx167V/2yZJYPkSBF2TcXcd9sDZTNqhsLL0pzc1nq8rjBV5iivfxFnNYRAVVvvcA5+jhmzar1TRL2+63arlMM45bobl/AX4toXY=</latexit>ij () = e 2iij (0)

ゲージ回転はHFでは見えない。超伝導状態で初めて現れる自由度<latexit sha1_base64="RGcAefkrGZJtKWXhaCd8kV0/hsU=">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</latexit>

ゲージ対称性の破れと対回転対称性を破った超伝導状態(物体固定系)

対称性を回復した状態(実験室系)BCS/HFBではu,v波動関数を実数としてゲージ角を固定
ゲージ回転した状態：HFB状態と縮退|<latexit sha1_base64="kzlJgiu9xfavqOOGpJdX6/QCVeg=">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</latexit>

0
HFB (

)i = ei N̂ |HFB i

ゲージ対称性が破れたHFB状態は無限に縮退

NGモード励起(対回転)(N mixed)
超伝導状態(複素位相がそろった状態)

1
E(N
)
=
(N
対回転エネルギー
2JN

N=N0 8
N=N0 6
N=N0 4
N=N0 2N=N0
粒子数演算子の固有状態

N0 )2 対回転バンド

NGモードは慣性系の変換(物体固定系 実験室系)

対回転の慣性モーメント
超伝導核(対凝縮)：ゲージ対称性の破れ
南部・ゴールドストーンモード：ゲージ回転運動(U(1))
QRPAによる慣性モーメント：Thouless-Valatinの慣性モーメントJTV = 2N (A + B) 1 N<latexit sha1_base64="YM3sf6mwC/tTcQ+Ap2q7bgErcvQ=">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</latexit>

実験値( N )2E(N ) = E(N0 ) + (N0 ) N +2J (N0 )<latexit sha1_base64="ZDCf95wyrsuAqwRVeZ7mmINgQ7I=">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</latexit>

Jexp (N ) =E(N + 2)42E(N ) + E(N2)<latexit sha1_base64="Z2Sva5r04pZJcWwIeRAYuRCFJCM=">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</latexit>

中性子の対回転慣性モーメントの計算例
Sn

N

Pb

N
NH and Nazarewicz, Phys. Rev. Lett. 116, 152502 (2016)

対相関による対称性の破れは偶々核の二重束縛エネルギー差からも見える
魔法数N=50, 82, 126では対凝縮していない

中性子・陽子両方のゲージ対称性が破れた場合

中性子と陽子のゲージ対称性が破れた場合(Δn 0 and Δp 0, U(1)nxU(1)p)[ĤHFB , N̂n ] 6= 0

[ĤHFB , N̂p ] 6= 0

中性子の対回転エネルギー
陽子の対回転エネルギー

QRPA固有状態の南部=ゴールドストンモードは中性子と陽子の線形結合(中性子ー陽子間の残留相互作用のため)
formal theory: Marshalek, Nucl. Phys. A 275,416 (1977)
first calculation: NH, Phys. Rev. C 92,034321 (2015)

N̂1 = N̂n cos ✓ + ↵N̂p sin ✓N̂2 =N̂n sin ✓ + ↵N̂p cos ✓E(N, Z) = E(N0 , Z0 ) +[ĤHFB , N̂1 ] 6= 0

1 (N0 , Z0 ) N1 +

2 (N0 , Z0 ) N2 +[ĤHFB , N̂2 ] 6= 0

( N1 ) 2( N2 ) 2+2J1 (N0 , Z0 ) 2J2 (N0 , Z0 )( N )22( N )( Z)( Z)2
Epairrot (N, Z) =++2Jnn (N0 , Z0 ) 2Jnp (N0 , Z0 ) 2Jpp (N0 , Z0 )陽子・中性子の空間での２次元回転中性子・陽子のゲージ対称性が破れた場合
Er(Z=68)同位体、N=100同中性子体(すべてプロレート変形)

p 3つの慣性モーメント(Jnn,Jnp,Jpp)は実験データとよく一致
中性子と陽子の対回転モードの混合の証拠

線形応答理論

QRPAのもう一つの定式化
時間に依存する外場⇣F̂ (t) = ⌘ F̂ e

i!t

† i!t

+ F̂ e

⌘

F̂ =

µ<⌫

<latexit sha1_base64="ttLB0goO3Oxhqftf0VUpMEDMlz4=">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</latexit>

