2022/9/18 熱場の量子論とその応用

fermionic Matrix Product Stateを用いた
Generalized Thouless Pumpの不変量について
基礎物理学研究所 大山修平
共同研究者：塩崎謙 佐藤昌利
based on arxiv:2206.01110(PRBにaccept済み)
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目次
• 研究の背景
SRE状態とGTPについて• 研究の⽬的と⽅法
MPSを⽤いたinteracting fermionic systemにおけるGTPの検証• 結果１：Pump不変量の定義
fundamental theorem for fMPS
Pump不変量の定義• 結果２：Pump不変量の計算
non-trivial phaseにおけるpump
trivial phaseにおけるpump
• まとめ

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ガチャガチャ模型の定義例) ガチャガチャ模型

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ガチャガチャ模型の定義例) ガチャガチャ模型
⽮印(⇅)を並べて間に丸(⽩/⿊)を置く．両端は上向きに固定．

4 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

5 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

6 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

7 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

8 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

9 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

⽮印を180度回す

10 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

⽮印を180度回す

11 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

⽮印を180度回す

12 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

⽮印を180度回す

13 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

⽮印を180度回す

14 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

⽮印を180度回す

15 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

⽮印を180度回す

系の⽚⽅の端からもう⽚⽅の端ににZ/2 chargeがpumpされた．

16 /92

ガチャガチャ模型の定義例) ガチャガチャ模型

上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

𝝅 rot.

⽮印を180度回す<latexit

sha1_base64="FIrxYIu8LOxuDc7YXQK9X0Gav64=">AAADJnichVI7SwNBEP5yvmJ8ayPYBINiIWFPBcUqaGNpEqOCUbk7N3p4L/Y2EQ3+Af+AgpWCoPgzbKzsgtjZiqWCjYWTy4GPkDjH3c58O9/sN3uje5bpS8aeIkpLa1t7R7Qz1tXd09vXPzC46rtFYfCc4VquWNc1n1umw3PSlBZf9wTXbN3ia/r+YnV/rcSFb7rOijz0+Kat7TpmwTQ0SdBWOa8X4vmipwnhHhxv9ydYkgUWr3fU0EkgtGV3IBJBHjtwYaAIGxwOJPkWNPj0bEAFg0fYJsqECfLMYJ/jGDHiFimLU4ZG6D59dynaCFGH4mpNP2AbdIpFryBmHGOswm7YG7tnt+yFfTasVQ5qVLUc0qrXuNzb7jsZzn78y7Jpldj7ZjXVLFHAXKDVJO1egFS7MGr80tHpW3Y+M1YeZ5fslfRfsCd2Rx04pXfjKs0z50306NS3RjGnmo1v7q/uybruGzMl5XjkqYTaTTN//okYYjQ16t8ZqXdWp5LqdHIqPZNILYTzE8UIRjFBJ84ihSUsI0e1Bc5wgUvlWnlQKspjLVWJhJwh/DLl+QsmPbTo</latexit>"
系の⽚⽅の端からもう⽚⽅の端ににZ/2 chargeがpumpされた．

17 /92

ガチャガチャ模型の定義例) ガチャガチャ模型

上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

𝝅 rot.

⽮印を180度回す<latexit

sha1_base64="FIrxYIu8LOxuDc7YXQK9X0Gav64=">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</latexit>"

系の⽚⽅の端からもう⽚⽅の端ににZ/2 chargeがpumpされた．
★ ⽮印をspin，丸をcpx. fermionに読み替えると以下のHam.で実現できる：

18 /92

ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

⽮印を180度回す

𝝅 rot.
<latexit"
sha1_base64="FIrxYIu8LOxuDc7YXQK9X0Gav64=">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</latexit>

系の⽚⽅の端からもう⽚⽅の端ににZ/2 chargeがpumpされた．
★ ⽮印をspin，丸をcpx. fermionに読み替えると以下のHam.で実現できる：
H(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

X
j

(a†j

†
x
aj )⌧j+(a1
j+1 + aj+1 )2Xj⌧jz,✓1 (12z,✓2a†j aj )⌧j+1
2

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ガチャガチャ模型の定義例) ガチャガチャ模型
上/下⽮印を並べて間に⿊/⽩丸を置く．両端は上向きに固定．

⽮印を180度回す

𝝅 rot.
<latexit"
sha1_base64="FIrxYIu8LOxuDc7YXQK9X0Gav64=">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</latexit>

系の⽚⽅の端からもう⽚⽅の端ににZ/2 chargeがpumpされた．
★ ⽮印をspin，丸をcpx. fermionに読み替えると以下のHam.で実現できる：
H(✓) =

j<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

ただし

X

(a†j

†
x
aj )⌧j+(a1
j+1 + aj+1 )2

z,✓i ✓2 ⌧jx z
i ✓2 ⌧jx⌧j = e⌧j e<latexit sha1_base64="PcQWA1hArg7cCqTw8i13IOoEVgQ=">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</latexit>Xj⌧jz,✓1 (12z,✓2a†j aj )⌧j+1

．このとき H(0) = H(⇡) .
<latexit sha1_base64="gVRcHWHY88yW65nYOgyJgMtoX6M=">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</latexit>

2

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ガチャガチャ模型の特徴例) ガチャガチャ模型
H(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

X
j

(a†j

†
x
aj )⌧j+1 (aj+1 + aj+1 )2Xj⌧jz,✓1 (12z,✓2a†j aj )⌧j+1
2

21 /92

ガチャガチャ模型の特徴例) ガチャガチャ模型
H(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

X
j

(a†j

†
x
aj )⌧j+1 (aj+1 + aj+1 )2Xj⌧jz,✓1 (12z,✓2a†j aj )⌧j+1
2

・各θに対してuniqueかつgappedなHamiltonianである．
このようなHam.のG.S.をShort-Range Entangled(SRE)状態という．

22 /92

ガチャガチャ模型の特徴例) ガチャガチャ模型
H(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

X
j

(a†j

†
x
aj )⌧j+1 (aj+1 + aj+1 )2Xj⌧jz,✓1 (12z,✓2a†j aj )⌧j+1
2

・各θに対してuniqueかつgappedなHamiltonianである．
このようなHam.のG.S.をShort-Range Entangled(SRE)状態という．
・各θに対して適当なunitary変換を⾏うと，θ=0にできる：

23 /92

ガチャガチャ模型の特徴例) ガチャガチャ模型
H(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

X

(a†j

j

†
x
aj )⌧j+1 (aj+1 + aj+1 )2X⌧jz,✓1 (12jz,✓
2a†j aj )⌧j+1
2

・各θに対してuniqueかつgappedなHamiltonianである．
このようなHam.のG.S.をShort-Range Entangled(SRE)状態という．
・各θに対して適当なunitary変換を⾏うと，θ=0にできる：
H(✓) = (<latexit sha1_base64="ESwNmDBGal1zXGTTcp7/s0YXhEQ=">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</latexit>Yjei ✓2 ⌧jx)H(0)(Yei ✓2 ⌧jx †)j24 /92

ガチャガチャ模型の特徴例) ガチャガチャ模型
H(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

X

(a†j

j

†
x
aj )⌧j+1 (aj+1 + aj+1 )2X⌧jz,✓1 (12jz,✓
2a†j aj )⌧j+1
2

・各θに対してuniqueかつgappedなHamiltonianである．
このようなHam.のG.S.をShort-Range Entangled(SRE)状態という．
・各θに対して適当なunitary変換を⾏うと，θ=0にできる：
H(✓) = (<latexit sha1_base64="ESwNmDBGal1zXGTTcp7/s0YXhEQ=">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</latexit>Yjei ✓2 ⌧jx)H(0)(Yei ✓2 ⌧jx †)jすなわち各θにおけるHam.は本質的に同じもの．

25 /92

ガチャガチャ模型の特徴例) ガチャガチャ模型
H(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

X

(a†j

j

†
x
aj )⌧j+1 (aj+1 + aj+1 )2X⌧jz,✓1 (12jz,✓
2a†j aj )⌧j+1
2

・各θに対してuniqueかつgappedなHamiltonianである．
このようなHam.のG.S.をShort-Range Entangled(SRE)状態という．
・各θに対して適当なunitary変換を⾏うと，θ=0にできる：
H(✓) = (<latexit sha1_base64="ESwNmDBGal1zXGTTcp7/s0YXhEQ=">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</latexit>Yjei ✓2 ⌧jx)H(0)(Yei ✓2 ⌧jx †)jすなわち各θにおけるHam.は本質的に同じもの．
つまりpumpはunique gapped Ham.の族としての⾮⾃明性と関係する．
26 /92

ガチャガチャ模型の特徴例) ガチャガチャ模型
H(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

X

(a†j

j

†
x
aj )⌧j+1 (aj+1 + aj+1 )2X⌧jz,✓1 (12jz,✓
2a†j aj )⌧j+1
2

・各θに対してuniqueかつgappedなHamiltonianである．
このようなHam.のG.S.をShort-Range Entangled(SRE)状態という．
・各θに対して適当なunitary変換を⾏うと，θ=0にできる：
H(✓) = (<latexit sha1_base64="ESwNmDBGal1zXGTTcp7/s0YXhEQ=">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</latexit>Yjei ✓2 ⌧jx)H(0)(Yei ✓2 ⌧jx †)jすなわち各θにおけるHam.は本質的に同じもの．
つまりpumpはunique gapped Ham.の族としての⾮⾃明性と関係する．
(Kitaevの予想から⼀般的に期待されている．[Kitaev10])

27 /92一般的な問い
d+1次元のSRE状態の族に対して，族としての⾮⾃明さを測る量は何か？
H(x)<latexit sha1_base64="W6s9E2Oi1RWcJ3wlr8Pvo7zQEbA=">AAADHHichVLLSsNQED2Nr1pfrW4EN2KpKEi5qYLFleDGpa9qoRZJ4lWDeZHcFmvxB1y4VRAEBRfiZ+hCEJcKfoK4VHDjwkka8FFaJyR35tw5c8/cjOoYuicYe45ILa1t7R3RzlhXd09vXzzRv+rZJVfjOc02bDevKh43dIvnhC4MnndcrpiqwdfU3Tl/f63MXU+3rRVRcXjRVLYtfUvXFOFD82N74xvxJEuzwIbrHTl0kghtwU5EIljHJmxoKMEEhwVBvgEFHj0FyGBwCCuiSphLnh7scxwgRtwSZXHKUAjdpe82RYUQtSj2a3oBW6NTDHpdYg4jxR7ZFXtjd+yavbDPhrWqQQ1fS4VWtcblzkbf4eDyx78sk1aBnW9WU80CW8gGWnXS7gSI34VW45f3j9+WZ5ZS1VF2wV5J/zl7ZjfUgVV+1y4X+dJpEz0q9a1QzKlm45v7q3uirvvGTEE5DnkyoWbTzJ9/IoYYTY38d0bqndVMWp5MZxankrPZcH6iGMIIxujEacxiHgvIUe0dHOEYJ9KZdCvdSw+1VCkScgbwy6SnLxbyr+M=</latexit>

S2X<latexit sha1_base64="XWvRdIEi5U9fbrIYe+wxr+LAO+s=">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</latexit>

2

S X ：parameter空間<latexit sha1_base64="XWvRdIEi5U9fbrIYe+wxr+LAO+s=">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</latexit>

H(x)：d+1次元 unique gapped Ham.
<latexit sha1_base64="W6s9E2Oi1RWcJ3wlr8Pvo7zQEbA=">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</latexit>

28 /92

目次• 研究の背景
SRE状態とGTPについて• 研究の⽬的と⽅法
MPSを⽤いたinteracting fermionic systemにおけるGTPの検証• 結果１：Pump不変量の定義
fundamental theorem for fMPS
Pump不変量の定義• 結果２：Pump不変量の計算
non-trivial phaseにおけるpump
trivial phaseにおけるpump• まとめ
40 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法
X=S^1の場合の先⾏研究

41 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法
X=S^1の場合の先⾏研究
free fermion

自明相

非自明相

fermion parity[Teo-Kane]fermion parity[Teo-Kane]

42 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
free fermion

自明相

非自明相

fermion parity[Teo-Kane]fermion parity[Teo-Kane]

43 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
自明相

非自明相

free fermion

fermion parity[Teo-Kane]fermion parity[Teo-Kane]interacting fermion

未検証

未検証

44 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
自明相

非自明相

free fermion

fermion parity[Teo-Kane]fermion parity[Teo-Kane]interacting fermion

未検証

未検証

⽬的

45 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
自明相

非自明相

free fermion

fermion parity[Teo-Kane]fermion parity[Teo-Kane]interacting fermion

未検証

未検証

⽬的

※⼀般には難しいので次元を1+1次元に限定する

46 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
自明相

非自明相

free fermion

fermion parity[Teo-Kane]fermion parity[Teo-Kane]interacting fermion

未検証

未検証

⽬的

※⼀般には難しいので次元を1+1次元に限定する
⽬的：
1+1次元interacting fermion系において⾃明相/⾮⾃明相のS1族を分類．

47 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
自明相

非自明相

free fermion

fermion parity[Teo-Kane]fermion parity[Teo-Kane]interacting fermion

未検証

未検証

⽬的

※⼀般には難しいので次元を1+1次元に限定する
⽬的：
1+1次元interacting fermion系において⾃明相/⾮⾃明相のS1族を分類．
＝SRE状態のS^1 familyに対して不変量を構成したい．

