原子核集団現象への平均場アプローチ
第２回：平均場理論

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. 7, 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

平均場理論とは
原子核はどんな形をしているか→平均ポテンシャルの形
平均ポテンシャルが核子の一粒子軌道を決める(調和振動子・Nilsson軌道など)
平均ポテンシャルはどのようにして決まるのか？
核子の配位(一粒子軌道)、密度で
平均ポテンシャルは決まる
平均ポテンシャル Γ(r)自己無撞着性(self-consistency)
核子の配位(一粒子軌道)・密度

平均ポテンシャル

原子核の有効相互作用(ハミルトニアン）

Ĥ =

X

1
†
tij ci cj +ij<latexit sha1_base64="OQiftS4B8ua1HtPu69ZCTpyHPi0=">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</latexit>

一体部分(運動エネルギー)

4Xv̄ijkl c†i c†j cl ck

ijkl

二体相互作用第二量子化による記法

第二量子化
ci , c†i 既知の一粒子軌道iの粒子の生成・消滅演算子<latexit sha1_base64="IZwIOw8SwyrD5USeI4Shp2pYito=">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</latexit>(調和振動子の一粒子軌道と思ってよい)

反交換関係をみたす(フェルミ粒子){ci , c†j } ⌘ ci c†j + c†j ci =ij<latexit sha1_base64="+kK1UczMRDi1Gvzy5d73ey9nUc8=">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</latexit>

一体場のみで対角的なHamiltonianX †Ĥ =ei ci ci

一粒子軌道iのエネルギーはei

一粒子軌道kに粒子が入った状態は一体場ハミルトニアンの固有状態|ki = c†k |0i<latexit sha1_base64="311s6PvWiQm1KV40UzR3NPEUAU8=">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</latexit>

原子核はフェルミ粒子の多体系
一粒子軌道cのエネルギー<latexit sha1_base64="OgiBiC4zGdtnnoVUK082SPJE0Vk=">AAACrnichVHBShtRFD2OrdW01bTdCG4GQ0oLJdyRglIoSN241MREIROHmeczPjOZGWYmgTj4A9106aKrCi7EdemuXXTjD7jIJ5QuU3DjojeTAVGpvcObd96599z3DtcJXBXFRP0xbfzBw4lHk1O5x0+eTs/knz2vRX4nFLIqfNcPtxw7kq7yZDVWsSu3glDabceVm05rZZjf7MowUr63EfcC2WjbTU/tKmHHTFn512aiC0u9Eda+eai/181EbJs7dpMpPUOjBFn5ApUoDf0uMDJQQBZrfv4bTOzAh0AHbUh4iBm7sBHxV4cBQsBcAwlzISOV5iUOkWNth6skV9jMtvjf5FM9Yz0+D3tGqVrwLS6vkJU6inRBpzSgczqjX3T1z15J2mP4lh7vzkgrA2vm42zl8r+qNu8x9q5V9ygcrr7fU4xdLKVeFHsLUmboUoz6dw+OBpV35WLyko7pN/v7Qn36yQ697h9xsi7Ln5HjARm3x3EX1BZKBpWM9beF5Q/ZqCYxh3m84nksYhmrWEOV7/2Er/iOHxppNa2hWaNSbSzTvMCN0Pb+Ajm5ntQ=</latexit>i<latexit sha1_base64="m3+RxpYY8RnjouHFTZMCQyfO0sw=">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</latexit>

一体系の場合{ci , cj } = {c†i , c†j } = 0

Ĥ|ki = ek |ki<latexit sha1_base64="+dY9O/OLGrXUPNO3GS5D5N0YlEc=">AAACpHichVFNLwNRFD3Gd30VG4mFRkOsmjsiIRKJsOnColRp0tLMjKcmnc5MZl6bUJY2/oCFFYmF2BI/wMYfsPATxJLExsLtdEIQ3Mmbd+5599z3Tq7uWqYviR6alOaW1rb2js5IV3dPb1+0f2DNdyqeITKGYzleVtd8YZm2yEhTWiLrekIr65ZY10uL9fP1qvB807FX5a4rNspa0Ta3TUOTTBWiI/kdTdaSB/ulvKfZRUvE5mKiUPpIC9E4JSiI2E+ghiCOMFJO9AZ5bMGBgQrKELAhGVvQ4POXgwqCy9wGasx5jMzgXOAAEdZWuEpwhcZsif9FznIha3Ne7+kHaoNvsXh5rIxhjO7pgp7pji7pkd5+7VULetTfssu73tAKt9B3NJR+/VdV5l1i51P1h0Ln6r89SWxjJvBisjc3YOoujUb/6t7xc3p2Zaw2Tmf0xP5O6YFu2aFdfTHOl8XKCSI8IPX7OH6CtcmESgl1eSo+vxCOqgPDGMUEz2Ma80gihQzfe4hLXOFaGVeWlLSSaZQqTaFmEF9C2XwH6kyc0Q==</latexit>

p 二粒子は同じ状態を取れない
p 粒子の入れ替えで波動関数の符号が変わる
AY
c†i |0i
Slater行列式 |SDi =i=1

一粒子軌道1からAまでに粒子が入っている状態|SD>は一体場ハミルトニアンの固有状態!
AX
固有エネルギーは
Ĥ|SDi =ei |SDi<latexit sha1_base64="vinHCHIngXz8p41S7djZvVJusds=">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</latexit>

一粒子軌道i=1,2<latexit sha1_base64="5jI1aJeXrFWu/CcazkpA3/DLnFE=">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</latexit>

i=1

第二量子化
非対角項がある一体場の場合 Ĥ =X

hij c†i cj

ij<latexit sha1_base64="G8mXUogxP2WS/HDMdTGqaoPwW0w=">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</latexit>|ki = c†k |0i 一粒子軌道kに粒子が入った状態は固有状態ではない一体系の場合<latexit sha1_base64="311s6PvWiQm1KV40UzR3NPEUAU8=">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</latexit>

線形結合をとって新しい一粒子軌道を考える(Dはユニタリ行列)
a†k =

X

Dlk c†l{ai , a†j } =<latexit sha1_base64="MO8zbe6OBAYOwKU94BuawH6KPSA=">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</latexit>lij{ai , aj } = {a†i , a†j } = 0<latexit sha1_base64="WxNYky55V0fhJYIe4hL4/hw02PA=">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</latexit>

Hamiltonianは軌道aについて対角的になるようにDを決める
X
X †
†
Ĥ =
hij ci cj =ei ai ai<latexit sha1_base64="zfNB8NkJ1Gz50zuJe+plmepl8w8=">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</latexit>

Slater行列式|SDi =

一粒子軌道aのエネルギー<latexit sha1_base64="nc1tzx2iTM9o9TMhMOBo3GE8wXc=">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</latexit>

A
Y

i=1

ij

i

a†i |0i ユニタリ行列Dは行列hを対角化することで得られる<latexit sha1_base64="1fKIF/Z0hji5le48gPnweFZ1RkY=">AAACyHichVFNTxNRFD2MorWiVN2YuJnYQFg1d4iJhoSE2A1xxYcFEgqTN49H+2C+MvNaUyfduPQPuHCliQviD/AHsGHnyoT+BOISEzcuvJ2OIUDAO5m55517z31zcr3Y16khGoxZN26O37pdulO+O3Hv/mTlwcO1NOokUjVk5EfJhidS5etQNYw2vtqIEyUCz1fr3n59WF/vqiTVUfja9GK1FYhWqHe1FIYpt1JvtoXJFvv2vN1MO4Gb6b2+3R4lud3cES1X29LdO6v3bcWU+FcTrnYrVapRHvZl4BSgiiKWoso3NLGDCBIdBFAIYRj7EEj52YQDQszcFjLmEkY6ryv0UWZth7sUdwhm9/nb4tNmwYZ8Hs5Mc7XkW3x+E1bamKIfdECndERf6YT+XDkry2cM/6XH2RtpVexOvn+8+vu/qoCzQftMdY3C4+7rPRns4kXuRbO3OGeGLuVofvfth9PVuZWpbJo+00/294kGdMgOw+4v+WVZrXxEmRfkXFzHZbA2W3Oo5iw/qy68LFZVwhM8xQzv4zkWsIglNPjeA3zHMQbWKyu23li9Uas1Vmge4VxY7/4CYS+qTw==</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="3mTgkcWFJnOrsOHdlHAR8R3MzyQ=">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</latexit>
sha1_base64="LhRxK1z6IVoYVTFugBXljVWfX/A=">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</latexit>|SD>は一体場ハミルトニアンの固有状態
固有エネルギーは

Ĥ|SDi =<latexit sha1_base64="5jI1aJeXrFWu/CcazkpA3/DLnFE=">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</latexit>

A
X
i=1

ei

!

|SDi

一粒子軌道i=1,2、ただしcではなくaHartree-Fock
原子核の有効相互作用(Hamiltonian）
X
1X
†
Ĥ =
tij ci cj +v̄ijkl c†i c†j cl ck
4
ij
ijkl
<latexit sha1_base64="OQiftS4B8ua1HtPu69ZCTpyHPi0=">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</latexit>

Slater行列式を使って一体場で近似

ĤHF =
新しい一粒子軌道X†ak =Dlk c†l<latexit sha1_base64="lrYXWKX3AFK0r+9jeYhHDmkOPNA=">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</latexit>l<latexit sha1_base64="zfNB8NkJ1Gz50zuJe+plmepl8w8=">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</latexit>

X
ij

|

HF i =<latexit sha1_base64="GSc6q1XEhVPiMjdMaJDYu6jAkx4=">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</latexit>

HF エネルギー

ĤHF |<latexit sha1_base64="UiT15Sq1sm6uxfJ23kOFPX+ljnY=">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</latexit>

HF i =

Slater行列式を使って二体相互作用部分を計算

hij c†i cj

A
X
i=1

ei

!