時間変化するHFB状態(Slater行列式)を考える↵µ† (t) =Xh

Ukµ (t)c†k + Vkµ (t)ckk<latexit sha1_base64="lyfT4OYfZjY0kefVLZ23emXY9tE=">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</latexit>↵µ (t) = {↵µ + ↵µ (t)}eiEµ t<latexit sha1_base64="zRuZw+Lzz1waYwR8Jq7m9h2aIZg=">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</latexit>

HFBでの準粒子からのずれ ↵µ (t) = ⌘TDHFB(時間依存HFB)方程式@
h (t)|Ĥ i~ | (t)i = 0
@t h

X
⌫

<latexit sha1_base64="vxx5qGLlQmIEzHPo5VsYMNfL16I=">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</latexit>

i

⇥
⇤
⇤
↵⌫† X⌫µ (!)e i!t + Y⌫µ(!)ei!t@↵µ (t)i= [Ĥ(t) + F̂ (t), ↵µ (t)]@t<latexit sha1_base64="a533fdJSEyfhDAslHN4g4nVTk10=">AAAC3XichVE9axRRFD0ZPxLXj2xME7B5uEQiyvImBCKCEBIIKfPhJgs7y/BmfJt95M0HM28X4rBlGpG0KawiWIg/wM4mTf6ARXobEW0i2Fh4Z3YgxGC8w8w799x77pvD9WKtUsP5yYh15eq166NjNyo3b92+M16duLuZRr3Elw0/0lHS9EQqtQplwyijZTNOpAg8Lbe8naW8vtWXSaqi8LnZjWU7ENuh6ihfGKLcalMxp5MIP3NikRglNHOEjrvCdYLejHk4OOPNgD1jLeZ0hclWBlRjj4bJcp48Pq9rM7da43VeBLsI7BLUUMZqVP0IBy8QwUcPASRCGMIaAik9LdjgiIlrIyMuIaSKusQAFdL2qEtShyB2h77blLVKNqQ8n5kWap9u0fQmpGSY5p/5e37Kj/kH/pX//uesrJiR/8sund5QK2N3/NXUxq//qgI6DbpnqksUHnVf7smggyeFF0Xe4oLJXfrD+f2XB6cbT9enswf8Lf9G/g75CT8ih2H/p/9uTa6/QYUWZP+9jotgc7Zu87q9NldbWCxXNYZ7uI8Z2sc8FrCCVTTo3k/4gu/4YbnWnvXa2h+2WiOlZhLnwjr4A8H4sbM=</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="oJBF53hdj4iFy83XOIHHaB5Gx8Y=">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</latexit>
sha1_base64="C2qwZB4/PWSmaYtynoTSSV8rnBQ=">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</latexit><latexit sha1_base64="/+G7N9MtOVVhJutC4eWfPNpLVnE=">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</latexit>

†

i!t

i!t

i

Ĥ(t) = Ĥ0 + ⌘ Ĥ(!)e+ Ĥ (!)e
X⇥
⇤
20
† †
02
Ĥ(!) =Hµ⌫ (!)↵µ ↵⌫ + Hµ⌫ (!)↵⌫ ↵µ<latexit sha1_base64="Sso7M+goAQOHPyq5jtrg21xCp4A=">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</latexit>µ<⌫ηの１次の項のみ取り出す<latexit sha1_base64="+pDxKmOjli/VDE8N37MiBYdZjio=">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</latexit>

線形応答の方程式(QRPAと同値)
✓
◆
A
B⇤

B
A⇤

✓

1
!
0

◆ ✓
◆
0
X(!)
=
1
Y (!)

✓

F 20
F 02

◆

X(ω)、Y(ω)はQRPA解ではない
QRPAはTDHFBの小振幅近似(η<<1)として得られる<latexit sha1_base64="M/LaW5ZkDaCYhHokTxz1mwPyhqQ=">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</latexit>

20 † †
02
Fµ⌫↵µ ↵⌫ + Fµ⌫↵⌫ ↵µ

Fは例えばアイソベクトル双極演算子、四重極演算子など
ηは小さい数<latexit sha1_base64="PXZhvyn/+PFkigEUkDKgOtCbvF4=">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</latexit>

準粒子が時間変化

X線形応答計算例
初期に外場を与え Vext (r, t) = ⌘F (r) (t)TDHFB方程式に従って時間発展<latexit sha1_base64="JpGMX/IMzlUg689Ndo2fCVnfUew=">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</latexit>

Canonical-basis TDHFB
16O isoscalar octupole mode

Ebata et al., Phys. Rev. C 82, 034306 (2010)

Nakatsukasa and Yabana, Phys. Rev. C 71, 024301 (2005)

有限振幅法
QRPA：AB行列の計算が大変
✓

A
B⇤

B
A⇤

<latexit sha1_base64="M/LaW5ZkDaCYhHokTxz1mwPyhqQ=">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</latexit>◆✓

Nakatsukasa, Inakura, Yabana, Phys. Rev. C 76, 024318 (2007)
Avogadro and Nakatsukasa, Phys. Rev. C 84, 014314 (2011)

◆ ✓
◆
0
X(!)
=
1
Y (!)