48 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
自明相

非自明相

free fermion

fermion parity[Teo-Kane]fermion parity[Teo-Kane]interacting fermion

未検証

未検証

⽬的

※⼀般には難しいので次元を1+1次元に限定する
⽬的：
1+1次元interacting fermion系において⾃明相/⾮⾃明相のS1族を分類．
＝SRE状態のS^1 familyに対して不変量を構成したい．
⽅法：
Z/2-graded central simpleなfermionic Matrix Product Stateの利⽤．

49 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
自明相

非自明相

free fermion

fermion parity[Teo-Kane]fermion parity[Teo-Kane]interacting fermion

未検証

未検証

⽬的

※⼀般には難しいので次元を1+1次元に限定する
⽬的：
1+1次元interacting fermion系において⾃明相/⾮⾃明相のS1族を分類．
＝SRE状態のS^1 familyに対して不変量を構成したい．
⽅法：
Z/2-graded central simpleなfermionic Matrix Product Stateの利⽤．

50 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
自明相

非自明相

free fermion

fermion parity[Teo-Kane]fermion parity[Teo-Kane]interacting fermion

未検証

未検証

⽬的

※⼀般には難しいので次元を1+1次元に限定する
⽬的：
1+1次元interacting fermion系において⾃明相/⾮⾃明相のS1族を分類．
＝SRE状態のS^1 familyに対して不変量を構成したい．
⽅法： 次のスライドで説明
Z/2-graded central simpleなfermionic Matrix Product Stateの利⽤．

51 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
自明相

非自明相

free fermion

fermion parity[Teo-Kane]fermion parity[Teo-Kane]interacting fermion

未検証

未検証

⽬的

※⼀般には難しいので次元を1+1次元に限定する
⽬的：
1+1次元interacting fermion系において⾃明相/⾮⾃明相のS1族を分類．
＝SRE状態のS^1 familyに対して不変量を構成したい．
⽅法： 次のスライドで説明
Z/2-graded central simpleなfermionic Matrix Product Stateの利⽤．

52 /92

[J.C.Y.Teo,C.L.Kane,”Topologicaldefectsandgaplessmodesininsulatorsandsuperconductors” Phys. Rev. B 82, 115120 (2010).]

研究の目的と方法※⾃明相：product stateと連続的に繋がる
⾮⾃明相：⾃明相以外

X=S^1の場合の先⾏研究
自明相

非自明相

free fermion

fermion parity[Teo-Kane]fermion parity[Teo-Kane]interacting fermion

未検証

未検証

⽬的

※⼀般には難しいので次元を1+1次元に限定する
⽬的：
1+1次元interacting fermion系において⾃明相/⾮⾃明相のS1族を分類．
＝SRE状態のS^1 familyに対して不変量を構成したい．
⽅法： 次のスライドで説明
次の次のスライドで説明
Z/2-graded central simpleなfermionic Matrix Product Stateの利⽤．

53 /92

目次• 研究の背景
SRE状態とGTPについて• 研究の⽬的と⽅法
MPSを⽤いたinteracting fermionic systemにおけるGTPの検証• 結果１：Pump不変量の定義
fundamental theorem for fMPS
Pump不変量の定義• 結果２：Pump不変量の計算
non-trivial phaseにおけるpump
trivial phaseにおけるpump• まとめ
54 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛

55 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
①

②

56 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
Mat2n (C)① Z/2-gr. CSA A 'は以下の２パターンのどちらかに同型(Wallの定理)：<latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit>②

57 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
Mat2n (C)① Z/2-gr. CSA A 'は以下の２パターンのどちらかに同型(Wallの定理)：(＋)-type：<latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit>②

58 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
Mat2n (C)① Z/2-gr. CSA A 'は以下の２パターンのどちらかに同型(Wallの定理)：(＋)-type： A ' Mat2n (C)<latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit>②

59 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
Mat2n (C)① Z/2-gr. CSA A 'は以下の２パターンのどちらかに同型(Wallの定理)：(＋)-type： A ' Mat2n (C)(−)-type：<latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">AAADSHichVJLSyNBEP4yvuMruhfBixgUFyT0RGGXZQ8uXrwIPjYqGAk9Y0cb57UznYAO8wc8C3vwpOBB9h/sVQ/i3YM/QbwsKMjCHqxMBnQNiTXMdNXX9VV/1VOGZ8lAMXab0traOzq7unvSvX39A4OZoeG1wK34piiYruX6GwYPhCUdUVBSWWLD8wW3DUusG3vztf31qvAD6Trf1b4ntmy+48iyNLkiqJTJFW2udk1uhd+iYiBt8SMGfDtc5CoqhXknmooRwwjno4+lTJblWGxjjY6eOFkktuQOpVIoYhsuTFRgQ8CBIt8CR0DPJnQweIRtISTMJ0/G+wIR0sStUJagDE7oHn13KNpMUIfiWs0gZpt0ikWvT8wxTLAbds4e2BX7xe7Yv6a1wrhGTcs+rUadK7zS4OHI6tO7LJtWhd0XVkvNCmV8jrVK0u7FSK0Ls86vHvx8WP2yMhFOslN2T/pP2C27oA6c6qN5tixWjlvoMahvTrGgms1v7q3u6YbumzMV5Xjk6YTaLTNf/4k00jQ1+tsZaXTW8jl9Jpdfns3OfU3mpxujGMcUnfgJc1jAEgpU+wi/cYFL7Vr7oz1pf+upWirhfMB/1qY9AwuCweI=</latexit>②

60 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
Mat2n (C)① Z/2-gr. CSA A 'は以下の２パターンのどちらかに同型(Wallの定理)：(＋)-type： A ' Mat2n (C)(−)-type：A ' Matn (C) Matn (C)<latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">AAADSHichVJLSyNBEP4yvuMruhfBixgUFyT0RGGXZQ8uXrwIPjYqGAk9Y0cb57UznYAO8wc8C3vwpOBB9h/sVQ/i3YM/QbwsKMjCHqxMBnQNiTXMdNXX9VV/1VOGZ8lAMXab0traOzq7unvSvX39A4OZoeG1wK34piiYruX6GwYPhCUdUVBSWWLD8wW3DUusG3vztf31qvAD6Trf1b4ntmy+48iyNLkiqJTJFW2udk1uhd+iYiBt8SMGfDtc5CoqhXknmooRwwjno4+lTJblWGxjjY6eOFkktuQOpVIoYhsuTFRgQ8CBIt8CR0DPJnQweIRtISTMJ0/G+wIR0sStUJagDE7oHn13KNpMUIfiWs0gZpt0ikWvT8wxTLAbds4e2BX7xe7Yv6a1wrhGTcs+rUadK7zS4OHI6tO7LJtWhd0XVkvNCmV8jrVK0u7FSK0Ls86vHvx8WP2yMhFOslN2T/pP2C27oA6c6qN5tixWjlvoMahvTrGgms1v7q3u6YbumzMV5Xjk6YTaLTNf/4k00jQ1+tsZaXTW8jl9Jpdfns3OfU3mpxujGMcUnfgJc1jAEgpU+wi/cYFL7Vr7oz1pf+upWirhfMB/1qY9AwuCweI=</latexit><latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">AAADSHichVJLSyNBEP4yvuMruhfBixgUFyT0RGGXZQ8uXrwIPjYqGAk9Y0cb57UznYAO8wc8C3vwpOBB9h/sVQ/i3YM/QbwsKMjCHqxMBnQNiTXMdNXX9VV/1VOGZ8lAMXab0traOzq7unvSvX39A4OZoeG1wK34piiYruX6GwYPhCUdUVBSWWLD8wW3DUusG3vztf31qvAD6Trf1b4ntmy+48iyNLkiqJTJFW2udk1uhd+iYiBt8SMGfDtc5CoqhXknmooRwwjno4+lTJblWGxjjY6eOFkktuQOpVIoYhsuTFRgQ8CBIt8CR0DPJnQweIRtISTMJ0/G+wIR0sStUJagDE7oHn13KNpMUIfiWs0gZpt0ikWvT8wxTLAbds4e2BX7xe7Yv6a1wrhGTcs+rUadK7zS4OHI6tO7LJtWhd0XVkvNCmV8jrVK0u7FSK0Ls86vHvx8WP2yMhFOslN2T/pP2C27oA6c6qN5tixWjlvoMahvTrGgms1v7q3u6YbumzMV5Xjk6YTaLTNf/4k00jQ1+tsZaXTW8jl9Jpdfns3OfU3mpxujGMcUnfgJc1jAEgpU+wi/cYFL7Vr7oz1pf+upWirhfMB/1qY9AwuCweI=</latexit><latexit sha1_base64="5bcaUpGLIra+awIV/D3UEhwtthw=">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</latexit>②

61 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
Mat2n (C)① Z/2-gr. CSA A 'は以下の２パターンのどちらかに同型(Wallの定理)：(＋)-type： A ' Mat2n (C)(−)-type：A ' Matn (C) Matn (C)<latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="5bcaUpGLIra+awIV/D3UEhwtthw=">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</latexit>② Z/2-gr. CSAには u2 / 1 を満たす元 u が存在する．
<latexit sha1_base64="pGnpYw0VZ7vAOBwgMftJbmWFHug=">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</latexit><latexit sha1_base64="fLJS3pGyQUFo7BO2FbRJyxbXYJM=">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</latexit>

62 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
Mat2n (C)① Z/2-gr. CSA A 'は以下の２パターンのどちらかに同型(Wallの定理)：(＋)-type： A ' Mat2n (C)(−)-type：A ' Matn (C) Matn (C)<latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="5bcaUpGLIra+awIV/D3UEhwtthw=">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</latexit>② Z/2-gr. CSAには u2 / 1 を満たす元 u が存在する．
上の⾏列表⽰において u は以下の⾏列で与えられる：<latexit sha1_base64="pGnpYw0VZ7vAOBwgMftJbmWFHug=">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</latexit><latexit sha1_base64="fLJS3pGyQUFo7BO2FbRJyxbXYJM=">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</latexit><latexit sha1_base64="pGnpYw0VZ7vAOBwgMftJbmWFHug=">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</latexit>

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Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
Mat2n (C)① Z/2-gr. CSA A 'は以下の２パターンのどちらかに同型(Wallの定理)：(＋)-type： A ' Mat2n (C)(−)-type：A ' Matn (C) Matn (C)<latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="5bcaUpGLIra+awIV/D3UEhwtthw=">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</latexit>② Z/2-gr. CSAには u2 / 1 を満たす元 u が存在する．
上の⾏列表⽰において u は以下の⾏列で与えられる：✓◆
1n(＋)-type：u /1n<latexit sha1_base64="pGnpYw0VZ7vAOBwgMftJbmWFHug=">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</latexit><latexit sha1_base64="fLJS3pGyQUFo7BO2FbRJyxbXYJM=">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</latexit><latexit sha1_base64="pGnpYw0VZ7vAOBwgMftJbmWFHug=">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</latexit><latexit sha1_base64="h7K2UiRvLGIai1tIDSdNqWhxonM=">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</latexit>

64 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
Mat2n (C)① Z/2-gr. CSA A 'は以下の２パターンのどちらかに同型(Wallの定理)：(＋)-type： A ' Mat2n (C)(−)-type：A ' Matn (C) Matn (C)<latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="5bcaUpGLIra+awIV/D3UEhwtthw=">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</latexit>② Z/2-gr. CSAには u2 / 1 を満たす元 u が存在する．
上の⾏列表⽰において u は以下の⾏列で与えられる：
✓
◆
✓
◆
1n
1n
(−)-type： u /(＋)-type：u /1n1n<latexit sha1_base64="pGnpYw0VZ7vAOBwgMftJbmWFHug=">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</latexit><latexit sha1_base64="fLJS3pGyQUFo7BO2FbRJyxbXYJM=">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</latexit><latexit sha1_base64="pGnpYw0VZ7vAOBwgMftJbmWFHug=">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</latexit><latexit sha1_base64="itJovyQ5jIAamaO1gwOk0Asa5TI=">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</latexit><latexit sha1_base64="h7K2UiRvLGIai1tIDSdNqWhxonM=">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</latexit>