A
Y

i=1

|

a†i |0i

HF i

Hartree-Fock近似：波動関数をSlater行列式の形に限定することが近似
二体相互作用の部分が一体場の平均ポテンシャルを与える

二体相互作用の計算|<latexit sha1_base64="GSc6q1XEhVPiMjdMaJDYu6jAkx4=">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</latexit>

HF i =

AYi=1

a†i |0ih

HF |Ĥ|Hartree-Fock状態(Slater行列式)
HF i =<latexit sha1_base64="JGZJ/Jhqd/qsgxyMxTwBoItqlP0=">AAADdXichVHNahRBEK7Z8SeOP1n1IojQuCYIwtojAUUQgoLscZO4m0AmDj2d3t3e7fmhZ3Yhzs5RD76AB08KHsQH8AG8+AIe8gjiRYiQiwdrfiCsWWMNPVX1dX1fdVFepGScULpv1MxTp8+cXThnnb9w8dJi/fKVbhyONRcdHqpQb3ksFkoGopPIRImtSAvme0pseqMn+f3mROhYhsGzZC8SOz7rB7InOUsQcus/HcWCvhLEaQ+kmzraJ62n2ZQ4A5akrWw6Azu6qLUeEcuJx76bymFGktJVOrMy/Lmzy/quJNwdzpUi1h3i9DTjqZ2lK6hSyY5UZjke0+kkq1JsOa8DOWpRBkPspfCMyPy3u/UGbdLCyPHAroIGVNYO65/BgV0IgcMYfBAQQIKxAgYxfttgA4UIsR1IEdMYyeJeQAYWcsdYJbCCITrCfx+z7QoNMM8144LNsYvCo5FJYIl+ox/pAf1KP9Hv9Pc/tdJCI3/LHnqv5IrIXXx9bePwvywffQKDI9YJDA+rT54pgR48KGaROFtUIPmUvNSfvHhzsPFwfSldpu/pD5zvHd2nX3DCYPKLf1gT628hX5D99zqOB917TZs27bWVxurjalULcB1uwm3cx31YhRa0oQPc6BpT46XxqnZo3jBvmctlac2oOFdhxsy7fwAZYejY</latexit>

X
ij

tij h

†
HF |ci cj |１体密度行列 ⇢ij = h HF |cj ci | HF i =†<latexit sha1_base64="9BOZQS0E14OL4xqXkQGDSUIGVCk=">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</latexit>h

HF |Ĥ|HF i =<latexit sha1_base64="ZTpG7sqVpjqCbG3IomK4HNDnOqk=">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</latexit>

X

AX

HF i +

ij

1X
v̄ijkl (⇢ki ⇢lj
4
ijkl

ik =

X

v̄ijkl ⇢lj

1X
h HF |Ĥ| HF i =tij ⇢ji +2ij<latexit sha1_base64="G5moFxj11ET5D132VOv7lJ0b0+U=">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</latexit><latexit sha1_base64="lrYXWKX3AFK0r+9jeYhHDmkOPNA=">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</latexit>ik ⇢ki

ik@EHF [⇢]= tij +@⇢ji

ĤHF =⇢kj ⇢li )

jl

X

hij =

HF i

Wickの定理⇤
Dik Djk<latexit sha1_base64="IEIlsJ3bx2FiGruw3XPhGQsciOk=">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</latexit><latexit sha1_base64="LL/ZD2a929cZinlYzku714f7RXU=">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</latexit>

ijkl

† †
HF |ci cj cl ck |i=1

tij ⇢ji +Hartree-Fock ポテンシャル

1X
v̄ijkl h
4X

ij [⇢]

平均ポテンシャル

hij c†i cj

ij

HFエネルギー、HFポテンシャルは密度行列の汎関数

Hartree-Fock法のアルゴリズム
インプット：有効相互作用、一粒子基底(調和振動子基底、座標基底など)
Slater行列式(ユニタリ行列D)の初期値密度行列を計算

⇢ij =

A
X

⇤
Dik Djki=1<latexit sha1_base64="cRYo41BZVzbTEYlTvZim2VkNENM=">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</latexit>

Hartree-Fock ポテンシャルを計算
ik =<latexit sha1_base64="IEIlsJ3bx2FiGruw3XPhGQsciOk=">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</latexit>Xv̄ijkl ⇢ljjl

Hartree-Fock Hamiltonianを対角化@EHF [⇢]hij == tij + ij [⇢]@⇢ji<latexit sha1_base64="G5moFxj11ET5D132VOv7lJ0b0+U=">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</latexit>

X<latexit sha1_base64="mKZa2bYKooJIlmNFptnCrIBqF5E=">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</latexit>

hij Djk = "k Dik

ユニタリ行列Dを対角化より得る

jp Hartree-Fockは自己無撞着(self-consistent)な方程式(反復法で解く)
p ベストな(最もエネルギーの低い)Slater行列式(あるいは一粒子軌道)を決定
p 初期値依存あり

座標表示Hartree-Fock方程式
X

Hartree-Fock方程式<latexit sha1_base64="mKZa2bYKooJIlmNFptnCrIBqF5E=">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</latexit>X

hij Djk = "k Dikj<latexit sha1_base64="UQKmGbDownRX8KdJIkBcaaVj5Mo=">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</latexit>†
座標基底： cr |0i = |ri{cr , c†r0 } = (r⇤v̄injl Dlm Dnm

ln m=1jr0 )<latexit sha1_base64="H+nkPrMqtSJ69vCD5Kcyt15OyWA=">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</latexit><latexit sha1_base64="fLEhfrPe/Kg2FytTKX1jOkdqFG4=">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</latexit>

tij +

A
XX

!

Djk = "k Dik{cr , cr0 } = {c†r , c†r0 } = 0<latexit sha1_base64="pkXDrOmgEVxJTvp83ZZtQOzUO0M=">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</latexit>

座標空間でのHartree-Fock方程式(簡単のためスピン・アイソスピンの自由度を無視)<latexit sha1_base64="RCzZEkYgxQtT8LOSfcfybn0WOPU=">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</latexit>

A Z
X
~2
'k (r) +dr 0 v(r, r 0 )'⇤j (r 0 )['j (r 0 )'k (r)2m
j=1

HartreeポテンシャルZ

dr 0 v(r, r 0 )H (r) =

j=1

Fockポテンシャル<latexit sha1_base64="F/qan/5NH/hYt2RoC63s4NXmlLE=">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</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="QZCd4EtLmYVSgR0MP3nxZhrHaF8=">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</latexit>
sha1_base64="WZgum3e0TvBiTF9uKymOnzQicys=">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</latexit>0F (r, r ) =v(r, r 0 )Hartree-Fock方程式<latexit sha1_base64="a5mjw25BxUE84bgVlwrU+lieqMM=">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</latexit>

~
2m
2

+

A
X

|'j (r 0 )|2 ='⇤j (r 0 )'j (r) =Z

dr 0 v(r, r 0 )⇢(r 0 )v(r, r 0 )⇢(r, r 0 )j=1<latexit sha1_base64="xcXPiBQEKbq3IpkxLkK6taG0YYU=">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</latexit>✓AX'j (r)'k (r 0 )] = "k 'k (r)

H (r)

◆

'k (r) +Z

dr 0 F (r, r 0 )'k (r 0 ) = "k 'k (r)有効相互作用
簡単な相互作用：四重極力

原子核で最も重要な相関は四重極相関(変形)。これをモデル化
2
X

ĤQQ =

2 µ= 2<latexit sha1_base64="WI2/ltihm/fG4VLlwNo6IINuB2g=">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</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="Slu2loPpvcq2+gzaXi7mW2gxSrs=">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</latexit>
sha1_base64="nvdbLT42EoSWbCSJYsRInh2wSYc=">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</latexit>Q̂†2µ Q̂2µ

Q̂2µ =

X
ij<latexit sha1_base64="PxP2vtmJmXcui2KoeV6w20L9Ub0=">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</latexit>

現実的なものでよく使われているもの

hi|r2 Y2µ |jic†i cj c は調和振動子基底ip Skyrme型有効相互作用(ゼロレンジ)
p Gogny型有効相互作用(有限レンジ)p 相対論的平均場理論
特に原子核を記述するうえで不可欠な項は・スピンー軌道力：魔法数を再現するのに不可欠・密度依存力：状態方程式を出すのに不可欠
有効相互作用のパラメターは原子核・核物質の性質に合わせて調整
Gogny力r12 ⌘ r1
<latexit sha1_base64="WzJRgTLzT7WO+5juu1kcQeC20bY=">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</latexit>P̂ =<latexit sha1_base64="Qru3EEG9OqIH7znmPtKkA11Vv3s=">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</latexit>

1
(1 +
2

パラメター

v̂Gogny (r12 ) =r2<latexit sha1_base64="cQywGq6kAiGm5cMKW86d1nY5nN0=">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</latexit>

1·

2)

P̂⌧ =

r2 )

1
(1 + ⌧1 · ⌧2 )2<latexit sha1_base64="QlNyLS16rrG28bjnUiPrHBeFGVo=">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</latexit><latexit sha1_base64="nSfeau6jhhqBG8VtDdKBoAB0QrQ=">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</latexit>2

e (r12 /µj ) (Wj + Bj P̂

Hj P̂⌧

Mj P̂ P̂⌧ )

i=1

i
(r1
2

k̂ =

2
X

r1 + r2)
2+ iWls ( 1 + 2 ) · k̂† ⇥ (r12 )k̂+ t3 (1 + x0 P̂ ) (r12 )⇢↵ (D1S (Decharge and Gogny, 1980), D1M (Goriely, et al., 2009)など

有効相互作用Skyrme力

スピンー軌道力

密度依存項
テンソル力

r12 ⌘ r1
<latexit sha1_base64="WzJRgTLzT7WO+5juu1kcQeC20bY=">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</latexit>P̂ =<latexit sha1_base64="Qru3EEG9OqIH7znmPtKkA11Vv3s=">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</latexit>

r2

1
(1 +
2

パラメター

k̂ =
<latexit sha1_base64="cQywGq6kAiGm5cMKW86d1nY5nN0=">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</latexit>