✓

20
!)Xµ⌫ (!) + Hµ⌫
(!) =

20
Fµ⌫

02
Hµ⌫
(!) =

02
Fµ⌫

1
!
0

F 20
F 02

◆

線形応答の方程式を反復法で解く(Eµ + E⌫

準粒子の変位X(!), Y (!)<latexit sha1_base64="DyFF4vJ9IwRmdK3AeokQvQDzHh4=">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</latexit>(Eµ + E⌫ + !)Yµ⌫ (!) +<latexit sha1_base64="VpyNBBWtw0UjorALuRvyrcC5HJ0=">AAADSnichVFPS9xQEJ9Erbq1dbWXgpdHF2XL4jJZBEtBEEvFo6tdXd3YkMTnGsw/kpcFG/YL+AU89KTgQfwAfgAReui1Bz9C6a0WSmkPTrLpH13cTniZeb+Z38z7MYZvW6FAvJLkvv6BB4NDw7mHI48ej+bHxtdCLwpMXjM92wvqhh5y23J5TVjC5nU/4Lpj2Hzd2HuV5NdbPAgtz30j9n2+5ehN19qxTF0QpOUvi+y1pjoRKyXejdg0Uz2HN/XnrK7FlCCsXfwNlZi6zW2hs6W3cQXb3QVzbHrxdoqpau7ujNKfGRu9Z2Dl3hn/ppiWL2AZU2PdgZIFBchs2cufgwrb4IEJETjAwQVBsQ06hPQ1QAEEn7AtiAkLKLLSPIc25IgbURWnCp3QPfo36dbIUJfuSc8wZZs0xaYTEJPBJH7CU7zGD3iGn/HXvb3itEfyln3yRofLfW304Onq9/+yHPICdv+yejAMqu6tScAOvEi1WKTNT5FEpdnp33p3eL36cmUynsJj/EL6jvAKL0ih2/pmnlT5ynvI0YKUu+voDtYqZQXLSnWmML+QrWoIJuAZFGkfszAPS7AMNTClklSVNqWG/FH+Kv+Qf3ZKZSnjPIFb1td/A8vk1C4=</latexit>

H 20 (!), H 02 (!)(通常の線形応答の計算)
20
Hµ⌫
(!) =

X

平均場Hamiltonianの変位(一体場)<latexit sha1_base64="gNKvn/nBpiApznqqkn2qNbB+sKg=">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</latexit>[Aµ⌫µ0 ⌫ 0(Eµ + E⌫ ) µµ0 ⌫⌫ 0 ]Xµ0 ⌫ 0 (!) + Bµ⌫µ0 ⌫ 0 Yµ0 ⌫ 0 (!)[A⇤µ⌫µ0 ⌫ 0⇤(Eµ + E⌫ ) µµ0 ⌫⌫ 0 ]Yµ0 ⌫ 0 (!) + Bµ⌫µ0 ⌫ 0 Xµ0 ⌫ 0 (!)µ0 <⌫ 0
02
Hµ⌫
(!) =
<latexit sha1_base64="Mw+bVvd0zjRo7stD3B6+RZ6xCKc=">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</latexit>Xµ0 <⌫ 0一体密度の変位を計算⇢(!) =U X(!)V T + V ⇤ Y T (!)U †一体平均ポテンシャルの変位を計算

h(!) =(h[⇢ + ⌘ ⇢]

(+) (!) =U X(!)U T + V ⇤ Y T (!)V †( ) (!) =V ⇤ X † (!)V † + U Y ⇤ (!)U T

Bogoliubov変換h[⇢])/⌘<latexit sha1_base64="jRU3zoaCdfbfCkIAIBUr1CKKZQQ=">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</latexit>(±)(!) =( [⇢ + ⌘ ⇢,  + ⌘ (±) , ⇤ + ⌘ (⌥)⇤ ]<latexit sha1_base64="qrJsRSM+FM4sWuC+jQdTPI/RRpY=">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</latexit><latexit sha1_base64="+x16cnzOLrsFdMLtgYkS1PzPfMI=">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</latexit>