65 /92

Z/2-graded Central Simple Algebra(CSA)
Z/2-gr. CSAは抽象的に定義されるが今回のトークでは使わないので割愛
⼤事な点
Mat2n (C)① Z/2-gr. CSA A 'は以下の２パターンのどちらかに同型(Wallの定理)：(＋)-type： A ' Mat2n (C)(−)-type：A ' Matn (C) Matn (C)<latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="FvKWbuaXIG4Ht22Sy36TSBbatCM=">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</latexit><latexit sha1_base64="5bcaUpGLIra+awIV/D3UEhwtthw=">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</latexit>② Z/2-gr. CSAには u2 / 1 を満たす元 u が存在する．
上の⾏列表⽰において u は以下の⾏列で与えられる：
✓
◆
✓
◆
1n
1n
(−)-type： u /(＋)-type：u /1n1n<latexit sha1_base64="pGnpYw0VZ7vAOBwgMftJbmWFHug=">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</latexit><latexit sha1_base64="fLJS3pGyQUFo7BO2FbRJyxbXYJM=">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</latexit><latexit sha1_base64="pGnpYw0VZ7vAOBwgMftJbmWFHug=">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</latexit><latexit sha1_base64="itJovyQ5jIAamaO1gwOk0Asa5TI=">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</latexit><latexit sha1_base64="h7K2UiRvLGIai1tIDSdNqWhxonM=">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</latexit>

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N. Bultinck, et.al., Phys.Rev.B 95 (2017) 7, 075108.

fermionic Matrix Product State (fMPS)
Z/2-gr. central simple matrices： {Ai }： 2n ⇥ 2n ⾏列の組
<latexit sha1_base64="orUbvzFF9Okb/IE9+5GjkktRDik=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>

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N. Bultinck, et.al., Phys.Rev.B 95 (2017) 7, 075108.

fermionic Matrix Product State (fMPS)
Z/2-gr. central simple matrices： {Ai }： 2n ⇥ 2n ⾏列の組
<latexit sha1_base64="orUbvzFF9Okb/IE9+5GjkktRDik=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>

s.t. A = Span({Ai }) がZ/2-gr. CSA<latexit sha1_base64="mlJs6JiuBQ1gu4bjnzQEnKrYmeE=">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</latexit><latexit sha1_base64="c+G6mIg+1yeWka1WPbrs+ppjHh0=">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</latexit>

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N. Bultinck, et.al., Phys.Rev.B 95 (2017) 7, 075108.

fermionic Matrix Product State (fMPS)
Z/2-gr. central simple matrices： {Ai }： 2n ⇥ 2n ⾏列の組
<latexit sha1_base64="orUbvzFF9Okb/IE9+5GjkktRDik=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">AAADIXichVI7LwRRFP52vNebRqIRG6KQzR0kRBREo/Raj1hkZtzlxrwyc3cTJvsHtAoFCkQhfoaGTiQK8QtESaJRODs7icdm153M3HO+e75zvnPn6K4pfMnYU0ypqq6pratviDc2Nbe0trV3LPlO1jN4ynBMx1vRNZ+bwuYpKaTJV1yPa5Zu8mV9d7pwvpzjni8ce1HuuXzd0rZtkRGGJglaTQdTG4HIp/ObbQmWZOHqKTXUyEggWrNOeyyGNLbgwEAWFjhsSLJNaPDpWYMKBpewdQSEeWSJ8JwjjzhxsxTFKUIjdJe+2+StRahNfiGnH7INqmLS6xGzB33skV2xN3bLrtkL+yybKwhzFLTs0a4XudzdbD3oWvj4l2XRLrHzzaqoWSKDsVCrIO1uiBS6MIr83P7R28L4fF/Qz87ZK+k/Y0/shjqwc+/G5RyfP66gR6e+NfI55Sx/c391D5Z0X54pKcYlSyXUqhj580/EEaepUf/OSKmxNJRUh5NDcyOJyYlofurRjV4MUMVRTGIGs0iFtQ9xglPlQrlT7pWHYqgSizid+LWU5y/8C7K4</latexit>

s.t. A = Span({Ai }) がZ/2-gr. CSA with some technical conditions.
<latexit sha1_base64="mlJs6JiuBQ1gu4bjnzQEnKrYmeE=">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</latexit><latexit sha1_base64="c+G6mIg+1yeWka1WPbrs+ppjHh0=">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</latexit>

69 /92

N. Bultinck, et.al., Phys.Rev.B 95 (2017) 7, 075108.

fermionic Matrix Product State (fMPS)
Z/2-gr. central simple matrices： {Ai }： 2n ⇥ 2n ⾏列の組
<latexit sha1_base64="orUbvzFF9Okb/IE9+5GjkktRDik=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>

some technical conditions.
s.t. A = Span({Ai }) がZ/2-gr. CSA withX
<latexit sha1_base64="mlJs6JiuBQ1gu4bjnzQEnKrYmeE=">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</latexit><latexit sha1_base64="c+G6mIg+1yeWka1WPbrs+ppjHh0=">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</latexit>i|{A, u}i =
このとき次で状態を定める：<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit>{ik }tr(uAi1 · · · AiL ) |i1 , ..., iL i

70 /92

N. Bultinck, et.al., Phys.Rev.B 95 (2017) 7, 075108.

fermionic Matrix Product State (fMPS)
Z/2-gr. central simple matrices： {Ai }： 2n ⇥ 2n ⾏列の組
<latexit sha1_base64="orUbvzFF9Okb/IE9+5GjkktRDik=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>

some technical conditions.
s.t. A = Span({Ai }) がZ/2-gr. CSA withX
<latexit sha1_base64="mlJs6JiuBQ1gu4bjnzQEnKrYmeE=">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</latexit><latexit sha1_base64="c+G6mIg+1yeWka1WPbrs+ppjHh0=">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</latexit>

i
i1
iL
|{A
,
u}i
=
tr(uA
·
·
·
A
) |i1 , ..., iL i
このとき次で状態を定める：
Xi
i1iL {ik }=tr(uA SRE状態を与える· · · A ) |i1 , ..., iL i[Bultinck,et.al.,]
すると |{A , u}i はfermionic<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">AAADdXichVLLahRBFD0z7SOOj0zMRhBhcEyIMHSqo2AQAxE3Llzk4SSB6dh0dyqxmH7RXT0Qm/oBfyALVwouxK8QN/6Ai3yBhGyUCNm48PYD1AwzVtNd9557z61zq68TeSKRjB3W6tq58xcuTlxqXL5y9dpkc+r6RhKmscu7buiF8ZZjJ9wTAe9KIT2+FcXc9h2Pbzr9J3l8c8DjRITBc7kf8W3f3gvErnBtSZDV7Jl9LjMze/wiE6qTmkotmUnqWwQJK+srAjIz9lsyVnNpnmRlhlKmuxPKpFX6z5S6W1QpYh1d1zslajXbTGfFag0bRmW0Ua2VcKpWg4kdhHCRwgdHAEm2BxsJPT0YYIgI20ZGWEyWKOIcCg3ippTFKcMmtE/fPfJ6FRqQn9dMCrZLp3j0xsRsYYZ9ZR/YCfvCPrIj9mtkrayokWvZp90puTyyJl/fWD/9L8unXeLlH9ZYzRK7WCy0CtIeFUjehVvyB68OTtYfrs1ks+wdOyb9b9kh+0wdBIOf7vtVvvZmjB6H+rbJ51Rz9M2d1d0Z6n40U1JORJZBqD828+8/0UCDpsY4OyPDxsaCbtzTF1bvt5cfVfMzgZu4jTk68QGW8RQr6FLtT/iG7/hRP9VuaXe02TK1Xqs40/hnafO/AVz+0bQ=</latexit><latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit>{ik }71 /92

N. Bultinck, et.al., Phys.Rev.B 95 (2017) 7, 075108.

fermionic Matrix Product State (fMPS)
Z/2-gr. central simple matrices： {Ai }： 2n ⇥ 2n ⾏列の組
<latexit sha1_base64="orUbvzFF9Okb/IE9+5GjkktRDik=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>

some technical conditions.
s.t. A = Span({Ai }) がZ/2-gr. CSA withX
<latexit sha1_base64="mlJs6JiuBQ1gu4bjnzQEnKrYmeE=">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</latexit><latexit sha1_base64="c+G6mIg+1yeWka1WPbrs+ppjHh0=">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</latexit>

i
i1
iL
|{A
,
u}i
=
tr(uA
·
·
·
A
) |i1 , ..., iL i
このとき次で状態を定める：
Xi
i1iL {ik }=tr(uA SRE状態を与える· · · A ) |i1 , ..., iL i[Bultinck,et.al.,]
すると |{A , u}i はfermionic
X {ik }iL
i
1|{Ai , u}i ：fermionic=tr(uAiSRE
···A
)
|i
,
...,
i
{A
state 1⇄ L}i：Z/2-gr. CSA<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit><latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit>{ik }72 /92

N. Bultinck, et.al., Phys.Rev.B 95 (2017) 7, 075108.

fermionic Matrix Product State (fMPS)
Z/2-gr. central simple matrices： {Ai }： 2n ⇥ 2n ⾏列の組
<latexit sha1_base64="orUbvzFF9Okb/IE9+5GjkktRDik=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>

some technical conditions.
s.t. A = Span({Ai }) がZ/2-gr. CSA withX
<latexit sha1_base64="mlJs6JiuBQ1gu4bjnzQEnKrYmeE=">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</latexit><latexit sha1_base64="c+G6mIg+1yeWka1WPbrs+ppjHh0=">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</latexit>

i
i1
iL
|{A
,
u}i
=
tr(uA
·
·
·
A
) |i1 , ..., iL i
このとき次で状態を定める：
Xi
i1iL {ik }=tr(uA SRE状態を与える· · · A ) |i1 , ..., iL i[Bultinck,et.al.,]
すると |{A , u}i はfermionic
X {ik }iL
i
1|{Ai , u}i ：fermionic=tr(uAiSRE
···A
)
|i
,
...,
i
{A
state 1⇄ L}i：Z/2-gr. CSA<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>{ik }例)periodic Kitaev chain：H =<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="aQodVYA8SB7AVEu9nyH8oTSGdOo=">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</latexit>Xa†j+1 aj+1

a†j+1 aj + a†j a†j+1 + aj+1 aj

j

73 /92

N. Bultinck, et.al., Phys.Rev.B 95 (2017) 7, 075108.

fermionic Matrix Product State (fMPS)
Z/2-gr. central simple matrices： {Ai }： 2n ⇥ 2n ⾏列の組
<latexit sha1_base64="orUbvzFF9Okb/IE9+5GjkktRDik=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>

some technical conditions.
s.t. A = Span({Ai }) がZ/2-gr. CSA withX
<latexit sha1_base64="mlJs6JiuBQ1gu4bjnzQEnKrYmeE=">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</latexit><latexit sha1_base64="c+G6mIg+1yeWka1WPbrs+ppjHh0=">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</latexit>

i
i1
iL
|{A
,
u}i
=
tr(uA
·
·
·
A
) |i1 , ..., iL i
このとき次で状態を定める：
Xi
i1iL {ik }=tr(uA SRE状態を与える· · · A ) |i1 , ..., iL i[Bultinck,et.al.,]
すると |{A , u}i はfermionic
X {ik }iL
i
1|{Ai , u}i ：fermionic=tr(uAiSRE
···A
)
|i
,
...,
i
{A
state 1⇄ L}i：Z/2-gr. CSA<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">AAADdXichVLLahRBFD0z7SOOj0zMRhBhcEyIMHSqo2AQAxE3Llzk4SSB6dh0dyqxmH7RXT0Qm/oBfyALVwouxK8QN/6Ai3yBhGyUCNm48PYD1AwzVtNd9557z61zq68TeSKRjB3W6tq58xcuTlxqXL5y9dpkc+r6RhKmscu7buiF8ZZjJ9wTAe9KIT2+FcXc9h2Pbzr9J3l8c8DjRITBc7kf8W3f3gvErnBtSZDV7Jl9LjMze/wiE6qTmkotmUnqWwQJK+srAjIz9lsyVnNpnmRlhlKmuxPKpFX6z5S6W1QpYh1d1zslajXbTGfFag0bRmW0Ua2VcKpWg4kdhHCRwgdHAEm2BxsJPT0YYIgI20ZGWEyWKOIcCg3ippTFKcMmtE/fPfJ6FRqQn9dMCrZLp3j0xsRsYYZ9ZR/YCfvCPrIj9mtkrayokWvZp90puTyyJl/fWD/9L8unXeLlH9ZYzRK7WCy0CtIeFUjehVvyB68OTtYfrs1ks+wdOyb9b9kh+0wdBIOf7vtVvvZmjB6H+rbJ51Rz9M2d1d0Z6n40U1JORJZBqD828+8/0UCDpsY4OyPDxsaCbtzTF1bvt5cfVfMzgZu4jTk68QGW8RQr6FLtT/iG7/hRP9VuaXe02TK1Xqs40/hnafO/AVz+0bQ=</latexit><latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>

{ik }

X

a†j+1 aj+1 a†j+1 aj + a†j a†j+1 + aj+1 aj
✓
✓
◆
✓
◆
j
1
1
1
0
u=
A
=
A
=
このとき⾏列の組として
111例)periodic Kitaev chain：H =<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="aQodVYA8SB7AVEu9nyH8oTSGdOo=">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</latexit>

X=
tr(uAi1 · · · AiL ) |i1 , ..., iL iとすると |{Ai , u}i は基底状態を与える．
<latexit sha1_base64="mqMsomwpqQZnnJ9m8+uxruqZr9A=">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</latexit><latexit sha1_base64="xZDC0y4l+BERXJ9jZUMeKN4Pl84=">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</latexit><latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit>{ik }<latexit sha1_base64="B9oYSF7LlENh3gJ99plhmmRqd18=">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</latexit>◆
1