1·

2)

P̂⌧ =
<latexit sha1_base64="nSfeau6jhhqBG8VtDdKBoAB0QrQ=">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</latexit>

i
(r1
2

r2 )

1
(1 + ⌧1 · ⌧2 )2

SIII (Beiner et al., 1975)
SkM* (Bartel et al., 1982)
SkP (Dobaczewski et al., 1984)
SLy4 (Chabanat et al., 1998)
SVmin(Klupfel et al., 2009)
UNEDF (Kortelainen et al., 2010-)

Hartree-Fockと対称性
対称性：Hamiltonian演算子と交換する演算子で表される[Ĥ, Ŝ] = 0<latexit sha1_base64="w1xACCZrFZkckv5l8mIivHw4teA=">AAACm3ichVHLSsNAFD3Gd31VBRFECBbFhZQbERRBEN2IuKjWqlBLSeJUg2kSkmlBa3/AH3DhSsGF1L0f4MYfcOEniEsFNy68TQOiot5hZs6cuefOHK7h2VYgiR6blOaW1rb2js5YV3dPb1+8f2ArcEu+KTKma7v+jqEHwrYckZGWtMWO5wu9aNhi2zhcrt9vl4UfWK6zKY88kSvq+45VsExdMpWPD2V3D3RZWalOqSFIV3Pqgkr5eIKSFIb6E2gRSCCKlBu/xS724MJECUUIOJCMbegIeGShgeAxl0OFOZ+RFd4LVBFjbYmzBGfozB7yus+nbMQ6fK7XDEK1ya/YPH1WqhinB7qmF7qnGj3R+6+1KmGN+l+OeDcaWuHl+06H02//qoq8Sxx8qv5QGJz9tyeJAuZCLxZ780Km7tJs1C8fn72k5zfGKxN0Sc/s74Ie6Y4dOuVX82pdbJwjxg3SvrfjJ9iaTmqU1NZnEotLUas6MIIxTHI/ZrGIFaSQ4XdPcIkabpRRZVlZVdYaqUpTpBnEl1AyH4THl+g=</latexit>

多体問題の厳密解では対称性は保たれる(エネルギーと対称性の同時固有状態)
Ĥ =

X

1
tij c†i cj +
4

ij

X

v̄ijkl c†i c†j cl ck

ijkl

Hartree-Fock(平均場近似)：二体相互作用部分をSlater行列式を使って近似<latexit sha1_base64="OQiftS4B8ua1HtPu69ZCTpyHPi0=">AAAC8HichVHLahRBFL3dvuL4yEQ3gpvCISIIw+0QUIRA0E2WeThJIB2b6krNpKarH1TXDMSmf8AfUHAhEVyoH+AHuHHrwkU+QVxGyMaFd7pbgwZjNVX33FP33OrDDTOtcot44Lhnzp47f2HqYuvS5StXp9sz19bzdGSE7IlUp2Yz5LnUKpE9q6yWm5mRPA613AijR5P7jbE0uUqTx3Yvk9sxHySqrwS3RAXtyN/ltlgq2QLz81EcFGpYMlsH8cTf4YNAMREM2V3G/L7hovDKYr78XRxpwiE3xbj8lR7LajAkvaYdBe0OdrFa7CTwGtCBZi2n7Q/gww6kIGAEMUhIwBLWwCGnbws8QMiI24aCOENIVfcSSmiRdkRVkio4sRGdA8q2GjahfNIzr9SCXtG0DSkZzOIXfIuH+Anf41f88c9eRdVj8i97FMNaK7Ng+tmNtaP/qmKKFnaPVacoQqo+3ZOFPtyvvCjyllXMxKWo+4+fPj9ce7A6W9zG1/iN/O3jAX4kh8n4u3izIldfQosG5P09jpNgfa7rYddbme8sPmxGNQU34RbcoXncg0VYgmXo0buf4cgBx3GN+8J95e7Xpa7TaK7DH8t99xMnFLkC</latexit>

Slater行列式に厳密解が持つべき対称性を課すべきか？

原子核の有効相互作用ではJ=2の四重極相関が重要(引力)2†Q̂ Q̂ ⇠2X

r2 Y2 (ij)r2 Y2 (kl)[[c†i cj ]J=2 [c†k cl ]J=2 ]J=0

ijkl

Slater行列式に回転対称性を課すとSlater行列式は角運動量の固有状態
Hartree項J=2の行列要素はゼロになる hJ = 0|[c† cj ]J=2 |J = 0i
<latexit sha1_base64="b/b18V+Ib+0LcJYmUBwinXPm4Ks=">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</latexit><latexit sha1_base64="07/RgrCECrHc8KBRDlWs9KkmS0g=">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</latexit>

i

対称性を破ると(この場合は回転対称性を破って変形すると)相関がとりこめる|<latexit sha1_base64="5dSC0s206ehCSeiYuRRYwLDnjPs=">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</latexit>

def.HF i = c(0)|J = 0i + c(2)|J = 2i + c(4)|J = 4i + · · ·h<latexit sha1_base64="spd5Og7u2aVn7OwRqUlS2lWMrmI=">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</latexit>†
J=2|
def.HF |[ci cj ]6 0
def.HF i =Slater行列式の範囲内(平均場近似)で四重極相関を記述するためには
回転対称性を破ることが本質的

Hartree-Fock計算で破れる対称性
四重極変形(回転対称性)

八重極変形(回転対称性＋パリティ対称性)

HF+BCS計算： Ebata and Nakatsukasa, Phys. Scr. 92 (2017) 064005

拘束付きHartree-Fock

Hartree-Fock: ある系(原子核)で最低エネルギーのSlater行列式(平均場)を求める

Slater行列式のエネルギー

変形度を変えたときのエネルギーの振る舞いは？ (E(β))
線形拘束(linear constraint)Ĥ ! ĤQ̂

<latexit sha1_base64="aFcU7daR8rnIKcna4iu5nCkZ9pU=">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</latexit>

λ：Lagrange未定定数
Q: 拘束する変形などの演算子(四重極変形Q20など)
拘束条件 h CHF |Q̂20 | CHF i = q<latexit sha1_base64="tkWSHCfloizfG7/W5Z4xtEqozus=">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</latexit>を満たすようにλは反復ごとに決定

HF状態
q

解では拘束条件が満たされる。
収束させるのが難しいことがある

Q20

線形拘束で軸を傾ける

二次拘束(quadratic constraint)1
Ĥ ! Ĥ + C(hQ̂i2q)2qを中心として二次の拘束ポテンシャルを追加
収束した状態の変形度はqから少しずれる<latexit sha1_base64="Zk6DxKzzsjqPdh5/gzgWhwJllyo=">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</latexit>q

Q20

Augmented Lagrange method

線形拘束と二次拘束を組み合わせたもの
Staszczak et al., Eur. Phys. J. A 46, 85 (2010)拘束付きHartree-Fock計算例

quadratic constrained Skyrme HF計算
Inakura, et al., Nucl. Phys. A 710 (2002) 261.

quadratic/Augmented Lagrange method (HFB)計算
Staszczak et al., Eur. Phys. J. A 46, 85 (2010)

Hartree-Fock状態と真空
Hartree-Fock状態|

HF i =<latexit sha1_base64="xQ7Vyn9/Zj7feri6aBInjJQhv60=">AAACvnichVHNStxQGD1G29rpj9O6KXQTOlhcDV9KQVsQRgvictSOCkbDTebOeDF/3GSGapwX8AW6cFFa6KL0AfoAIrjSVRc+QulSwU0X/SYTKFZqv5Dcc88957s5fG7sqyQlOhsyhkdu3b4zerd07/6Dh2PlR49XkqijPdnwIj/Sa65IpK9C2UhV6su1WEsRuL5cdbff9M9Xu1InKgrfpjux3AhEO1Qt5YmUKaf8as+ubykns3VgLsz3bC3Cti/NGdOOddR0MjVj9TZnTbFpN0XbUeaeSYXGKVeoSnmZ14FVgAqKqkflb7DRRAQPHQSQCJEy9iGQ8LMOC4SYuQ1kzGlGKj+X6KHE3g6rJCsEs9v8bfNuvWBD3vd7Jrnb41t8fjU7TUzQd/pC53RMX+kH/fpnryzv0f+XHV7dgVfGztj+k+XL/7oCXlNs/XHd4HBZfXOmFC1M51kUZ4tzpp/SG/Tv7r4/X369NJE9p0/0k/N9pDM65IRh98L7vCiXDlDiAVl/j+M6WHlRtahqLb6s1OaKUY3iKZ5hkucxhRoWUEeD7/2AI5zg1KgZLSMwooHUGCo847hSxrvfFvCmMQ==</latexit>

A
Y

i=1

a†i |0i Hartree-Fock軌道にA個粒子が詰まった状態

一粒子軌道aのエネルギー

粒子・空孔演算子

a†i =

粒子空間

(

d†i
bi

<latexit sha1_base64="i26ETsfbW+AI0TmyxZ7BAgQIup4=">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</latexit>

空孔空間

d: 粒子演算子。粒子空間に粒子を生成
b: 空孔演算子。空孔空間に空孔を生成

Hartree-Fock状態は粒子・空孔演算子に関して真空
真空：消滅演算子を作用させたときにゼロとなる状態

di |<latexit sha1_base64="sz2zLoKeVVxgfpbeZuXEnswuDnQ=">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</latexit>

HF i = 0ĥHF = EHF +<latexit sha1_base64="l3jvWMbTrdrrDnK7WF9HjbYoMyE=">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</latexit>Xp(i > A)(1  i  A)bi |<latexit sha1_base64="Y/Adp4ZkVGUaBbbJIMpiCv38zIo=">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</latexit>"p d†p dp