DFTではAB行列が出てくる二体行列要素の計算を回避することができる[⇢, , ⇤ ])/⌘

有限振幅法の計算例
双極子巨大共鳴Oishi et al., Phys. Rev. C 93, 034329 (2016)

四重極巨大共鳴(非軸対称原子核)

Washiyama and Nakatsukasa, Phys. Rev. C 96, 014304(R) (2017)

行列対角化によるQRPA計算：O(N6) (Nは準粒子の次元~103程度)
有限振幅法によるQRPA計算：O(N2)の反復計算

pnQRPA
[Ĥ 0 , Ô† ] = ⌦ Ô†[Ĥ 0 , Ô ] =<latexit sha1_base64="s2SpvMS8HkgkogGA78bfL+2AqzM=">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</latexit><latexit sha1_base64="m3NuC9LENUVclSafRMzuTCzQNAc=">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</latexit>⌦ Ôフォノン演算子として中性子・陽子準粒子対の形を考える
X
†
(n)
Ô =
Xµ⌫ ↵µ(n)† ↵⌫(p)† Yµ⌫ a(p)
⌫ aµ
µ<⌫

<latexit sha1_base64="Vt2RBIzYL8EisSfd9mRx76+M7bU=">AAADHHichVFda9RAFL2JX+360a2+FHwJLpUKdrkpQkUUSn3xzX647UqnGybZ6W7oZBKSyUIN+QP9A30QBAUfxB/gD9AHQXxU6E8QHyv4ouDNhyxarBOSuXPuOWdyuG4k/UQjHhrmqdNnzp6bmGycv3Dx0lRz+vJGEqaxJzpeKMO46/JESF+Jjva1FN0oFjxwpdh0d+8X/c2RiBM/VI/0XiS2Az5Q/o7vcU2Q0zxgQ64z5nFpPcx7rM8HDpMk73PrnsWSNHAyFqR3mUpzq9vL6l5eoiXIuIyGvJfNqRuFOneoMQaj36BKrXnrca/Wj+UVqSJULpWD02xhG8tlHS/sumhBvVbC5htg0IcQPEghAAEKNNUSOCT0bIENCBFh25ARFlPll30BOTRImxJLEIMTukvfAZ22alTRufBMSrVHt0h6Y1JaMIuf8BUe4Xt8jV/wxz+9stKj+Jc92t1KKyJnan9m/ft/VQHtGoZj1QkKl9gnZ9KwA7fLLD5li0qkSOlV/qMnB0frd9Zms+v4Ar9Svud4iG8poRp9816uirWn0KAB2X+P43ixsdC2sW2v3motLdejmoCrcA3maB6LsAQPYAU6dO9Po2XcNObNZ+Y784P5saKaRq25An8s8/Mvj5XKaw==</latexit>

pnQRPA励起状態：基底状態が(N,Z)なら励起状態は(N 1, Z 1)
ベータ崩壊、ガモフ＝テラー巨大共鳴、二重ベータ崩壊など
残留相互作用：中性子ー陽子相互作用のみ
原子核密度汎関数法の場合は中性子ー陽子混合密度の導入が必要
対相関：中性子ー陽子対相関
アイソベクトル型(T=1, S=0)
アイソスカラー型(T=0, S=1)
アイソベクトル型はアイソスピン対称性が使える(Vn=Vp=Vnp)
アイソスカラー型の対相関については(対凝縮の存在も含めて)よくわかっていない
最近のレビュー：Frauendorf and Macchiavelli Prog. Part. Nucl. Phys. 78, 24 (2014)

pnQRPA計算例
β崩壊の半減期

ガモフ＝テラー共鳴

Yoshida, PTEP 2013, 113D02

Zr

Mustonen et al., Phys. Rev. C 90, 024308 (2014)

まとめ
p 励起状態の記述
p 平均場の集団励起は残留相互作用によっておこる
p 集団励起を扱う理論
p Tamm-Dancoff近似
p RPA, QRPA(準粒子乱雑位相近似)
p 1p1hまたは二準粒子励起の重ね合わせで集団励起を記述
p 遷移強度と巨大共鳴・和則
p QRPAとHFB解の安定性
p 対称性の破れとQRPA
p 平均場近似で破れた対称性はQRPAレベルで回復
p QRPAにゼロエネルギー解が発生(南部・ゴールドストーンモード)
p 回転対称性の破れと集団回転運動、慣性モーメントp ゲージ対称性の破れと対回転、二重束縛エネルギー差
p 線形応答理論・有限振幅法
p 時間依存平均場理論の小振幅近似としてのQRPA
p 行列対角化より効率的
p pnQRPA (β崩壊、ガモフテラー遷移)