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N. Bultinck, et.al., Phys.Rev.B 95 (2017) 7, 075108.

fermionic Matrix Product State (fMPS)
Z/2-gr. central simple matrices： {Ai }： 2n ⇥ 2n ⾏列の組
<latexit sha1_base64="orUbvzFF9Okb/IE9+5GjkktRDik=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>

some technical conditions.
s.t. A = Span({Ai }) がZ/2-gr. CSA withX
<latexit sha1_base64="mlJs6JiuBQ1gu4bjnzQEnKrYmeE=">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</latexit><latexit sha1_base64="c+G6mIg+1yeWka1WPbrs+ppjHh0=">AAADO3ichVK/SxxBFP5uo8ZcjJ6xCdicORQFOWZNQAkGFJuU6uVUcM0xuxl1cH+xO3dglq0D+QdSWClYhJBWKysb/wELayuTdCppUvhub8Efx51v2Z33vnnfm+/NPtO3ZagYO8toTzo6u552P8s+73nR25frf7kUetXAEmXLs71gxeShsKUrykoqW6z4geCOaYtlc2uuvr9cE0EoPfej2vbFmsM3XLkuLa4IquSGDIerTYvb0Wz8PjICJ1/yuRuPGtHsp0jGRjxWyRVYkSWWb3b01CkgtXmvP5OBgc/wYKEKBwIuFPk2OEJ6VqGDwSdsDRFhAXky2ReIkSVulbIEZXBCt+i7QdFqiroU12uGCduiU2x6A2LmMcxO2Q92yU7YT3bB/resFSU16lq2aTUbXOFX+r69Kv17lOXQqrB5y2qrWWEdU4lWSdr9BKl3YTX4tS/fL0vvFoejEbbH/pD+XXbGjqkDt3Zl7S+IxZ02ekzqm1MsqGbrm3uoe7yp+9ZMRTk+eTqhTtvMu38iiyxNjf5wRpqdpYmi/qY4sfC2MDOdzk83BvEao3TiJGbwAfMoU+2v+IUDHGpH2rn2W/vbSNUyKWcA90y7vgH75ry7</latexit>

i
i1
iL
|{A
,
u}i
=
tr(uA
·
·
·
A
) |i1 , ..., iL i
このとき次で状態を定める：
Xi
i1iL {ik }=tr(uA SRE状態を与える· · · A ) |i1 , ..., iL i[Bultinck,et.al.,]
すると |{A , u}i はfermionic
X {ik }iL
i
1|{Ai , u}i ：fermionic=tr(uAiSRE
···A
)
|i
,
...,
i
{A
state 1⇄ L}i：Z/2-gr. CSA<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">AAADIXichVI7LwRRFP52vNebRqIRG6KQzR0kRBREo/Raj1hkZtzlxrwyc3cTJvsHtAoFCkQhfoaGTiQK8QtESaJRODs7icdm153M3HO+e75zvnPn6K4pfMnYU0ypqq6pratviDc2Nbe0trV3LPlO1jN4ynBMx1vRNZ+bwuYpKaTJV1yPa5Zu8mV9d7pwvpzjni8ce1HuuXzd0rZtkRGGJglaTQdTG4HIp/ObbQmWZOHqKTXUyEggWrNOeyyGNLbgwEAWFjhsSLJNaPDpWYMKBpewdQSEeWSJ8JwjjzhxsxTFKUIjdJe+2+StRahNfiGnH7INqmLS6xGzB33skV2xN3bLrtkL+yybKwhzFLTs0a4XudzdbD3oWvj4l2XRLrHzzaqoWSKDsVCrIO1uiBS6MIr83P7R28L4fF/Qz87ZK+k/Y0/shjqwc+/G5RyfP66gR6e+NfI55Sx/c391D5Z0X54pKcYlSyXUqhj580/EEaepUf/OSKmxNJRUh5NDcyOJyYlofurRjV4MUMVRTGIGs0iFtQ9xglPlQrlT7pWHYqgSizid+LWU5y/8C7K4</latexit>

{ik }

X

a†j+1 aj+1 a†j+1 aj + a†j a†j+1 + aj+1 aj
✓
✓
◆
✓
◆
j
1
1
1
0
u=
A
=
A
=
このとき⾏列の組として
111例)periodic Kitaev chain：H =<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="aQodVYA8SB7AVEu9nyH8oTSGdOo=">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</latexit>

X=
tr(uAi1 · · · AiL ) |i1 , ..., iL iとすると |{Ai , u}i は基底状態を与える．
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1

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N. Bultinck, et.al., Phys.Rev.B 95 (2017) 7, 075108.

fermionic Matrix Product State (fMPS)
Z/2-gr. central simple matrices： {Ai }： 2n ⇥ 2n ⾏列の組
<latexit sha1_base64="orUbvzFF9Okb/IE9+5GjkktRDik=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">AAADIXichVI7LwRRFP52vNebRqIRG6KQzR0kRBREo/Raj1hkZtzlxrwyc3cTJvsHtAoFCkQhfoaGTiQK8QtESaJRODs7icdm153M3HO+e75zvnPn6K4pfMnYU0ypqq6pratviDc2Nbe0trV3LPlO1jN4ynBMx1vRNZ+bwuYpKaTJV1yPa5Zu8mV9d7pwvpzjni8ce1HuuXzd0rZtkRGGJglaTQdTG4HIp/ObbQmWZOHqKTXUyEggWrNOeyyGNLbgwEAWFjhsSLJNaPDpWYMKBpewdQSEeWSJ8JwjjzhxsxTFKUIjdJe+2+StRahNfiGnH7INqmLS6xGzB33skV2xN3bLrtkL+yybKwhzFLTs0a4XudzdbD3oWvj4l2XRLrHzzaqoWSKDsVCrIO1uiBS6MIr83P7R28L4fF/Qz87ZK+k/Y0/shjqwc+/G5RyfP66gR6e+NfI55Sx/c391D5Z0X54pKcYlSyXUqhj580/EEaepUf/OSKmxNJRUh5NDcyOJyYlofurRjV4MUMVRTGIGs0iFtQ9xglPlQrlT7pWHYqgSizid+LWU5y/8C7K4</latexit>

some technical conditions.
s.t. A = Span({Ai }) がZ/2-gr. CSA withX
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i
i1
iL
|{A
,
u}i
=
tr(uA
·
·
·
A
) |i1 , ..., iL i
このとき次で状態を定める：
Xi
i1iL {ik }=tr(uA SRE状態を与える· · · A ) |i1 , ..., iL i[Bultinck,et.al.,]
すると |{A , u}i はfermionic
X {ik }iL
i
1|{Ai , u}i ：fermionic=tr(uAiSRE
···A
)
|i
,
...,
i
{A
state 1⇄ L}i：Z/2-gr. CSA<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">AAADdXichVLLahRBFD0z7SOOj0zMRhBhcEyIMHSqo2AQAxE3Llzk4SSB6dh0dyqxmH7RXT0Qm/oBfyALVwouxK8QN/6Ai3yBhGyUCNm48PYD1AwzVtNd9557z61zq68TeSKRjB3W6tq58xcuTlxqXL5y9dpkc+r6RhKmscu7buiF8ZZjJ9wTAe9KIT2+FcXc9h2Pbzr9J3l8c8DjRITBc7kf8W3f3gvErnBtSZDV7Jl9LjMze/wiE6qTmkotmUnqWwQJK+srAjIz9lsyVnNpnmRlhlKmuxPKpFX6z5S6W1QpYh1d1zslajXbTGfFag0bRmW0Ua2VcKpWg4kdhHCRwgdHAEm2BxsJPT0YYIgI20ZGWEyWKOIcCg3ippTFKcMmtE/fPfJ6FRqQn9dMCrZLp3j0xsRsYYZ9ZR/YCfvCPrIj9mtkrayokWvZp90puTyyJl/fWD/9L8unXeLlH9ZYzRK7WCy0CtIeFUjehVvyB68OTtYfrs1ks+wdOyb9b9kh+0wdBIOf7vtVvvZmjB6H+rbJ51Rz9M2d1d0Z6n40U1JORJZBqD828+8/0UCDpsY4OyPDxsaCbtzTF1bvt5cfVfMzgZu4jTk68QGW8RQr6FLtT/iG7/hRP9VuaXe02TK1Xqs40/hnafO/AVz+0bQ=</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">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</latexit>

{ik }

X

a†j+1 aj+1 a†j+1 aj + a†j a†j+1 + aj+1 aj
✓
✓
◆
✓
◆
j
1
1
1
0
u=
A
=
A
=
このとき⾏列の組として
111例)periodic Kitaev chain：H =<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">AAADdXichVLLahRBFD0z7SOOj0zMRhBhcEyIMHSqo2AQAxE3Llzk4SSB6dh0dyqxmH7RXT0Qm/oBfyALVwouxK8QN/6Ai3yBhGyUCNm48PYD1AwzVtNd9557z61zq68TeSKRjB3W6tq58xcuTlxqXL5y9dpkc+r6RhKmscu7buiF8ZZjJ9wTAe9KIT2+FcXc9h2Pbzr9J3l8c8DjRITBc7kf8W3f3gvErnBtSZDV7Jl9LjMze/wiE6qTmkotmUnqWwQJK+srAjIz9lsyVnNpnmRlhlKmuxPKpFX6z5S6W1QpYh1d1zslajXbTGfFag0bRmW0Ua2VcKpWg4kdhHCRwgdHAEm2BxsJPT0YYIgI20ZGWEyWKOIcCg3ippTFKcMmtE/fPfJ6FRqQn9dMCrZLp3j0xsRsYYZ9ZR/YCfvCPrIj9mtkrayokWvZp90puTyyJl/fWD/9L8unXeLlH9ZYzRK7WCy0CtIeFUjehVvyB68OTtYfrs1ks+wdOyb9b9kh+0wdBIOf7vtVvvZmjB6H+rbJ51Rz9M2d1d0Z6n40U1JORJZBqD828+8/0UCDpsY4OyPDxsaCbtzTF1bvt5cfVfMzgZu4jTk68QGW8RQr6FLtT/iG7/hRP9VuaXe02TK1Xqs40/hnafO/AVz+0bQ=</latexit><latexit sha1_base64="aQodVYA8SB7AVEu9nyH8oTSGdOo=">AAADgnichVLLSiNBFD2xfcZX1M2AG5mgiNFQrYIyOCC4cekrKhgN1W0ZW/tFdyXgNP0D/oALVwouxM9w48qdCz9BZqngxhm86TT4iInVdNe9p865dW51aa5p+JKx+0ST0tzS2tbekezs6u7pTfX1r/tOydNFTndMx9vUuC9MwxY5aUhTbLqe4JZmig3tcKGyvlEWnm849po8csW2xYu2sWfoXBJUSO0v/s77JasQHIQTfCfI7/JiUXgh5Rk15NXp64Uw8xEOa1gZ/o5dSKVZlkVjqDZQ4yCNeCw5fYkE8tiFAx0lWBCwISk2weHTswUVDC5h2wgI8ygyonWBEEnSlogliMEJPaRvkbKtGLUpr9T0I7VOu5j0eqQcwjC7Y5fskd2wK/bAXurWCqIaFS9HNGtVrXALvcc/Vp+/VVk0S+y/qRp6ltjDbOTVIO9uhFS60Kv68p+Tx9VfK8PBCDtnf8n/Gbtn19SBXX7SL5bFymkDPxr1zSkXVLP+yX32PV7TfX2lJI5LkUqo1ZD5/k8kkaRbo36+I7XB+mRWncpOLk+n52fj+9OOQfzEKO04g3ksYgk5qn2LJ/zDf6VZGVNUZapKbUrEmgF8GMrcK4AI1eM=</latexit>