HF i = 0Xh"h b†h bh対相関と対称性
対相関：J=0に組んだ粒子対による強い引力(単極子対相関)GP̂ † P̂ ⇠G

Xh[c†i c†ī ]J=0 [cj̄ cj ]J=0

ij<latexit sha1_base64="WHJlbrFRykk5rBHxBCUQ7KCSmVo=">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</latexit>

相関の主要な項(時間反転状態との対)は対称性(粒子数保存)を保ったSlater行列式では拾えない

hA|[c†i c†ī ]J=0 |Ai

iJ=0|A>:粒子数演算子の固有状態<latexit sha1_base64="pAbw45kbW/yng3IFYJVBjeJldxs=">AAACvHichVHNStxQGD1Gbe3U1qndFLoJDkpXwxcRLIUWtZviyr9RYTION5nreJmbH5LMgMZ5gb5AF6WLCi6kD9AHcNOFLl34CKVLC25c+E0mpajUfiG55557zndz+JxQqzghOh8wBoeGHzwceVR4PPrk6Vjx2fh6HLQjV1bcQAfRpiNiqZUvK4lKtNwMIyk8R8sNp/W+d77RkVGsAn8t2Q1lzRNNX20rVyRM1YuzthZ+U0tz3tyvmu6W3RDNuvoDUtsRUaq63dpWuviWuvvmvB1l+nqxRGXKyrwLrByUkNdSUPwOGw0EcNGGBwkfCWMNgZifKiwQQuZqSJmLGKnsXKKLAnvbrJKsEMy2+NvkXTVnfd73esaZ2+VbNL8RO01M0hkd0QX9oG/0k67+2SvNevT+ZZdXp++VYX3s44vVy/+6PF4T7Px13eNwWH1/pgTbeJ1lUZwtzJheSrffv7P36WL1zcpkOkUH9IvzfaVzOuaEfue3e7gsVz6jwAOybo/jLlifLltUtpZnSnML+ahG8BITeMXzmMUcPmAJFb73C45xglPjndEwWobXlxoDuec5bpTRuQYxE6Xq</latexit>

c+c+は粒子数を2変えるので期待値はゼロになる

対称性を破るSlater行列式を考えるとこの相関を取り込める|<latexit sha1_base64="1KBwk4yvlJjtBIyQD+wEPPivtTs=">AAADDXichVFLSxxBEK4ZTaJrElfNQZDIkMVgGLLULIIiBHxA8Lg+VgVHlpm2XRvnxUzvgq7eJZccPXhS8CCeQy7ixYP5Azn4E0TwYiCXHFLzCMlGYqqZ7qqv6quaj7IDR0QS8UpR29ofPX7S0ZnrevrseXe+p3cx8ush4xXmO364bFsRd4THK1JIhy8HIbdc2+FL9uZ0nF9q8DASvrcgtwK+6lo1T6wLZkmCqvm9HbO8IapNM3S1mfdTu2ZoeTWHa+80k635MtJ0jQ1Pvh15s0PXr2SKlWKs1IIR0hLrcY3eWqPHvfQ/eqWDqvkCFjEx7b5jZE4BMiv7+c9gwhr4wKAOLnDwQJLvgAURnRUwACEgbBWahIXkiSTPYRdyxK1TFacKi9BNumsUrWSoR3HcM0rYjKY49IXE1GAIv+IJ3uEXPMVr/PHPXs2kR/wvW/TaKZcH1e4P/fPf/8ty6ZWw8Zv1AMOm6oc1SViHsUSLIG1BgsQqWdq/sb1/Nz8+N9R8jUd4Q/oO8QovSKHX+MaOZ/ncAeRoQcbf67jvLJaKBhaN2ZHCxFS2qg4YgFcwTPsYhQmYgTJUaO6t8kJ5qQyqH9VP6pl6npaqSsbpgxZTL38CKN27zQ==</latexit>

HFB i = · · · + c(A4)|A4i + c(Ah<latexit sha1_base64="my+wrfzmXViPGd3IJRy+yHNAv+g=">AAAC3XichVE9axRRFD0Zv+L6kVUbwWZwiVgtd4KgBIQQQYLVJnGThZ3N8GbysnnkzUfezC7EyZQ2IrYprBQsxB9gZ2PjH7BIbyOiTQQbC+/OjogJxjvMvPPOu+e8OVw/0SrNiPYnrBMnT50+M3m2du78hYtT9UuXV9J4YALZDmIdm44vUqlVJNuZyrTsJEaK0Ndy1d+6NzpfHUqTqjh6mO0ksheKfqQ2VCAyprx6x9Ui6mtpu61N5eWuCe2F+/OFvdu1gzV3XfQ99Rvkri9Mroqit5Y/uEvF7iGNayqnSG7b5NUb1KSy7KPAqUADVbXi+lu4WEeMAAOEkIiQMdYQSPnpwgEhYa6HnDnDSJXnEgVqrB1wl+QOwewWf/u861ZsxPuRZ1qqA75F82tYaWOaPtJrOqAP9IY+089/euWlx+hfdnj1x1qZeFNPri7/+K8q5DXD5h/VMQqfu4/PlGEDd8osirMlJTNKGYz9h4/2DpZnl6bzG/SSvnC+F7RP7zlhNPwevFqUS89R4wE5h8dxFKzMNB1qOou3GnPz1agmcQ3XcZPncRtzWEALbb73HT7hK75ZnvXYemo9G7daE5XmCv4qa+8X/hiywA==</latexit>2)|A

2i + c(A)|Ai + c(A + 2)|A + 2i + c(A + 4)|A + 4i + · · ·† † J=0|
HFB |[ci cī ]6 0
HFB i =

HFB状態：粒子数ゲージ空間での”変形”状態に対応
Slater行列式の範囲内(平均場近似)で対相関を記述するためには粒子数に関する対称性を破ることが本質的

Bogoliubov準粒子
HF状態を作る変換は粒子数を保存
HF状態：粒子・空孔演算子の真空a†k =<latexit sha1_base64="zfNB8NkJ1Gz50zuJe+plmepl8w8=">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</latexit>

di |

X

Dlk c†ll

HF i = 0<latexit sha1_base64="sz2zLoKeVVxgfpbeZuXEnswuDnQ=">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</latexit>bi |<latexit sha1_base64="Y/Adp4ZkVGUaBbbJIMpiCv38zIo=">AAACoXichVFNSxtRFD2ZtjZNq0l1U9DF0KB0Fe6IUBEKQaHEXT6MCkkYZsZnfGS+mJkEYpqNy/6BLrpqQWhx1W78Ad34B1z4E6RLC25c9GYyUDRo7+O9d95599z3Dtf0bRlGRBcp5dHjJ1NP088yz19Mz2RzL2e3Q68bWKJuebYX7JpGKGzpinokI1vs+oEwHNMWO2ZnY3S/0xNBKD13K+r7ouUYbVfuS8uImNJzC6YuPzTLB1IfNANHLb0fNgPDbdtCfaeSnstTgeJQJ4GWgDySKHu5UzSxBw8WunAg4CJibMNAyKMBDQSfuRYGzAWMZHwvMESGtV3OEpxhMNvhtc2nRsK6fB7VDGO1xa/YPANWqlikc/pOV3RGJ3RJN/fWGsQ1Rn/p826OtcLXsx9f1a7/q3J4j3DwT/WAwuTshz1F2Mdq7EWyNz9mRi6tcf3e4aer2lp1cbBEX+k3+/tCF/SLHbq9P9ZxRVQ/I8MN0u62YxJsLxc0KmiVlXxxPWlVGvN4jTfcj7coooQy6vzuEb7hB34qeWVTKSvVcaqSSjRzuBVK4y/gEZqb</latexit>

HF i = 0

対相関をどうやって一体場に取り込むか？
“粒子”の概念を拡張
Bogoliubov準粒子

u2i + vi2 = 1↵i† =ui a†i vi aī
↵ī† =ui a†ī + vi ai

Bogoliubov変換
粒子の生成・消滅演算子の線形結合で準粒子を定義†{↵i , ↵j } = {↵i† , ↵j† } = 0
準粒子も同じ反交換関係を満たす {↵i , ↵j } = ij
<latexit sha1_base64="I5SmPTp6P0uV4uq/NyyxrGTMbHE=">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</latexit><latexit sha1_base64="KOIF1kzaoOo36j99HFikJ+aSXhQ=">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</latexit>

BCS(HFB)状態：準粒子の真空↵µ |<latexit sha1_base64="cYa8XLzxa7BWMdO73j8SG7qiXwY=">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</latexit><latexit sha1_base64="AOJBM1tyGgvo/j18DPcr2kIUEiI=">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</latexit><latexit sha1_base64="DhdJ9bmfDZalYOqFU0D+vDJZ0VE=">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</latexit>|

BCS i = 0<latexit sha1_base64="XCx5eK00dh1t7eHW06IOPB0mMcc=">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</latexit>↵i ↵ī |0i = vi (ui + vi a†i a†ī )|0i

BCS i =

Y|<latexit sha1_base64="xYYg2OIHz9KPSXfMUpecGPAWVXE=">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</latexit>

BCS i =

i↵i ↵ī |0i =Yi(ui + vi a†i a†ī )|0i

|0>は粒子の真空

ci |0i = 0<latexit sha1_base64="6Kn2t5kmfPLbKW+lavHqCoAwzZQ=">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</latexit>

Yi↵i ↵ī |0i<latexit sha1_base64="5/37hOvsg5FhwgDgjIkSIW0Tgok=">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</latexit>(BCS form)

HF軌道対(i)に対して粒子が入っていない成分(ui)と2粒子が入っている成分(vi)の重ね合わせ
ui2:非占有確率 vi2:占有確率

HF+BCS
二体相互作用項をWickの定理で分解<latexit sha1_base64="I2mqa4PECMfTzCoO56IHJHKCqOA=">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</latexit>

1X
v̄ijkl h4† †HFB |ci cj cl ck |ijkl

HFB i =

1Xv̄ijkl (⇢ki ⇢lj
4⇢kj ⇢li + ⇤ij kl ) 対ポテンシャル

ijkl

対密度 ij = h HFB |cj ci | HFB i

Hartree-Fockポテンシャル(a,a+を決めるときに取り込まれている)<latexit sha1_base64="IchgrAS0nkCXz1A9h6NXdsERc2o=">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</latexit>

対ポテンシャルはHF基底で対角的と仮定(BCS, Bardeen Cooper Schrieffer)
対相互作用はparticle-hole相互作用とは別のものを使うことが多い(Skyrme)

Ĥ =

X

"i a†i ai +

i

X

† †
v̄ipairaa a a
īj j̄ i ī j̄ jij>0

一体場で近似(BCSハミルトニアン)X †X⇤ĥBCS =
"i a i a i +iī aī ai +<latexit sha1_base64="YND4gauIjJSNLxntVx63qwrkUGk=">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</latexit>i<latexit sha1_base64="4XlYUZ/FYzm8L6JndvfgpqHRu84=">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</latexit>↵i† =ui a†i

BCS |ĥBCS |<latexit sha1_base64="FIqscCWfhUAPC+0d2azQ+Cs9QiI=">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</latexit>vi aī

↵ī† =ui a†ī + vi ai<latexit sha1_base64="OrpRz+/sRwO/qAwhyXyNn5Cijao=">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</latexit><latexit sha1_base64="4ALivtzqjw9bLglosL+GNJvYf78=">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</latexit>† †iī ai aī

i>0

Bogoliubov変換

ĥBCS = h

対ポテンシャル

BCS i +X

BCS状態のエネルギーi

X pairv̄iīj j̄ haj̄ aj i =
iī =
j>0

X pairv̄iīj j̄ uj vj
j>0

相互作用がi,jによらない場合は定数になる
v̄ipair=īj j̄

G<latexit sha1_base64="bjoh2GQBGMVMoQEKDQGfu8Xh+Fc=">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</latexit>

Ei ↵i† ↵i
Ei: 準粒子エネルギー=G<latexit sha1_base64="uITY2od6LmOnzA+ozM0DI6zKM5c=">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</latexit>X

uj v jj>0α+α+やααの項(危険な項)が消滅するようにBogoliubov変換を選ぶ

HF+BCS
BCS：粒子数を保存しないため粒子数の平均を拘束する必要がある
X
X
†
†
0
⇤
ĥBCS = ĥBCSN̂ =("i)(ai ai + aī aī ) +iī aī ai +i>0<latexit sha1_base64="meuO0p/y0fzm4D/ggGRMIVoa3Tc=">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</latexit>† †iī ai aī

i>0

BCS Hamiltonianを準粒子で書いたときα+α+、ααの項がない

2x2の固有値問題✓<latexit sha1_base64="QzKctcXJWKBsGuZlJfnYcGf72XA=">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</latexit>

"i
iī

◆✓ ◆
✓ ◆
ui
ui
iī
= Ei
"i
vi
vi

"

1
"i
p
1
+
非占有率
2
("i
)2 +
"
1
"i
1 p
占有率 vi2 =
2
("i
)2 +
u2i =

<latexit sha1_base64="Z9/+4C4kzS1SrQItG1X8sbdel70=">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</latexit>

u2i + vi2 = 1<latexit sha1_base64="aCnq2ebW8svTdmVzbJk2R9PaNNs=">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</latexit>

準粒子のエネルギー Ei =<latexit sha1_base64="msbTvLBw2+DZInh/6uyLb0PsKrE=">AAACu3ichVHLShxBFD12Hurk4Wg2QjZFBoMhZLg9iEpAEJOAS1+jgq1NdVuaijXdbXfNgBn8AX8gC90kkIXkA7JPNoLZZuEnBJcGsskid3oaQiIxt6muU+fec6sON0iMzizRWY9z7fqNm719/aVbt+/cHSgPDi1ncTMNVT2MTZyuBjJTRkeqbrU1ajVJlWwERq0EO886+ZWWSjMdR0t2L1HrDbkd6S0dSsuUXx5/4WsxJbxsN7VtMeq1ZKqSTBvOafFEeIZbbcpHGzXxWHjPlbHS1xu1fb9coSrlIS4DtwAVFDEXlz/CwyZihGiiAYUIlrGBRMbfGlwQEubW0WYuZaTzvMI+SqxtcpXiCsnsDv+3+bRWsBGfOz2zXB3yLYZXykqBEfpKx3RBJ/SBvtHPf/Zq5z06b9njPehqVeIPHAwv/vivqsG7xcvfqisUAVdf7cliC5O5F83ekpzpuAy7/Vuv31wsPl0YaT+kd3TO/t7SGX1mh1Hre/h+Xi0cosQDcv8ex2WwXKu6VHXnxyrTM8Wo+nAfDzDK85jANGYxhzrfe4RPOMUXZ8oJnVeO6ZY6PYXmHv4Ip/kL2cmj/Q==</latexit>

λは粒子数の平均値の拘束条件 h BCS |N̂ | BCS i = 2を満たすように反復ごとに決定<latexit sha1_base64="ChtgTILN8nwDWoNue5A+3+REoZw=">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</latexit>Xiqvi2 = N0

("i

)2 +

2
i

2
i

2
i

#
#

BCSの計算アルゴリズム
HF方程式を解く(HF軌道が求まる)
Δiに初期値を設定
粒子数拘束条件 h BCS |N̂ | BCS i = 2を満たすようにλを決定<latexit sha1_base64="ChtgTILN8nwDWoNue5A+3+REoZw=">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</latexit>

u,v(波動関数)を計算

Δiをアップデート<latexit sha1_base64="FIqscCWfhUAPC+0d2azQ+Cs9QiI=">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</latexit>

vi2 = N0

i

"

1
"i
1+ p
2
("i
)2 +
"
1
"i
vi2 =
1 p
2
("i
)2 +

u2i =

<latexit sha1_base64="Z9/+4C4kzS1SrQItG1X8sbdel70=">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</latexit>

X

X pairv̄iīj j̄ haj̄ aj i =
iī =
j>0

2
i

2
i

#
#

X pairv̄iīj j̄ uj vjj>0|<latexit sha1_base64="xQ7Vyn9/Zj7feri6aBInjJQhv60=">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</latexit>

HF i =

A
Y

i=1

BCS状態
a†i |0i|

BCS i =

<latexit sha1_base64="xYYg2OIHz9KPSXfMUpecGPAWVXE=">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</latexit>i↵i ↵ī |0i =Y(ui + vi a†i a†ī )|0ii

BCS(HFB)状態

粒子空間
空孔空間

一粒子軌道の占有率(vi2)
Hartree-Fock状態ではエネルギーの低い順に軌道に粒子を詰める
占有率は1(空孔空間)か0(粒子空間)

一粒子軌道aのエネルギー

Hartree-Fock状態
一粒子軌道aのエネルギーY

λ

一粒子軌道の占有率(vi2)
BCS状態では
すべての一粒子軌道は部分占有

HFBとBogoliubov変換
HF+BCS: 対相関はHF軌道の性質を変えない
HFB(Hartree-Fock-Bogoliubov)
HF部分と対相関部分をまとめて解く
Bogoliubov変換

↵µ† =

✓

=

W=

✓

↵µ
↵µ†

<latexit sha1_base64="2qRUsMfC/F9FXEb6hYaRcjEwhtU=">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</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="L3BPXKOlOGTZpPx75vQJQo3a5GU=">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</latexit>
sha1_base64="Hclf8GM/+IbCGNi4t0qCDiOpZas=">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</latexit>

X ✓U†

VT

k

⇤

U
V

c: 基底の生成消滅演算子

Ukµ c†k + Vkµ ck

k◆<latexit sha1_base64="R7nKopG5phaLZlEnjgA6cA7ov8w=">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</latexit><latexit sha1_base64="he2dknrvYutz8jiLCgJanDaHibA=">AAADPnichVHLahRBFL3dvmJrzKgbwYWNQ0JWw20RFEEIiuBK8+pJIJ001TWVSdFPuqsHYzM/4A+4cKXgQvwAcSe4yQ9kkR8IBBEXEZKFC28/MGaC8RZdde6pe0715XpJIDOFuKPpZ86eO39h7KJx6fL4lYnW1WvdLM5TLmweB3G67LFMBDIStpIqEMtJKljoBWLJ8x+X90sDkWYyjhbVZiJWQ9aP5LrkTBHltj47nujLqEhCplL5Ymg4LEg2mOuEuek45lG25vRY3zQcEfX+FJvGQ2KyPHR9Y9THrgVTZrcGZNZdW6Tcpv2Yi1uUj/nDUQuTu34p45VBif9WGW6rjR2swjwJrAa0oYnZuPUJHOhBDBxyCEFABIpwAAwyWitgAUJC3CoUxKWEZHUvYAgGaXOqElTBiPVp71O20rAR5aVnVqk5vRLQl5LShEncxg+4j1v4Effw1z+9isqj/JdNOr1aKxJ34tWNhYP/qkI6FWwcqU5ReFR9ek8K1uF+1Yuk3pKKKbvktf/g5ev9hQfzk8UUvsNv1N9b3MGv1GE0+Mnfz4n5N1AOyBodx0nQvdOxsGPN3W3PPGpGNQY34TZM0zzuwQw8hVmwgWu3tCfaM+25/kXf1b/rP+pSXWs01+FY6Ie/Adxd1ew=</latexit>

ユニタリ行列

X

V
U⇤

†

V
UT

◆

✓ ◆
ck
†
c
k
µk

◆

WW † = W † W = 1<latexit sha1_base64="RGUVqPNmLcG3XwSEnNWwjWxGEBk=">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</latexit>