X=
tr(uAi1 · · · AiL ) |i1 , ..., iL iとすると |{Ai , u}i は基底状態を与える．

<latexit sha1_base64="B9oYSF7LlENh3gJ99plhmmRqd18=">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</latexit><latexit sha1_base64="mqMsomwpqQZnnJ9m8+uxruqZr9A=">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</latexit><latexit sha1_base64="xZDC0y4l+BERXJ9jZUMeKN4Pl84=">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</latexit>{ik }{Ai } の⽣成する代数 A は= Span({AA ' C i })C よりZ/2-gr. CSAでtype(−).
<latexit sha1_base64="3L8mYysUoon9ssqv9bg3JAr+Rd4=">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</latexit><latexit sha1_base64="2q0EvUGsAs3gXo0VhJgQi66Q0Zo=">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</latexit><latexit sha1_base64="YAK8EznmEtC9Dfe87tQBS3RMIbE=">AAADIXichVI7LwRRFP52vNebRqIRG6KQzR0kRBREo/Raj1hkZtzlxrwyc3cTJvsHtAoFCkQhfoaGTiQK8QtESaJRODs7icdm153M3HO+e75zvnPn6K4pfMnYU0ypqq6pratviDc2Nbe0trV3LPlO1jN4ynBMx1vRNZ+bwuYpKaTJV1yPa5Zu8mV9d7pwvpzjni8ce1HuuXzd0rZtkRGGJglaTQdTG4HIp/ObbQmWZOHqKTXUyEggWrNOeyyGNLbgwEAWFjhsSLJNaPDpWYMKBpewdQSEeWSJ8JwjjzhxsxTFKUIjdJe+2+StRahNfiGnH7INqmLS6xGzB33skV2xN3bLrtkL+yybKwhzFLTs0a4XudzdbD3oWvj4l2XRLrHzzaqoWSKDsVCrIO1uiBS6MIr83P7R28L4fF/Qz87ZK+k/Y0/shjqwc+/G5RyfP66gR6e+NfI55Sx/c391D5Z0X54pKcYlSyXUqhj580/EEaepUf/OSKmxNJRUh5NDcyOJyYlofurRjV4MUMVRTGIGs0iFtQ9xglPlQrlT7pWHYqgSizid+LWU5y/8C7K4</latexit><latexit sha1_base64="c+G6mIg+1yeWka1WPbrs+ppjHh0=">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</latexit>◆
1

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目次• 研究の背景
SRE状態とGTPについて• 研究の⽬的と⽅法
MPSを⽤いたinteracting fermionic systemにおけるGTPの検証• 結果１：Pump不変量の定義
fundamental theorem for fMPS
Pump不変量の定義• 結果２：Pump不変量の計算
non-trivial phaseにおけるpump
trivial phaseにおけるpump• まとめ
77 /92

不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

78 /92

不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

0+1次元系におけるBerry位相：

79 /92

不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

0+1次元系におけるBerry位相：
ファイバー

C ：状態の位相#L#
X ：parameter空間<latexit sha1_base64="+/csAw7twFUG78CcVMfyF/ASVfc=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

全空間<latexit sha1_base64="O3bTm2KaIGDpz3gEtK7P8HscJH8=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

低空間<latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit>

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不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

0+1次元系におけるBerry位相：

C ：状態の位相#L
全空間#
X ：parameter空間
低空間
Berry接続/曲率 = L のtopology
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81 /92

不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

0+1次元系におけるBerry位相：

1+1次元系におけるPump不変量：

C ：状態の位相#L
全空間#
X ：parameter空間
低空間
Berry接続/曲率 = L のtopology
ファイバー<latexit sha1_base64="+/csAw7twFUG78CcVMfyF/ASVfc=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit><latexit sha1_base64="O3bTm2KaIGDpz3gEtK7P8HscJH8=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit><latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit><latexit sha1_base64="O3bTm2KaIGDpz3gEtK7P8HscJH8=">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</latexit>

82 /92

不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

0+1次元系におけるBerry位相：

1+1次元系におけるPump不変量：

C ：状態の位相#L
全空間#
X ：parameter空間
低空間
Berry接続/曲率 = L のtopology

ファイバー

ファイバー

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全空間<latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit><latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit>

A ：Z/2-gr. CSA#A#
X ：parameter空間<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

低空間<latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit><latexit sha1_base64="O3bTm2KaIGDpz3gEtK7P8HscJH8=">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</latexit>

83 /92

不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

0+1次元系におけるBerry位相：

1+1次元系におけるPump不変量：

C ：状態の位相#L
全空間#
X ：parameter空間
低空間
Berry接続/曲率 = L のtopology

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全空間<latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit><latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit>

A ：Z/2-gr. CSA#A#
X ：parameter空間<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

低空間<latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit>

Pump不変量 = A のtopology<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit>

84 /92

不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

0+1次元系におけるBerry位相：

1+1次元系におけるPump不変量：

C ：状態の位相#L
全空間#
X ：parameter空間
低空間
Berry接続/曲率 = L のtopology

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ファイバー

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A の変換関数は何か？

<latexit sha1_base64="8FZL1cvikCziplX+4+LKl3rtA4w=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

全空間<latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit><latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">AAADGXichVK7SgNRED1Z3/GtjWAjBsVCwt0oKFaijWUe5gEqsrve6JJ9sXsT0OAPaKtYpFKwEL9CbMRWLPIJYhnBxsLJZkFNSJxl986cO2fumbujOobuCcaqIamru6e3r38gPDg0PDI6Nj6R8eyiq/G0Zhu2m1MVjxu6xdNCFwbPOS5XTNXgWbWwWd/Plrjr6ba1LY4dvmcqh5ae1zVFEJTI7Y9FWJT5NtPqyIETQWBxezwUwi4OYENDESY4LAjyDSjw6NmBDAaHsD2UCXPJ0/19jlOEiVukLE4ZCqEF+h5StBOgFsX1mp7P1ugUg16XmDOYY6/sjtXYE7tnb+yrba2yX6Ou5ZhWtcHlzv7o2VTq81+WSavA0Q+ro2aBPFZ9rTppd3yk3oXW4JdOrmqpteRceZ7dsHfSf82q7JE6sEof2m2CJysd9KjUt0Ixp5rtb65Z92JL9+2ZgnIc8mRCzY6Zv/9EGGGaGrl5RlqdTCwqL0VjieXI+kYwP/2YxiwW6MQVrGMLcaSpNsc5LnApVaQH6Ul6bqRKoYAziT8mvXwDsRWvFg==</latexit>

A ：Z/2-gr. CSA#A#
X ：parameter空間<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

低空間<latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit>

Pump不変量 = A のtopology<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit><latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit>

85 /92

不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

0+1次元系におけるBerry位相：

1+1次元系におけるPump不変量：

C ：状態の位相#L
全空間#
X ：parameter空間
低空間
Berry接続/曲率 = L のtopology

ファイバー

ファイバー

<latexit sha1_base64="+/csAw7twFUG78CcVMfyF/ASVfc=">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</latexit>

A ：Z/2-gr. CSA#A#
X ：parameter空間<latexit sha1_base64="8FZL1cvikCziplX+4+LKl3rtA4w=">AAADGXichVI9SwNBEH05v2L8jDaCTTAoFhL2VFCsojaWxphEUAl356qH98XdJhCDf0BbxcJKwUL8FWIjtmKRnyCWEWwsnFwO1ITEOe525u282Td7ozqG7gnGKiGpo7OruyfcG+nrHxgcGo6OZD274Go8o9mG7W6piscN3eIZoQuDbzkuV0zV4Dn1aLW2nyty19Nta1OUHL5rKgeWvq9riiAotZwfjrME8y3W7MiBE0dg63Y0FMIO9mBDQwEmOCwI8g0o8OjZhgwGh7BdlAlzydP9fY4TRIhboCxOGQqhR/Q9oGg7QC2KazU9n63RKQa9LjFjmGSv7I5V2RO7Z2/sq2Wtsl+jpqVEq1rncic/dDqW/vyXZdIqcPjDaqtZYB+LvladtDs+UutCq/OLx5fV9NLGZHmK3bB30n/NKuyROrCKH9ptim9ctdGjUt8KxZxqtr65Rt0zTd23ZgrKcciTCTXbZv7+ExFEaGrkxhlpdrKzCXkuMZuajydXgvkJYxwTmKYTF5DEGtaRodocZzjHhXQlPUhP0nM9VQoFnFH8MenlG3LSrv8=</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit><latexit sha1_base64="O3bTm2KaIGDpz3gEtK7P8HscJH8=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">AAADInichVK7SgNRED1ZXzE+o41gIwbFQsJdFRQr0cYyPmKCGsLu5qqL+2L3JkGDP2AtWIiCooX4GTZiJ2Ih+ANiqWBj4WSzoCYkzrJ7Z86dM/fM3VEdQ/cEY88hqam5pbUt3B7p6Ozq7umN9q15dt7VeFKzDdtNq4rHDd3iSaELg6cdlyumavCUurtQ3k8VuOvptrUq9hyeMZVtS9/SNUUQtL6Zs4uW4rp2MdsbY3Hm21CtIwdODIEl7GgohE3kYENDHiY4LAjyDSjw6NmADAaHsAxKhLnk6f4+xwEixM1TFqcMhdBd+m5TtBGgFsXlmp7P1ugUg16XmEMYYU/smr2zO3bDXtlX3Volv0ZZyx6taoXLnWzP4cDK578sk1aBnR9WQ80CW5jxteqk3fGRchdahV/YP35fmV0eKY2yC/ZG+s/ZM7ulDqzCh3a1xJdPGuhRqW+FYk41699cte7xmu7rMwXlOOTJhJoNM3//iQgiNDVy9YzUOmsTcXkyPrE0FZubD+YnjEEMY4xOnMYcFpFAkmpbOMIpzqRL6V56kB4rqVIo4PTjj0kv36ujs1c=</latexit>

全空間<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit><latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit><latexit sha1_base64="O3bTm2KaIGDpz3gEtK7P8HscJH8=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

低空間<latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">AAADGXichVK7SgNRED1Z3/GtjWAjBsVCwt0oKFaijWUe5gEqsrve6JJ9sXsT0OAPaKtYpFKwEL9CbMRWLPIJYhnBxsLJZkFNSJxl986cO2fumbujOobuCcaqIamru6e3r38gPDg0PDI6Nj6R8eyiq/G0Zhu2m1MVjxu6xdNCFwbPOS5XTNXgWbWwWd/Plrjr6ba1LY4dvmcqh5ae1zVFEJTI7Y9FWJT5NtPqyIETQWBxezwUwi4OYENDESY4LAjyDSjw6NmBDAaHsD2UCXPJ0/19jlOEiVukLE4ZCqEF+h5StBOgFsX1mp7P1ugUg16XmDOYY6/sjtXYE7tnb+yrba2yX6Ou5ZhWtcHlzv7o2VTq81+WSavA0Q+ro2aBPFZ9rTppd3yk3oXW4JdOrmqpteRceZ7dsHfSf82q7JE6sEof2m2CJysd9KjUt0Ixp5rtb65Z92JL9+2ZgnIc8mRCzY6Zv/9EGGGaGrl5RlqdTCwqL0VjieXI+kYwP/2YxiwW6MQVrGMLcaSpNsc5LnApVaQH6Ul6bqRKoYAziT8mvXwDsRWvFg==</latexit>

Pump不変量 = A のtopology<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit>

0i
i
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0i 0
{A
} の関係は？
{A
}
,
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u}i
/
|{A
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A の変換関数は何か？
の時と<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit><latexit sha1_base64="YEvL6GzZ9XXCWOWoq6cEWzIRL4o=">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</latexit><latexit sha1_base64="rJV2lAqxKXEdBrvvR7Mi0jMcMLI=">AAADIXichVI7S8RAEP4uvs+3NoKNeCgWcmxUUKx8NJa+zgeeShL3dDEvkr0DDfcHbC0s1ELFQvwZNtqJYCH+ArFUsLFwLhfwcdy5IdmZb+eb+WYzumsKXzL2FFOqqmtq6+ob4o1NzS2tbe0dS76T9QyeMhzT8VZ0zeemsHlKCmnyFdfjmqWbfFnfnS6cL+e45wvHXpR7Ll+3tG1bZIShSYJW08HkRiDy6fxmW4IlWbh6Sg01MhKI1qzTHoshjS04MJCFBQ4bkmwTGnx61qCCwSVsHQFhHlkiPOfII07cLEVxitAI3aXvNnlrEWqTX8jph2yDqpj0esTsQR97ZFfsjd2ya/bCPsvmCsIcBS17tOtFLnc3Ww+6Fj7+ZVm0S+x8sypqlshgLNQqSLsbIoUujCI/t3/0tjA+3xf0s3P2SvrP2BO7oQ7s3LtxOcfnjyvo0alvjXxOOcvf3F/dgyXdl2dKinHJUgm1Kkb+/BNxxGlq1L8zUmosDSXV4eTQ3EhiYiqan3p0oxcDVHEUE5jBLFJh7UOc4FS5UO6Ue+WhGKrEIk4nfi3l+Qv92bK+</latexit><latexit sha1_base64="oxZk14Y3FLQIaA2A9cczqDAptgU=">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</latexit>

86 /92

不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

0+1次元系におけるBerry位相：

1+1次元系におけるPump不変量：

C ：状態の位相#L
全空間#
X ：parameter空間
低空間
Berry接続/曲率 = L のtopology

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ファイバー

<latexit sha1_base64="+/csAw7twFUG78CcVMfyF/ASVfc=">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</latexit>

A ：Z/2-gr. CSA#A#
X ：parameter空間<latexit sha1_base64="8FZL1cvikCziplX+4+LKl3rtA4w=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit><latexit sha1_base64="O3bTm2KaIGDpz3gEtK7P8HscJH8=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

全空間<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit><latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit><latexit sha1_base64="O3bTm2KaIGDpz3gEtK7P8HscJH8=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