U † U + V † V = 1,

U U † + V ⇤V T = 1

U T V + V T U = 0,

U V † + V ⇤U T = 0<latexit sha1_base64="eYvhg+oJWW/HTDtqf06WTaqLcyo=">AAAC8nichVFNSxtRFL0Za7WpbWLdFNw8DEqhJdwRoUUQRDddapIZBScNbybPOGS+nHkJ2OAf8A8IurKlC9G9P8BN96WL/ATpMoVS6MI7L0NLDbV3mLnnnXvOfXO5duS5iUTs57SxB+MPJyYf5R9PPXlaKE4/M5OwEzvCcEIvjLdtngjPDYQhXemJ7SgW3Lc9sWW319P6VlfEiRsGNXkQibrPW4G76zpcEtUo+sY7q8lbzGAvmTmEJlth+itm7Xd4k/hMoMo8kZRqqYBZVt4gaKpKjYQrDH+7zDsmQ5mwUSxhGVWwUaBnoARZbITFK7CgCSE40AEfBAQgCXvAIaFnB3RAiIirQ4+4mJCr6gIOIU/eDqkEKTixbfq26LSTsQGd056Jcjt0i0dvTE4G8/gVz3GAn/ECb/DXP3v1VI/0Xw4o20OviBqFo+fVH/91+ZQl7P1x3eOwSX3/TBJ24Y2axaXZIsWkUzrD/t33x4PqcmW+t4Af8BvNd4Z9vKYJg+5359OmqJxCnhak313HKDAXyzqW9c2l0upatqpJmIU5eEH7eA2r8BY2wKB7v8DPnJYb06R2op1pH4dSLZd5ZuCv0C5vAf02sKg=</latexit>

Bloch-Messiahの定理
ユニタリ行列Wは左の形に分解できるW=<latexit sha1_base64="p1BYvXOIUP4HBwgc/HDkDJgv+k0=">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</latexit>

D, Cはユニタリ行列
0
u1
B0
B
B
B
B
Ū = B
B
B
B
@

u2
0

0
u2
..

.
un
0

<latexit sha1_base64="HrZo/Q4fPwpgGUnTI/MNT45GK4A=">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</latexit>

D
0

0
D⇤

<latexit sha1_base64="uaCFe7BQwOvP2So+zk2Zhf5xFjg=">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</latexit>

0

0
B v1
B
B
B
B
V̄ = B
B
B
B
@

C
C
C
C
C
C
C
C
C
0A
un

◆✓

Ū
V̄

V̄
Ū

◆✓
C
0

0
C⇤

◆

V = D⇤ V̄ C

U = DŪ C,
1

0
u1

✓

1

v1
0
0
v2

v2
0
..

.
0
vn

<latexit sha1_base64="pfkXSnEdYUP5zQ04CCrvgnOhqPE=">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</latexit>

C
C
C
C
C
C
C
C
C
vn A
0

Bloch-Messiahの定理によるとBogoliubov変換は３つの部分に分けられる(c, c† ) ! (a, a† ) ! ( ,<latexit sha1_base64="VL25UOKiu/nEeigEgHai1MNViy0=">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</latexit>

DŪ , V̄

正準基底
基底(調和振動子など)<latexit sha1_base64="jl+aln2oS8C3cFYtbBYE5rtPFM8=">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</latexit>†) ! (↵, ↵† )準粒子

C準粒子の組み換え

変換DはHFで行ってきたようなもの。正準基底→準粒子はBCS
変換Cは状態は変えないが(β|HFB>=α|HFB>=0)
HFB Hamiltonianの準粒子エネルギー部分の対角化に必要

HFB方程式Ĥ =X

1
tij c†i cj +4ij<latexit sha1_base64="OQiftS4B8ua1HtPu69ZCTpyHPi0=">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</latexit>Xv̄ijkl c†i c†j cl ck

ijkl

一体場Hamiltonianで近似(粒子数に関する線形拘束も追加)
HFポテンシャルij [⇢] =X<latexit sha1_base64="I9NaUCR3GzMjfrPIK0TevOlxgAU=">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</latexit>

HFB i

ij = h

HFB |cj ci |

HFB i

† †
ij []ci cj +⇤ij []cj cikl<latexit sha1_base64="RGcAefkrGZJtKWXhaCd8kV0/hsU=">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</latexit>N̂ =†HFB |cj ci |<latexit sha1_base64="63NssL9Nb6Er7kslrVTCZAb18ic=">AAACzHichVFNaxRBEH0ZPxLXj6x6EbwMLhFPS40ISkAIEUxOskncJJCJw0yns9tJzwc9vQtxstcc8gdy8KTgQcSzP8CLd/EQ0B8gHiN48WDt7IDEYKyhp1+/qlfdj4oyrXJLdDjmnDl77vz4xIXaxUuXr0zWr15bztOeEbItUp2a1SjMpVaJbFtltVzNjAzjSMuVaPvRML/SlyZXafLU7mRyPQ47idpUIrRMBfU533TToFBbg4eur8Oko6Xrt7oqKHwTu/OPZwe7rnjmb4SdYMsVgXJ3j2d9M9IE9QY1qQz3JPAq0EAVrbT+Hj42kEKghxgSCSxjjRA5f2vwQMiYW0fBnGGkyrzEADXW9rhKckXI7Db/O3xaq9iEz8OeeakWfIvmZVjpYoo+0xs6oo/0lr7Rr3/2Ksoew7fs8B6NtDILJvdvLP38ryrm3aL7R3WKIuLq0z1ZbOJB6UWxt6xkhi7FqH//+cHR0vTiVHGbXtF39veSDukDO0z6P8TrBbn4AjUekPf3OE6C5btNj5rewr3GzGw1qgncxC3c4Xncxwzm0UKb732HT/iCr84TxzqFMxiVOmOV5jqOhbP3G7ZQrAc=</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="XbmmThslzoI6tPpkVxZBdUXExSQ=">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</latexit>
sha1_base64="5Ga4QixQRwV3AD2GAvpAzGSBV5g=">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</latexit>

1X
対ポテンシャル
[]=v̄ijkl kl
ij
2
kl
Xh
<latexit sha1_base64="tjxUcvOO74VaSJZ0JOkQqnTLR28=">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</latexit>ĥHFB = ĥ⇢ij = hv̄ikjl ⇢lk(tij +

ij [⇢]<latexit sha1_base64="IchgrAS0nkCXz1A9h6NXdsERc2o=">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</latexit>†ij )ci cj +ijBogoliubov変換✓<latexit sha1_base64="he2dknrvYutz8jiLCgJanDaHibA=">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</latexit>

↵µ
↵µ†

◆

=

X ✓U†
k

VT

V†
UT

◆

✓ ◆
ck
c†k
µk

Ei: 準粒子エネルギー
HFB状態のエネルギー⌘X⇣††ĥHFB = h HFB |ĥHFB | HFB i +Eµ ↵µ ↵µ + Eµ̄ ↵µ̄ ↵µ̄iµ>0<latexit sha1_base64="ghkNnuE44iIIeqLFShnkFb1ZR6Q=">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</latexit>

α+α+やααの項(危険な項)が消滅するようにBogoliubov変換を選ぶ

HFB方程式✓<latexit sha1_base64="6mOaZKc8IuAQ6+ZMg0pRx7wkFao=">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</latexit>

h
⇤

◆✓ ◆
✓ ◆
Uµ
Uµ
= Eµ
h⇤ +
Vµ
Vµ

HFB方程式を解くアルゴリズム
ρ、κ行列に初期値を設定
HF, 対ポテンシャルを計算

ij [⇢] =Xv̄ikjl ⇢lkkl<latexit sha1_base64="tjxUcvOO74VaSJZ0JOkQqnTLR28=">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</latexit>

粒子数拘束条件 h HFB |N̂ | HFB i = 2

X<latexit sha1_base64="RGcAefkrGZJtKWXhaCd8kV0/hsU=">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</latexit>

1X
v̄ijkl klij [] =
2
kl

⇤
Vkµ
Vkµ = N0k,µ>0を満たすようにHFB方程式のλを決定<latexit sha1_base64="c1Fse+H2Pc0IPNBrBBAu1LEiQFA=">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</latexit>✓t + [⇢]⇤[<latexit sha1_base64="E80heGxlN/uBMiNU5j9ViAo0wVk=">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</latexit>

t⇤

◆✓ ◆
✓ ◆
[]
Uµ
Uµ
=
E
µ
⇤
[⇢] +
Vµ
Vµ

U,V(固有ベクトル)を使って粒子密度、対密度をアップデートX†⇤⇤⇢ij = h HFB |cj ci | HFB i =(ViµVjµ + Viµ̄
Vj µ̄ )µ>0

<latexit sha1_base64="sJsvYy5EJz88bpd5V+7i5U+bWhI=">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</latexit>

ij = h<latexit sha1_base64="/npRCxORxm8SJhpX9nVbALa/QQk=">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</latexit>

HFB |cj ci |HFB i =

X

µ>0

⇤
⇤
(Viµ
Ujµ + Viµ̄
Uj µ̄ )

HFB方程式の性質✓<latexit sha1_base64="6mOaZKc8IuAQ6+ZMg0pRx7wkFao=">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</latexit>

h
⇤

◆✓ ◆
✓ ◆
Uµ
Uµ
=
E
µ
h⇤ +
Vµ
Vµ

(U,V), Eが解であれば (V*, U*), -Eも解✓
<latexit sha1_base64="zBGl3eTwW/Vp6Mm+pIBkz25YdfM=">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</latexit>

h
⇤

◆✓
Uµ
h⇤ +
Vµ

Vµ⇤
Uµ⇤

◆

=

✓

Uµ
Vµ

Vµ⇤
Uµ⇤

◆✓
Eµ
0

準粒子αとα＋を入れ替える対称性に対応

0
Eµ

◆

対相互作用
単極子対相関力V̂0pair =<latexit sha1_base64="SgpilwBVxFNfusW0oSSaRO2hi7g=">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</latexit>

G0 P̂0† P̂0P̂0† =

i>0<latexit sha1_base64="+rZuGq4tj8hq9vD2TMZGCOnHTQ8=">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</latexit>

四重極対相関力V̂2pair =

G2

2
X

†
P̂2µ
P̂2µ

µ= 2

<latexit sha1_base64="rkgjCxZEIG7knWlvgX61K9wLmoo=">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</latexit>X † †ci cī

†
P̂2µ
=

X
ij

<latexit sha1_base64="gT4goKfW5waC7ebzLNoSlGisMpI=">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</latexit>hi|r2 Y2µ |jic†i c†j̄

Gogny：同じGogny型有効相互作用から対ポテンシャルを導出
Skyrme: 対ポテンシャルは以下の形のデルタ型対相互作用から導出することが多い(SkPを除く)v̂pair (r12 ) =

V0
(1
2

P̂ ) 1<latexit sha1_base64="IuNCvlj2IbBTJQV60BaD3s4Z7qE=">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</latexit>

密度依存

"

⌘

✓

⇢0 (r)⇢c◆ #(r12 )

η=0: volume pairing
η=1: surface pairing
η=0.5: mixed pairing

ゼロレンジの対相関は一粒子空間を増やすと発散：準粒子エネルギーなどにcutoffを導入
有限レンジでは収束する

密度汎関数理論(density functional theory)
Hohenberg-Kohnの定理
基底状態のエネルギー(やその他の観測量)は(核子の)密度の汎関数で書ける⇢(r) $ | i $ v(r) 密度から相互作用が決定できるE[⇢]<latexit sha1_base64="iYDStmpGu2XTebcuV/BSlJCrBp8=">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</latexit>

エネルギー汎関数を最小にする密度は基底状態の密度⇢Z<latexit sha1_base64="nLhweHFNmLlFX5vn4EF9huElkgM=">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</latexit>

E[⇢]

µ

dr⇢(r)

N=0

Egs = min E[⇢]⇢<latexit sha1_base64="yOSTtc4WHdF+d6VmFW4OwFYUszM=">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</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="ivfvixNsAcxTn4ZS/GzNXAB26Bg=">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</latexit>
sha1_base64="au1Rd8YyjuZ2XHJDS6SgVhgUhnM=">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</latexit>

Kohn-Shamの定理<latexit sha1_base64="3EEAFshPNcnRci90gnSgmyOG6vU=">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</latexit>

E[⇢] = Ts [⇢] + (E[⇢]<latexit sha1_base64="3wDXby3VQ/I7BLKd/6tUCUWAd1M=">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</latexit>

Ts [⇢]) = Ts [⇢] + V [⇢]相互作用する系の密度は一体ポテンシャル中の自由粒子系の密度として計算できる。
Z~2
Ts [⇢] =dr'⇤k (r)r2 'k (r) 自由粒子系での運動エネルギー2m<latexit sha1_base64="naCRay8DgLoNvgZrJiFq+x/e5a0=">AAAC53ichVHNahRBEP4y/sWNmjVeBC+DSyQeXHqWgCIEgrl4zN8mgZ3doXvSm212/ujpXYjDvoAXbyaQUwQP4gP4AF7yAh7yCCJ4SSAePFgzOxI0GGvoqaqv6qvujxJJoFLD2PGEdeXqtes3Jm9Wpm7dvjNdvTuzkcYD7cumHwex3hI8lYGKZNMoE8itREseikBuiv5SXt8cSp2qOFo3u4lsh3wnUl3lc0OQVxXrXtpydS9u2wv2E7eruZ+5PcF1pzHKGuHIdlVk7G1XhJmmZMh10lMdl6fG68+N0ce5cyMuAj7qNH73nJe9ao3VWWH2xcApgxpKW46rn+BiGzF8DBBCIoKhOABHSl8LDhgSwtrICNMUqaIuMUKFuAPqktTBCe3Tf4eyVolGlOcz04Lt0y0BHU1MG7PsC/vATtgR+8i+sp//nJUVM/K37JIXY65MvOnX99fO/ssKyRv0zlmXMAR1X67JoItnhRZF2pICyVX64/nDV3sna89XZ7NH7B37RvoO2TH7TAqj4an/fkWuHqBCC3L+XsfFYKNRd1jdWZmvLb4oVzWJB3iIOdrHUyziJZbRpHuP8B1n+GEp64311toft1oTJece/jDr8BdSWLgK</latexit>

Kohn-Sham方程式✓<latexit sha1_base64="eUdj4elF0Z9hA7c3OsReQHnNIZk=">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</latexit>

◆
~2 2
V [⇢]
r +
(r) 'k (r) = "k 'k (r)2m⇢⇢(r) =<latexit sha1_base64="V8x+f8rOOH1NBPl6FDIROL55TEs=">AAACyXichVFNS9xQFD3Gtur0w6ndCN2EDordDDeloBQEsS6EQvFrVDAakvTpPCZfvrwZ1DArd/4BF65a6KLtD/AHuOmymy7Gf1C6tNBNF97JhEortTck77zzzrkvh+slgUw1UafP6L91+87A4FDp7r37D4bLD0dW07ipfFHz4yBW656bikBGoqalDsR6ooQbeoFY8xovu+drLaFSGUcrej8Rm6G7E8lt6buaKac8Z6t6PGF7YabaT81p006boZM1pq321mvTbrkqqcst20210/itKugrxilXqEp5mdeBVYAKilqIy6ew8QYxfDQRQiCCZhzARcrPBiwQEuY2kTGnGMn8XKCNEnubrBKscJlt8HeHdxsFG/G+2zPN3T7fEvCr2GlijL7SB7qgz/SJvtGvf/bK8h7df9nn1et5ReIMH40u//yvK+RVo37lusHhsfrmTBrbmMqzSM6W5Ew3pd/r3zo4vlh+sTSWjdM7+s753lKHzjhh1Prhv18USyco8YCsv8dxHaw+q1pUtRafV2Zmi1EN4jGeYILnMYkZzGMBNb73I76gg3PjlbFr7BkHPanRV3ge4Y8yDi8BnMarSQ==</latexit>NX'⇤k (r)'k (r)'k (r) Kohn-Sham軌道<latexit sha1_base64="eUA5PsSGuF4lJFwxhMXwry9VM4o=">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</latexit>

k=1

Hartree-Fock方程式と同じ形：HFは近似なのに対しDFTは厳密なエネルギーを与える
Kohn-Sham軌道：物理的に意味はないエネルギー密度汎関数(energy density functional)
エネルギー密度汎関数(EDF)：電子系の場合

E[⇢] = Ts [⇢] + V [⇢] = Ts [⇢] +Ze
drvext (r)⇢(r) +2外場(原子核-電子のポテンシャル)<latexit sha1_base64="L1t6RWUwWMBsqdagKx1gZyL9CRE=">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</latexit><latexit sha1_base64="KC5heWrYAeudnr5D15UMv+NfHWg=">AAAC5nichVHLahRBFD1pX3GiZtSN4KZxiESE4XYQFEEIuskyD2cSmB6G6kpNpkm/qK4ZEpv+gSyykyzcqOBC/AA/wIX+gIt8goirSLJx4e2eVtFovE11nXvqnlt1uF4S+Kkh2p+wTp0+c/bc5Pna1IWLl6brl6+003iopWrJOIj1midSFfiRahnfBGot0UqEXqBWvc1HxfnqSOnUj6PHZjtR3VBsRH7fl8Iw1asLt6+FzGx3XQVG2O2OqwdxN8/GeZHk9gN71MtcHdpqy+SzrhdmOr9l3/7BLvyF25I/yV69QU0qwz4OnAo0UMViXH8LF+uIITFECIUIhnEAgZS/DhwQEua6yJjTjPzyXCFHjbVDrlJcIZjd5P8GZ52KjTgveqalWvItAS/NShsz9JFe0wF9oDf0ib79s1dW9ijess27N9aqpDe9c23l6L+qkHeDwS/VCQqPq0/2ZNDHvdKLz96SkilcynH/0ZO9g5X7yzPZTXpJn9nfC9qnd+wwGn2Vr5bU8jPUeEDOn+M4DtpzTYeaztKdxvzDalSTuI4bmOV53MU8FrCIFt/7Hl9wiCNrYO1aT629cak1UWmu4rewnn8HKwC2pQ==</latexit>

2 Z

V [⇢]
= vext (r) + vH (r) + vxc (r)⇢電子間ポテンシャル(Hartree項) vH (r) = e

2<latexit sha1_base64="6QUmm0AdLPLW4ph0p3WhfvbyZpQ=">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</latexit>

相関ー交換ポテンシャル

vxc (r) =<latexit sha1_base64="KnVrD7CvJ626N2wiPsdSGfB+TRE=">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</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="umlaOz1sdDgArm4qWjVxh49lPpA=">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</latexit>
sha1_base64="zfE0LoHRkKg9xutHXLucYNUVmHo=">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</latexit>Z00 ⇢(r)⇢(r )drdr+ Exc [⇢]
0
|r r |

0
⇢(r
)
0
dr
|r r 0 |Exc⇢

原子核密度汎関数
1

2
X~E[⇢, ⇢˜, ⇢˜⇤ ] =⌧0 (r) +2m<latexit sha1_base64="KZUCmQSuqF012piUAFQF6v78N60=">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</latexit>k=0

密度(中性子・陽子)と対密度の汎関数

phk [⇢] +

X

t=n,p

⇤
˜pp
[⇢,
⇢
˜
,
⇢
˜
]
t

Skyrme型EDF(particle-hole)
原子核密度汎関数

1X~2E[⇢, ⇢˜, ⇢˜ ] =⌧0 (r) +2m⇤

phk [⇢] +˜pp
˜, ⇢˜⇤ ]t [⇢, ⇢

t=n,p

k=0

<latexit sha1_base64="KZUCmQSuqF012piUAFQF6v78N60=">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</latexit>