低空間<latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">AAADGXichVK7SgNRED1Z3/GtjWAjBsVCwt0oKFaijWUe5gEqsrve6JJ9sXsT0OAPaKtYpFKwEL9CbMRWLPIJYhnBxsLJZkFNSJxl986cO2fumbujOobuCcaqIamru6e3r38gPDg0PDI6Nj6R8eyiq/G0Zhu2m1MVjxu6xdNCFwbPOS5XTNXgWbWwWd/Plrjr6ba1LY4dvmcqh5ae1zVFEJTI7Y9FWJT5NtPqyIETQWBxezwUwi4OYENDESY4LAjyDSjw6NmBDAaHsD2UCXPJ0/19jlOEiVukLE4ZCqEF+h5StBOgFsX1mp7P1ugUg16XmDOYY6/sjtXYE7tnb+yrba2yX6Ou5ZhWtcHlzv7o2VTq81+WSavA0Q+ro2aBPFZ9rTppd3yk3oXW4JdOrmqpteRceZ7dsHfSf82q7JE6sEof2m2CJysd9KjUt0Ixp5rtb65Z92JL9+2ZgnIc8mRCzY6Zv/9EGGGaGrl5RlqdTCwqL0VjieXI+kYwP/2YxiwW6MQVrGMLcaSpNsc5LnApVaQH6Ul6bqRKoYAziT8mvXwDsRWvFg==</latexit>

Pump不変量 = A のtopology<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit>

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A の変換関数は何か？
の時と<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit><latexit sha1_base64="YEvL6GzZ9XXCWOWoq6cEWzIRL4o=">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</latexit><latexit sha1_base64="rJV2lAqxKXEdBrvvR7Mi0jMcMLI=">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</latexit><latexit sha1_base64="oxZk14Y3FLQIaA2A9cczqDAptgU=">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</latexit>⇒fermionic MPSのfundamental theoremを証明する必要がある． 87 /92

fundamental thm. for Z/2-gr. CS MPS

88 /92

fundamental thm. for Z/2-gr. CS MPS

変換関数がZ/2-gradingを保つunitary変換のみであることを証明した．

89 /92

fundamental thm. for Z/2-gr. CS MPS

変換関数がZ/2-gradingを保つunitary変換のみであることを証明した．
不変量の具体的な計算法(解析的な⽅法)：

90 /92

fundamental thm. for Z/2-gr. CS MPS

変換関数がZ/2-gradingを保つunitary変換のみであることを証明した．
不変量の具体的な計算法(解析的な⽅法)：
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)
2
{H(✓)}✓2[0,2⇡] ! {A (✓), u(✓)}を周期的にとる：{Ã (0), ũ(0)}Ã (2⇡),ũ(2⇡)}C ",=(✓){=, C #, (✓2[0,2⇡]①ゲージ変換で✓i sin ( )<latexit sha1_base64="fCMH01Qx1S1W3Dj3pTRK3tC1lnQ=">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</latexit><latexit sha1_base64="MxfQ6EW37E40ooXLuw6AxVLiMzw=">AAADg3icjVLLbtNAFD2pKbTh0RQ2SGwqoqIioug6VGqFQGrFpss+SFspEyLbnSSj+iV7EqlY+QF+gAWrVmKB+Aw27Fix6CcglqXqpircOEZAowSuZc+ZM/fcOXc8duiqWBMd5yaMK5NXr01N56/fuHlrpjB7ezsOOpEjq07gBtGubcXSVb6saqVduRtG0vJsV+7Y+8/76ztdGcUq8F/og1DWPavlq6ZyLM1Uo6BEsrYgdFtq66HoNZIBFMoXrmzqGpUqIlQiUq22rveEDkSy+jJRvV+SUue/xY1CkcqUxtwwMDNQRBbrwWwuB4E9BHDQgQcJH5qxCwsxPzWYIITM1ZEwFzFS6bpED3nWdjhLcobF7D5/WzyrZazP837NOFU7vIvLb8TKOczTF3pPJ/SJPtBXOh9ZK0lr9L0c8GgPtDJszLy+u3X2T5XHo0b7t2qsZ40mllOvir2HKdPvwhnou6/enGw92ZxPHtARfWP/h3RMH7kDv/vdebchN9+O8WNz3xbPJdccfXKXfZeGuh+t1JwTMjKZ9cZm/vkn8sjzrTEv35FhsF0pm4/LlY3F4srT7P5M4R7uY4F3XMIK1rCOKtf+jFNc4IcxaTwyKsbiIHUil2nu4K8wnv0ELg3Vvw==</latexit>

C ",• (✓) =<latexit sha1_base64="YfSLoB3/ROClgMsc2ucR56EcAVE=">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</latexit>✓

i sin ( ✓2 )2◆✓cos ( 2 ), C #,• (91 /92

fundamental thm. for Z/2-gr. CS MPS

変換関数がZ/2-gradingを保つunitary変換のみであることを証明した．
不変量の具体的な計算法(解析的な⽅法)：
✓
◆
✓
i
i
i
cos
(
)
2
{H(✓)}✓2[0,2⇡] ! {A (✓), u(✓)}を周期的にとる：{Ã (0), ũ(0)}Ã (2⇡),ũ(2⇡)}C ",=(✓){=, C #, (✓2[0,2⇡]①ゲージ変換で✓i sin ( )Z<latexit sha1_base64="fCMH01Qx1S1W3Dj3pTRK3tC1lnQ=">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</latexit><latexit sha1_base64="MxfQ6EW37E40ooXLuw6AxVLiMzw=">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</latexit>✓

2
2 cos ( ✓ )◆2
, C #,• (dlog(tr(ũ(✓) )) 2 Z② tr(ũ(✓)2 ) は周期的なので巻きつき数は量⼦化： ntop. =C ",• (✓) =i sin ( ✓ )<latexit sha1_base64="8Z+rrdJxwx8vwEC+Nkuw0DsOxAc=">AAADOXichVK/T9tAGH0xLQ1pC6FdKnUgakQFEorOKRIRExJLR5I0ASlJI9u5wAn/kn2JRK2MXfoPdGACiQFlpVPHLvwDHTJ2g3ZMpS4MfHYstSVK+Cz73r373nfvO5/umsKXjA0SysyDh7OPknOpx0+ezi+kF59VfafjGbxiOKbj7emaz01h84oU0uR7rsc1Szf5rn64Ha7vdrnnC8d+J49c3rC0fVu0haFJoprppaDuWRnp9VbqUpgtHnRCdMCltvo+yPdWm+ksy7EoMuNAjUEWcew4i4kE6mjBgYEOLHDYkIRNaPDpqUEFg0tcAwFxHiERrXP0kCJth7I4ZWjEHtJ3n2a1mLVpHtb0I7VBu5j0eqTMYJl9Z+dsyC5Zn12zm4m1gqhG6OWIRn2k5W5z4dOL8p97VRaNEgd/VVM9S7RRiLwK8u5GTNiFMdJ3P3weljdLy8Frdsp+kf8TNmDfqAO7+9s4K/LS8RQ/OvWt0ZxTzcknd9f32lj3k5WSclxCKrHW1Mx//0QKKbo16t07Mg6q+Zz6Jpcvrme3CvH9SeIlXmGFdtzAFt5iBxWq/RF9XOCL8lX5oVwpP0epSiLWPMd/oQxvATK+u8E=</latexit><latexit sha1_base64="vs4WSsToFzMuzF42UUssK60YTO4=">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</latexit><latexit sha1_base64="YfSLoB3/ROClgMsc2ucR56EcAVE=">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</latexit>2

92 /92

fundamental thm. for Z/2-gr. CS MPS

変換関数がZ/2-gradingを保つunitary変換のみであることを証明した．
不変量の具体的な計算法(解析的な⽅法)：
✓
◆
✓
i
i
i
cos
(
)
2
{H(✓)}✓2[0,2⇡] ! {A (✓), u(✓)}を周期的にとる：{Ã (0), ũ(0)}Ã (2⇡),ũ(2⇡)}C ",=(✓){=, C #, (✓2[0,2⇡]①ゲージ変換で✓i sin ( )Z<latexit sha1_base64="fCMH01Qx1S1W3Dj3pTRK3tC1lnQ=">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</latexit><latexit sha1_base64="MxfQ6EW37E40ooXLuw6AxVLiMzw=">AAADg3icjVLLbtNAFD2pKbTh0RQ2SGwqoqIioug6VGqFQGrFpss+SFspEyLbnSSj+iV7EqlY+QF+gAWrVmKB+Aw27Fix6CcglqXqpircOEZAowSuZc+ZM/fcOXc8duiqWBMd5yaMK5NXr01N56/fuHlrpjB7ezsOOpEjq07gBtGubcXSVb6saqVduRtG0vJsV+7Y+8/76ztdGcUq8F/og1DWPavlq6ZyLM1Uo6BEsrYgdFtq66HoNZIBFMoXrmzqGpUqIlQiUq22rveEDkSy+jJRvV+SUue/xY1CkcqUxtwwMDNQRBbrwWwuB4E9BHDQgQcJH5qxCwsxPzWYIITM1ZEwFzFS6bpED3nWdjhLcobF7D5/WzyrZazP837NOFU7vIvLb8TKOczTF3pPJ/SJPtBXOh9ZK0lr9L0c8GgPtDJszLy+u3X2T5XHo0b7t2qsZ40mllOvir2HKdPvwhnou6/enGw92ZxPHtARfWP/h3RMH7kDv/vdebchN9+O8WNz3xbPJdccfXKXfZeGuh+t1JwTMjKZ9cZm/vkn8sjzrTEv35FhsF0pm4/LlY3F4srT7P5M4R7uY4F3XMIK1rCOKtf+jFNc4IcxaTwyKsbiIHUil2nu4K8wnv0ELg3Vvw==</latexit>✓

2
2 cos ( ✓ )◆2
, C #,• (ntop. =
dlog(tr(ũ(✓) )) 2 Z② tr(ũ(✓)2 ) は周期的なので巻きつき数は量⼦化：
C ",• (✓) =Zi sin ( ✓2 )2i✓n=dlog(tr(ũ(✓))) 2 Z/2Z
top.
ũ(✓)
!
e
ũ(✓)
③
は周期的な位相の付け替えでは偶数だけシフト：<latexit sha1_base64="8Z+rrdJxwx8vwEC+Nkuw0DsOxAc=">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</latexit><latexit sha1_base64="vs4WSsToFzMuzF42UUssK60YTO4=">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</latexit><latexit sha1_base64="YfSLoB3/ROClgMsc2ucR56EcAVE=">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</latexit><latexit sha1_base64="GUJiyAF36MfGci8RVF39e10NUaA=">AAADW3ichVJNS9xQFD0zqa1NbR2VguBGOlgslOHFFipdWbpx6UdHBWOHJHPVh/kieTNoQ/6Af6CLrloQEX+GGzcu2+KiP0BcKrhx4Z1MwI9hpjck775z77nv3Jdrh66MlRCnhaL2qO/xk/6n+rOB5y8GS0PDS3HQiByqOoEbRCu2FZMrfaoqqVxaCSOyPNulZXvrcyu+3KQoloH/Re2EtOZZG75cl46lGKqVPplKunVKGumkqTZJWW9MFZiKtlVWO4monib0NZHtaJp25NdKZVERmY13OkbulJHbXDBUKMBEHQEcNOCB4EOx78JCzM8qDAiEjK0hYSxiT2ZxQgqduQ3OIs6wGN3i7wbvVnPU532rZpyxHT7F5Tdi5jgmxG9xIC7EsTgUZ+K6a60kq9HSssOr3eZSWBvcHV28+i/L41Vh85bVU7PCOqYzrZK1hxnS6sJp85vfvl8sflyYSF6LX+Kc9f8Up+KIO/Cbl87ePC386KHH5r4t3hPX7H5zD3W/7ei+O1NxTsiewajXM/Pun9Ch89QYD2ek01maqhjvKlPz78sz0/n89GMMrzDJJ37ADGYxhyrX3scJ/uBv8Z+mabo20E4tFnLOCO6Z9vIGlyrH5Q==</latexit><latexit sha1_base64="Y+lgwgOifqe6+Kh+dnrNtBd1y8Q=">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</latexit>

93 /92

fundamental thm. for Z/2-gr. CS MPS

変換関数がZ/2-gradingを保つunitary変換のみであることを証明した．
不変量の具体的な計算法(解析的な⽅法)：
✓
◆
✓
i
i
i
cos
(
)
2
{H(✓)}✓2[0,2⇡] ! {A (✓), u(✓)}を周期的にとる：{Ã (0), ũ(0)}Ã (2⇡),ũ(2⇡)}C ",=(✓){=, C #, (✓2[0,2⇡]①ゲージ変換で✓i sin ( )Z<latexit sha1_base64="fCMH01Qx1S1W3Dj3pTRK3tC1lnQ=">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</latexit><latexit sha1_base64="MxfQ6EW37E40ooXLuw6AxVLiMzw=">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</latexit>✓