X

ph
⇢
⇢
2
⌧
J0 2
J1 2
J2 2
rJ
k (r) =Ck [⇢0 ]⇢k + Ck ⇢k ⇢k + Ck ⇢k ⌧k + Ck Jk + Ck Jk + Ck Jk + Ck ⇢k r · Jk
Cks [⇢0 ]s2k + Ck s sk · sk + CkT sk · Tk + Ckj jk2 + Ckrj sk · (r ⇥ jk ) + Ckrs (r · sk )2 + CkF sk · Fk

k=0:アイソスカラー(n+p), k=1 アイソベクター(n-p)

密度行列⇢ˆ(rst, r 0 s0 t0 ) = h |c†r0 s0 t0 crst | i

非局所密度⇢k (r, r 0 ) =X⇢ˆ(rst, r 0 st)⌧ttk

sk (r, r 0 ) =st<latexit sha1_base64="hZ0PdvhFqG+gs8ydW5sKu4iLMBI=">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</latexit>

局所密度(time-even)X⇢ˆ(rst, r 0 s0 t) ˆ s0 s ⌧ttk

ss0 t

<latexit sha1_base64="eJfxtjiD9E3uIY+bALDTx4Cgy4s=">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</latexit>0⌧ =tensor density

density⇢k (r) = ⇢k (r, r)

kinetic density⌧k (r) = (r · r0 )⇢k (r, r 0 )

1
(r
2i

Jk (r) =Jk (r) =

r=r 0

X

✓

r0 ) ⌦ sk (r, r 0 )

1
0

◆
0
1

1

⌧ =

✓

1
0

0
1

◆

r=r 0

Jkaa (r)aJka (r) =X"abc Jkbc (r)bcJkab (r) =局所密度(time-odd)
current density1
jk (r) = (r2i

11Jkab (r) + Jkba (r)2
2

1
Jk (r) ab3sk (r) = sk (r, r)

r0 )⇢k (r, r 0 )r=r 0

spin density
Tk (r) = (r · r0 )sk (r, r 0 )spin-kinetic density
r=r 0
tensor-kinetic density F (r) = 1 (r ⌦ r0 + r0 ⌦ r)s (r, r0 )
k

2

k

r=r 0

Skyrme型EDF(pairing)
原子核密度汎関数
t=n or p

1

X~2
E[⇢, ⇢˜, ⇢˜ ] =⌧0 (r) +2m⇤

phk [⇢] +

k=0

<latexit sha1_base64="KZUCmQSuqF012piUAFQF6v78N60=">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</latexit>X˜pp˜, ⇢˜⇤ ]t [⇢, ⇢

t=n,p

⇢
˜pp
⇢t |2 + C̃t ⇢ Re˜⇢⇤t ⇢˜t + C̃t⌧ Re˜⇢⇤t ⌧˜t + C̃tJ0 |J˜t |2 + C̃tJ1 |J˜t |2 + C̃tJ2 |J̃t |2 + C̃trJ Re˜⇢⇤t r · J˜tt (r) =C̃t [⇢0 ]|˜

密度行列

⇢ˆ
˜(rst, r 0 s0 t0 ) =非局所密度⇢˜t (r, r 0 ) =

局所密度

X

2s0 h |cr0 s0 t0 crst | i⇢ˆ˜(rst, r 0 st)

0s̃t (r, r ) =

s

X

⇢ˆ
˜(rst, r 0 s0 t) s0 s

ss0

tensor density

density⇢˜t (r) = ⇢˜t (r, r)

J̃t (r) =kinetic density⌧˜t (r) = (r · r0 )˜⇢t (r, r 0 )J˜t (r) =r=r 0

J˜ta (r) =

1
(r
2i

X
a
X

r0 ) ⌦ s̃t (r, r 0 )r=r 0J̃taa (r)"abc J̃tbc (r)bc

11J̃tab (r) = J̃tab (r) + J̃tba (r)22

1˜
Jt (r) ab3

neutron-proton pairingを考えるとcurrent, spin pair densityなども現れる
Skyrme EDF 一般形：Perlinska et al., Phys. Rev. C 69, 014316 (2004)

有効相互作用と密度汎関数との対応
Skyrme有効相互作用v̂pair (r12 ) =Skyrme EDF<latexit sha1_base64="HxeaMe11PJnaRZBIDVo0CyIuYg8=">AAADJXichVHLahRBFL3dahLbR0bdCG4Kh8gM6FA9+EIQgm5cThJnEpgam+pKzUyRflFdMxCb/gF/wIUbFUTED/AD3Lhxp0JWrsVlBDcuvP0AH8F4m+4699x7bvXh+kmgUkPpnmUfOXpsYXHpuHPi5KnTy40zZwdpPNNC9kUcxHrL56kMVCT7RplAbiVa8tAP5Ka/c7eob86lTlUc3Te7iRyFfBKpsRLcIOU1nrIpN9k89zKmQ5JwpfMW88NMI+N28za5TdhYc5ENPJpn3Zy0XHKFlKJe7rFUTULeJiyQYzN0SFkrcKtWMT2NPVpPbOdVLnKm1WRq2g+YLw13qmxE2LYMDEfpb/c7XqNJO7QMchC4NWhCHb248QYYbEMMAmYQgoQIDOIAOKT4DMEFCglyI8iQ04hUWZeQg4PaGXZJ7ODI7uB3gtmwZiPMi5lpqRZ4S4CvRiWBFfqRvqL79B19Tb/QH/+clZUzin/ZxdOvtDLxlh+d3/j+X1WIp4HpL9UhCh+7D/dkYAw3Sy8KvSUlU7gU1fz5w8f7G7fWV7JL9Dn9iv6e0T36Fh1G82/ixZpcfwLFgty/13EQDLodl3bctavN1Tv1qpbgAlyEFu7jBqzCPehBH4S1YF22rlnX7Zf2e/uD/alqta1acw7+CPvzT+PIyks=</latexit>

V0
(1
2

"

P̂ ) 1✓⇢0 (r)⇢c◆ #(r12 )

ph
⇢
⇢
2
⌧
J0 2
J1 2
J2 2
rJ
k (r) =Ck [⇢0 ]⇢k + Ck ⇢k ⇢k + Ck ⇢k ⌧k + Ck Jk + Ck Jk + Ck Jk + Ck ⇢k r · Jk
Cks [⇢0 ]s2k + Ck s sk · sk + CkT sk · Tk + Ckj jk2 + Ckrj sk · (r ⇥ jk ) + Ckrs (r · sk )2 + CkF sk · Fk⇢
˜pp⇢t |2 + C̃t ⇢ Re˜⇢⇤t ⇢˜t + C̃t⌧ Re˜⇢⇤t ⌧˜t + C̃tJ0 |J˜t |2 + C̃tJ1 |J˜t |2 + C̃tJ2 |J̃t |2 + C̃trJ Re˜⇢⇤t r · J˜tt (r) =C̃t [⇢0 ]|˜

p 有効相互作用からEDFの結合定数を導出できる
p 逆はできない場合もある(パラメータの数)
p EDFは一般にHamiltonian演算子と対応していない
p Skyrmeの場合はphとppチャネルで異なる相互作用：Hamiltonianとの対応していない

公開HFB計算コード(Fortran 90)
p Langanke, Maruhn, Koonin: Computational Nuclear Physics I (textbook)
Chap. 2 Skyrme Hartree-Fock (P.-G. Reinhard) (Fortran77コード)
p HFBRAD: 球対称(1次元)座標表示HFBコード
Bennaceur and Dobaczewski, Comp. Phys. Commun. 168, 96 (2005)
p HOSPHE: 1次元調和振動子基底HFBコード
Carlsson et al., Comp. Phys. Commun. 181, 1641 (2010)
p HFBTHO: 軸対称(2次元)調和振動子基底HFBコード
Navarro Perez et al., Comp. Phys. Commun. 220, 363 (2017) v3.00
p HFODD: 3次元調和振動子基底HFBコード
Schunck et al., Comp. Phys. Commun. 216, 145 (2017) v.2.73y

論文にダウンロードページへのリンクあり

課題：HFBTHOを動かしてみよう
HFBTHO: 軸対称(2次元)調和振動子基底HFBコード
Navarro Perez et al., Comp. Phys. Commun. 220, 363 (2017) v3.00
ダウンロード： https://data.mendeley.com/datasets/c5g2f92by3/1
解凍： tar zxvf HFBTHO̲V300.tgz
sources/にソースファイル
MakefileのFortranコンパイラを指定(gfortran, ifortなど),
MPI, OpenMPを使うかをUSE̲MPI, USE̲OPENMPで指定
LAPACK/BLASが必要
コンパイル make

で実行ファイルhfbtho̲mainを作る

入力ファイル例はruns/
入力ファイルはhfbtho̲NAMELIST.datという名前にして実行ファイルを実行

課題：HFBTHOを動かしてみよう
入力ファイル：hfbtho̲NAMELIST.dat

HFB計算のデータベース
Gogny HFB : (フランス、CEA)
AMEDEE database (Gogny D1S)

http://www-phynu.cea.fr/science̲en̲ligne/carte̲potentiels̲microscopiques/carte̲potentiel̲nucleaire̲eng.htm

mass explorer (アメリカ、MSU)
Skyrme HFB・相対論的平均場理論
http://massexplorer.frib.msu.edu/

まとめp 平均場理論p 平均ポテンシャルを自己無撞着に決定
p Hartree-Fock近似：二体相互作用を一体場で近似
p 一体場に相関を取り入れるためには対称性を破るのが本質的
p 回転対称性の破れ→変形解p BCS近似：粒子数保存の破れ→準粒子の導入
p 有効相互作用
p Hartree-Fock-Bogoliubov近似：HF部分と対部分をまとめて決定
p 原子核密度汎関数理論
p 公開コード HFBRAD, HFBTHO, HFODD, HOSPHEなど