2
2 cos ( ✓ )◆2
, C #,• (ntop. =
dlog(tr(ũ(✓) )) 2 Z② tr(ũ(✓)2 ) は周期的なので巻きつき数は量⼦化：
C ",• (✓) =Zi sin ( ✓2 )2i✓n=dlog(tr(ũ(✓))) 2 Z/2Z
top.
ũ(✓)
!
e
ũ(✓)
③
は周期的な位相の付け替えでは偶数だけシフト：例) ũ(✓) ! ei✓ ũ(✓) に対して ntop. 7! ntop. + 2 ．
<latexit sha1_base64="8Z+rrdJxwx8vwEC+Nkuw0DsOxAc=">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</latexit><latexit sha1_base64="vs4WSsToFzMuzF42UUssK60YTO4=">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</latexit><latexit sha1_base64="YfSLoB3/ROClgMsc2ucR56EcAVE=">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</latexit><latexit sha1_base64="GUJiyAF36MfGci8RVF39e10NUaA=">AAADW3ichVJNS9xQFD0zqa1NbR2VguBGOlgslOHFFipdWbpx6UdHBWOHJHPVh/kieTNoQ/6Af6CLrloQEX+GGzcu2+KiP0BcKrhx4Z1MwI9hpjck775z77nv3Jdrh66MlRCnhaL2qO/xk/6n+rOB5y8GS0PDS3HQiByqOoEbRCu2FZMrfaoqqVxaCSOyPNulZXvrcyu+3KQoloH/Re2EtOZZG75cl46lGKqVPplKunVKGumkqTZJWW9MFZiKtlVWO4monib0NZHtaJp25NdKZVERmY13OkbulJHbXDBUKMBEHQEcNOCB4EOx78JCzM8qDAiEjK0hYSxiT2ZxQgqduQ3OIs6wGN3i7wbvVnPU532rZpyxHT7F5Tdi5jgmxG9xIC7EsTgUZ+K6a60kq9HSssOr3eZSWBvcHV28+i/L41Vh85bVU7PCOqYzrZK1hxnS6sJp85vfvl8sflyYSF6LX+Kc9f8Up+KIO/Cbl87ePC386KHH5r4t3hPX7H5zD3W/7ei+O1NxTsiewajXM/Pun9Ch89QYD2ek01maqhjvKlPz78sz0/n89GMMrzDJJ37ADGYxhyrX3scJ/uBv8Z+mabo20E4tFnLOCO6Z9vIGlyrH5Q==</latexit><latexit sha1_base64="Y+lgwgOifqe6+Kh+dnrNtBd1y8Q=">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</latexit><latexit sha1_base64="GUJiyAF36MfGci8RVF39e10NUaA=">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</latexit><latexit sha1_base64="qukIE/fHJrRvr8AwyIv648kdj4I=">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</latexit>

94 /92

fundamental thm. for Z/2-gr. CS MPS

変換関数がZ/2-gradingを保つunitary変換のみであることを証明した．
不変量の具体的な計算法(解析的な⽅法)：
✓
◆
✓
i
i
i
cos
(
)
2
{H(✓)}✓2[0,2⇡] ! {A (✓), u(✓)}を周期的にとる：{Ã (0), ũ(0)}Ã (2⇡),ũ(2⇡)}C ",=(✓){=, C #, (✓2[0,2⇡]①ゲージ変換で✓i sin ( )Z<latexit sha1_base64="fCMH01Qx1S1W3Dj3pTRK3tC1lnQ=">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</latexit><latexit sha1_base64="MxfQ6EW37E40ooXLuw6AxVLiMzw=">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</latexit>✓

2
2 cos ( ✓ )◆2
, C #,• (ntop. =
dlog(tr(ũ(✓) )) 2 Z② tr(ũ(✓)2 ) は周期的なので巻きつき数は量⼦化：
C ",• (✓) =Zi sin ( ✓2 )2i✓n=dlog(tr(ũ(✓))) 2 Z/2Z
top.
ũ(✓)
!
e
ũ(✓)
③
は周期的な位相の付け替えでは偶数だけシフト：例) ũ(✓) ! ei✓ ũ(✓) に対して ntop. 7! ntop. + 2 ．
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Pump不変量：ntop. =<latexit sha1_base64="Y+lgwgOifqe6+Kh+dnrNtBd1y8Q=">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</latexit>

Z<latexit sha1_base64="qukIE/fHJrRvr8AwyIv648kdj4I=">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</latexit>

dlog(tr(ũ(✓)2 )) 2 Z/2Z

95 /92

fundamental thm. for Z/2-gr. CS MPS

変換関数がZ/2-gradingを保つunitary変換のみであることを証明した．
不変量の具体的な計算法(解析的な⽅法)：
✓
◆
✓
i
i
i
cos
(
)
2
{H(✓)}✓2[0,2⇡] ! {A (✓), u(✓)}を周期的にとる：{Ã (0), ũ(0)}Ã (2⇡),ũ(2⇡)}C ",=(✓){=, C #, (✓2[0,2⇡]①ゲージ変換で✓i sin ( )Z<latexit sha1_base64="fCMH01Qx1S1W3Dj3pTRK3tC1lnQ=">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</latexit><latexit sha1_base64="MxfQ6EW37E40ooXLuw6AxVLiMzw=">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</latexit>✓

2
2 cos ( ✓ )◆2
, C #,• (ntop. =
dlog(tr(ũ(✓) )) 2 Z② tr(ũ(✓)2 ) は周期的なので巻きつき数は量⼦化：
C ",• (✓) =Zi sin ( ✓2 )2i✓n=dlog(tr(ũ(✓))) 2 Z/2Z
top.
ũ(✓)
!
e
ũ(✓)
③
は周期的な位相の付け替えでは偶数だけシフト：例) ũ(✓) ! ei✓ ũ(✓) に対して ntop. 7! ntop. + 2 ．
<latexit sha1_base64="8Z+rrdJxwx8vwEC+Nkuw0DsOxAc=">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</latexit><latexit sha1_base64="vs4WSsToFzMuzF42UUssK60YTO4=">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</latexit><latexit sha1_base64="YfSLoB3/ROClgMsc2ucR56EcAVE=">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</latexit><latexit sha1_base64="GUJiyAF36MfGci8RVF39e10NUaA=">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</latexit><latexit sha1_base64="Y+lgwgOifqe6+Kh+dnrNtBd1y8Q=">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</latexit><latexit sha1_base64="GUJiyAF36MfGci8RVF39e10NUaA=">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</latexit>

Pump不変量：ntop. =<latexit sha1_base64="Y+lgwgOifqe6+Kh+dnrNtBd1y8Q=">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</latexit>

Z<latexit sha1_base64="qukIE/fHJrRvr8AwyIv648kdj4I=">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</latexit>

dlog(tr(ũ(✓)2 )) 2 Z/2Z

96 /92

fundamental thm. for Z/2-gr. CS MPS
不変量の具体的な計算法(代数的な⽅法)：
i
{U
}
{A
① Xの開被覆 ↵ を取り，各 U↵ 上のfMPS表⽰↵ , u↵ } を取る．
<latexit sha1_base64="3pKz91Rn0E9IcEEhmpDyVADVeX4=">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</latexit><latexit sha1_base64="gvxA3h3KW1kBn3QQ0vBrxsTXxA8=">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</latexit><latexit sha1_base64="JX1Ngw7+6xFb/r2P/TQ3TbiUDaI=">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</latexit>② u↵ (✓)の位相を取り替えて，u↵ (✓)2 = 1 を満たすようにとる．
③ U↵ と U の共通部分においてfundamental theoremを⽤いる：
Ai↵ (✓) = ei ↵, V↵, (✓)Ai (✓)V↵, (✓)†
2
2
u
(✓)
=
u
(✓)
= 1 を満たすように取った時，V↵, (✓) は以下を満たす：④ ↵u↵ (✓)V↵, (✓) = ⌘↵, V↵, (✓)u (✓)( ⌘↵, = ±1 )
Y
1
[⌘
]
2
H
(X; Z/2Z) )
⌘
=
⌘
2
{±1}
↵,
(
Pump不変量： top.
↵,<latexit sha1_base64="hMmlKuXmL1RLA4pDiUBwhB6gSjM=">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</latexit><latexit sha1_base64="yshHLOVQQjgz+POjfBVDKdoqjqY=">AAADMnichVJNSyNBEH2ZVVfjV1wvwl7ChogLEnqygsuCIHrxaNRowGjoGVszOF/MdAI6xB/gH/DgSXEPy6In/4EX/4AH9yaeRG8KXvawlcmAriGxhpmuel2v+lVPaa5p+JKx65jyoaOz62N3T7y3r39gMDH0adl3Kp4u8rpjOl5B474wDVvkpSFNUXA9wS3NFCva9mx9f6UqPN9w7CW544o1i2/Zxqahc0lQKTFSKQVFbrplXhsryrKQ/Ot6kK1NqaVEimVYaMlmR42cFCKbd4ZiMRSxAQc6KrAgYEOSb4LDp2cVKhhcwtYQEOaRZ4T7AjXEiVuhLEEZnNBt+m5RtBqhNsX1mn7I1ukUk16PmEmk2RX7xR7ZJfvN7tjflrWCsEZdyw6tWoMr3NLg/sji87ssi1aJ8gurrWaJTXwPtRqk3Q2Rehd6g1/dPXhc/LGQDkbZMbsn/Ufsml1QB3b1Sf+ZEwuHbfRo1DenWFDN1jf3Vvd4U/etmZJyXPJUQq22ma//RBxxmhr17Yw0O8vZjPotk81NpKZnovnpxmd8wRidOIlpzGEeeaq9hxOc4kw5V/4oN8ptI1WJRZxh/GfKwz9WXbi8</latexit><latexit sha1_base64="1l+yEgVtqajvtNvwns8kAYIaFwg=">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</latexit><latexit sha1_base64="13pPCxvXnydBQzuKb0mNUYXRrwM=">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</latexit><latexit sha1_base64="vhFdLQ5ioKOpROizCUQdeKD7/OY=">AAADmHichVLNTttAEB7jtoBbSigcUHtBjUBUQtEakFpVQkrhUHqD0AQkDNHaDMkK/6zsTSSw3AfoC/TACSQOqI/BhRfgwL2Xqkcq9dJDJ46jlkRJx7J39pv5Zr9Zjy1dESnGbrUh/cHDR8Mjo8bjJ2NPx3MTzypR0AgdLDuBG4Q7No/QFT6WlVAu7sgQuWe7uG0frbXi200MIxH4H9WxxD2P13xxKByuCKrmPr3bj0VSjS3uyjpP5i1VR8VfrSDBlqyLTmTBsglPkkoX0CF0ytwD+yXvx9YBr9UwTKq5PCuw1GZ6HTNz8pDZRjChaWDBAQTgQAM8QPBBke8Ch4ieXTCBgSRsD2LCQvJEGkdIwCBug7KQMjihR/St0W43Q33at2pGKduhU1x6Q2LOwCy7YZfsjl2zr+w7+923VpzWaGk5ptVuc1FWxz9Pb/36L8ujVUH9L2ugZgWH8CbVKki7TJFWF06b3zz5crf1tjQbz7Fz9oP0n7FbdkUd+M2fzsUmlk4H6LGpb057pJr9b65b90JP9/2ZinIkeSah3sDMf/+EAQZNjdk9I71OZbFgLhUWN5fzxdVsfkbgBbyEeTrxNRRhHTagTLW/aYY2qU3pz/Wi/l7/0E4d0jLOJNwzvfQHOFjcSg==</latexit><latexit sha1_base64="Eim8x59nUTnPZR21oYeRuykWkPo=">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</latexit><latexit sha1_base64="L1rLe0Vrz+FolZ+AjoETJX8aW94=">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</latexit><latexit sha1_base64="y/Aub9v2QDTD/5pFZsXeweeu6YE=">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</latexit><latexit sha1_base64="cce3QOqN2WBOkBcvn5G4Ix98K8A=">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</latexit><latexit sha1_base64="PfwKwYKJGlHgktbkKZ54X3BXkPQ=">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</latexit><latexit sha1_base64="l1cUS/pnb/6V7VwvZwZ/GPV1ZgQ=">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</latexit>

特に⼆つの不変量は等価で，⼀般に ⌘top. = ( 1)<latexit sha1_base64="lkMIAQtG1Kp1OJ+ZlaBLYH19t0w=">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</latexit>

nU
↵
top.
<latexit sha1_base64="C7iDcxWI5GevcXtSWpscODI+7qM=">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</latexit>Uという関係にある．
10<latexit sha1_base64="LmMgNT/j+TCUK8XmFT/zdqRqdwc=">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</latexit>/92

fundamental thm. for Z/2-gr. CS MPS
不変量の具体的な計算法(代数的な⽅法)：
i
{U
}
{A
① Xの開被覆 ↵ を取り，各 U↵ 上のfMPS表⽰↵ , u↵ } を取る．
<latexit sha1_base64="3pKz91Rn0E9IcEEhmpDyVADVeX4=">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</latexit><latexit sha1_base64="gvxA3h3KW1kBn3QQ0vBrxsTXxA8=">AAADJnichVI7SwNBEP5yvuNbG8FGDIqFhD0VFCvRxjJREwWjYe9czeG9uNsE9Mgf8A8oWCkIij/DxspOJJ2tWCrYWDi5HPgIiXPc7cy3881+szeaaxq+ZKwSU1pa29o7Orvi3T29ff0Dg0NZ3yl6usjojul4Wxr3hWnYIiMNaYot1xPc0kyxqR2uVPc3S8LzDcfekEeu2LH4gW3sGzqXBO3mgkw+yHHTLfByrpwfSLAkC22s3lEjJ4HIUs5gLIYc9uBARxEWBGxI8k1w+PRsQwWDS9gOAsI88oxwX6CMOHGLlCUogxN6SN8DirYj1Ka4WtMP2TqdYtLrEXMME+yR3bA3ds9u2Qv7bFgrCGtUtRzRqtW4ws33n4ysf/zLsmiVKHyzmmqW2MdCqNUg7W6IVLvQa/zS8enb+uLaRDDJLtkr6b9gFXZHHdild/0qLdbOm+jRqG9OsaCajW/ur+7puu4bMyXluOSphFpNM3/+iTjiNDXq3xmpd7IzSXU2OZOeSywtR/PTiVGMY4pOnMcSVpFChmp7OMMFLpVr5UF5VJ5qqUos4gzjlynPX2NXtP4=</latexit><latexit sha1_base64="JX1Ngw7+6xFb/r2P/TQ3TbiUDaI=">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</latexit>② u↵ (✓)の位相を取り替えて，u↵ (✓)2 = 1 を満たすようにとる．
③ U↵ と U の共通部分においてfundamental theoremを⽤いる：
Ai↵ (✓) = ei ↵, V↵, (✓)Ai (✓)V↵, (✓)†
2
2
u
(✓)
=
u
(✓)
= 1 を満たすように取った時，V↵, (✓) は以下を満たす：④ ↵u↵ (✓)V↵, (✓) = ⌘↵, V↵, (✓)u (✓)( ⌘↵, = ±1 )
Y
1
[⌘
]
2
H
(X; Z/2Z) )
⌘
=
⌘
2
{±1}
↵,
(
Pump不変量： top.
↵,<latexit sha1_base64="hMmlKuXmL1RLA4pDiUBwhB6gSjM=">AAADLHichVLLSsNAFD2N7/qqdSO4EYuiIGWiguJKdOPSV1WwUiZxtMG8SKbFWvwBf8CFICiIiuBPuPEDdCHiVhCXCm5ceJsGfJTWG5K598w9d86dXM01DV8y9hBR6uobGpuaW6Ktbe0dnbGu+Irv5DxdpHTHdLw1jfvCNGyRkoY0xZrrCW5ppljVdmZL+6t54fmGYy/Lgis2LL5tG1uGziVBmVg8lymmuelm+f5QWmaF5MOZWIIlWWB9lY4aOgmENu90RSJIYxMOdORgQcCGJN8Eh0/POlQwuIRtoEiYR54R7AvsI0rcHGUJyuCE7tB3m6L1ELUpLtX0A7ZOp5j0esTswwC7Z5fsjd2yK/bCPqvWKgY1SloKtGplrnAznQc9Sx//sixaJbLfrJqaJbYwGWg1SLsbIKUu9DI/v3f4tjS1OFAcZKfslfSfsAd2Qx3Y+Xf9bEEsHtXQo1HfnGJBNavf3F/dIxXdV2dKynHJUwm1amb+/BNRRGlq1L8zUumsjCbVseTownhieiacn2b0oh9DdOIEpjGHeaSo9i6OcY4L5Vq5Ux6Vp3KqEgk53fhlyvMX8a+2ig==</latexit><latexit sha1_base64="yshHLOVQQjgz+POjfBVDKdoqjqY=">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</latexit><latexit sha1_base64="1l+yEgVtqajvtNvwns8kAYIaFwg=">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</latexit><latexit sha1_base64="13pPCxvXnydBQzuKb0mNUYXRrwM=">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</latexit>

割愛<latexit sha1_base64="vhFdLQ5ioKOpROizCUQdeKD7/OY=">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</latexit><latexit sha1_base64="Eim8x59nUTnPZR21oYeRuykWkPo=">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</latexit><latexit sha1_base64="L1rLe0Vrz+FolZ+AjoETJX8aW94=">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</latexit><latexit sha1_base64="y/Aub9v2QDTD/5pFZsXeweeu6YE=">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</latexit><latexit sha1_base64="cce3QOqN2WBOkBcvn5G4Ix98K8A=">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</latexit><latexit sha1_base64="PfwKwYKJGlHgktbkKZ54X3BXkPQ=">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</latexit><latexit sha1_base64="l1cUS/pnb/6V7VwvZwZ/GPV1ZgQ=">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</latexit>

特に⼆つの不変量は等価で，⼀般に ⌘top. = ( 1)<latexit sha1_base64="lkMIAQtG1Kp1OJ+ZlaBLYH19t0w=">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</latexit>

nU
↵
top.
<latexit sha1_base64="C7iDcxWI5GevcXtSWpscODI+7qM=">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</latexit>Uという関係にある．
10<latexit sha1_base64="LmMgNT/j+TCUK8XmFT/zdqRqdwc=">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</latexit>

/92

目次
• 研究の背景
SRE状態とGTPについて• 研究の⽬的と⽅法
MPSを⽤いたinteracting fermionic systemにおけるGTPの検証• 結果１：Pump不変量の定義
fundamental theorem for fMPS
Pump不変量の定義• 結果２：Pump不変量の計算
non-trivial phaseにおけるpump
trivial phaseにおけるpump
• まとめ
11
/92

Pump不変量の計算例①例) ガチャガチャ模型
XH(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

ただし

(a†j

†
x
aj )⌧j+1 (aj+1 + aj+1 )2

jz,✓i ✓2 ⌧jx z
i ✓2 ⌧jx⌧j = e⌧j eXj⌧jz,✓1 (12z,✓2a†j aj )⌧j+1
2

．このとき H(0) = H(⇡) .
<latexit sha1_base64="gVRcHWHY88yW65nYOgyJgMtoX6M=">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</latexit><latexit sha1_base64="PcQWA1hArg7cCqTw8i13IOoEVgQ=">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</latexit>

11

/92

Pump不変量の計算例①例) ガチャガチャ模型
XH(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

ただし

(a†j

†
x
aj )⌧j+1 (aj+1 + aj+1 )2

jz,✓i ✓2 ⌧jx z
i ✓2 ⌧jx⌧j = e⌧j eXj⌧jz,✓1 (12z,✓2a†j aj )⌧j+1
2

．このとき H(0) = H(⇡) .
<latexit sha1_base64="gVRcHWHY88yW65nYOgyJgMtoX6M=">AAADI3ichVJNSxtRFD2ZamtHW7VuBDdiUBIo4Y4VlIIgdJOlXzGBJMjM9EUfzhczLwEb+ge6FxeCxYKL0p/hptCd6MKFP0BcKrhx4c1kwI+Q9A4z797z7rnv3DfXChwZKaKLlPaqr//1m4G3+uDQu/fDI6MfNiK/HtqiYPuOH5YsMxKO9ERBSeWIUhAK07UcUbR2vrT2iw0RRtL31tVuIKquueXJmrRNxVA5n6HsYj5TCWR2cyRNOYptstMxEieNxJb90VQKFXyFDxt1uBDwoNh3YCLipwwDhICxKpqMhezJeF/gO3Tm1jlLcIbJ6A5/tzgqJ6jHcatmFLNtPsXhN2TmJKbpnH7TDf2lP3RF911rNeMaLS27vFptrgg2h3+Mr939l+XyqrD9yOqpWaGGhVirZO1BjLS6sNv8xrf9m7XPq9PNGfpF16z/iC7ohDvwGrf28YpYPeihx+K+TY4F1+x+cy91f+zovjtTcU7AnsGo2zPz6Z/QofPUGC9npNPZmM0Zn3KzK3PppYVkfgYwgSlk+MR5LCGPZRTi2ns4xE/tWPunnWpn7VQtlXDG8My0ywfyDbHs</latexit><latexit sha1_base64="PcQWA1hArg7cCqTw8i13IOoEVgQ=">AAADsHiclVLLShxBFL1tx9ck6phsBDfiYBAch+oxEAkEhGyy9JHRAVub6vaOlvaL7ppBLfoH/ACzcGXAhfgZbvyBLPyEkKWBbFzkdk+D0WFGrKa77j11z6lzq8sOXRFLxm61Pv1V/8Dg0HDh9ZuR0bHi+Nv1OGhGDtacwA2ius1jdIWPNSmki/UwQu7ZLm7YB1/S9Y0WRrEI/G/yKMQtj+/6oiEcLgmyiqem5E1L7Sfb6rhsSjyUmaaKcCdRptxDyZPkM24rYTYi7qhuJaqaPCgdJllCkkkKEHv+hXSrWGIVlo2pzsDIgxLkYzkY1zQwYQcCcKAJHiD4ICl2gUNMzyYYwCAkbAsUYRFFIltHSKBA3CZVIVVwQg/ou0vZZo76lKeaccZ2aBeX3oiYUzDDfrJLdsdu2BX7xe67aqlMI/VyRLPd5mJojZ1MrP19luXRLGHvgdXTs4QGLGZeBXkPMyTtwmnzW8ff79Y+rc6o9+wH+03+z9ktu6YO/NYf52IFV896+LGpb045kmb3k3vqu9zRfXempJqQIoNQr2fl/3+iAAW6NcbTO9IZrFcrxkKluvKhtLSY358hmIRpmKUdP8ISfIVlqJH2vTatzWllvarXdUvn7dI+Lee8g0dD3/8HMPLsQA==</latexit>

𝝅 rot.
<latexit"
sha1_base64="FIrxYIu8LOxuDc7YXQK9X0Gav64=">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</latexit>

11

/92

Pump不変量の計算例①例) ガチャガチャ模型
XH(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

ただし

(a†j

X

†
x
aj )⌧j+1 (aj+1 + aj+1 )2

jz,✓i ✓2 ⌧jx z
i ✓2 ⌧jx⌧j = e⌧j ez,✓
2a†j aj )⌧j+1

⌧jz,✓1 (12j2．このとき H(0) = H(⇡) .
<latexit sha1_base64="gVRcHWHY88yW65nYOgyJgMtoX6M=">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</latexit><latexit sha1_base64="PcQWA1hArg7cCqTw8i13IOoEVgQ=">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</latexit>

この模型は⾃明相に属することが確認できる．
対応する周期的なfMPSは
✓
◆
✓
◆
✓
◆
C ",• (✓) =<latexit sha1_base64="YfSLoB3/ROClgMsc2ucR56EcAVE=">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</latexit>

i sin ( ✓2 )cos ( ✓2 )✓◆, C #,• (✓) =✓✓cos ( ✓2 )✓cos ( ✓2 )i sin ( ✓2 ),◆u(✓) =

1

1<latexit sha1_base64="J64pI/SYepKkOR2HIkcc4UFXTts=">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</latexit>

sha1_base64="FIrxYIu8LOxuDc7YXQK9X0Gav64=">AAADJnichVI7SwNBEP5yvmJ8ayPYBINiIWFPBcUqaGNpEqOCUbk7N3p4L/Y2EQ3+Af+AgpWCoPgzbKzsgtjZiqWCjYWTy4GPkDjH3c58O9/sN3uje5bpS8aeIkpLa1t7R7Qz1tXd09vXPzC46rtFYfCc4VquWNc1n1umw3PSlBZf9wTXbN3ia/r+YnV/rcSFb7rOijz0+Kat7TpmwTQ0SdBWOa8X4vmipwnhHhxv9ydYkgUWr3fU0EkgtGV3IBJBHjtwYaAIGxwOJPkWNPj0bEAFg0fYJsqECfLMYJ/jGDHiFimLU4ZG6D59dynaCFGH4mpNP2AbdIpFryBmHGOswm7YG7tnt+yFfTasVQ5qVLUc0qrXuNzb7jsZzn78y7Jpldj7ZjXVLFHAXKDVJO1egFS7MGr80tHpW3Y+M1YeZ5fslfRfsCd2Rx04pXfjKs0z50306NS3RjGnmo1v7q/uybruGzMl5XjkqYTaTTN//okYYjQ16t8ZqXdWp5LqdHIqPZNILYTzE8UIRjFBJ84ihSUsI0e1Bc5wgUvlWnlQKspjLVWJhJwh/DLl+QsmPbTo</latexit>✓

i sin ( ✓2 ), C #, (✓) =𝝅 rot.
<latexit

C ", (✓) =i sin ( ✓2 )"cos ( ✓2 )

.
✓

✓

✓

を⽤いて C̃ i,j (✓) = e i 2 ei 2 x C i,j (✓)e i 2 x , ũ(✓) = e i 2 ei 2 x u(✓)e i 2 x で与えられる．
<latexit sha1_base64="C6uvGihf9Y98XiDjHQs9RMSeomo=">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</latexit>

11

/92

Pump不変量の計算例①例) ガチャガチャ模型
XH(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">AAAEDHicjVJNaxNBGH6360ddP5q2F8FLMVQS0oTZWGgpFApeeuyHaQPdNsxOJuna/WJ3NrQd1qviwasHTwoexKs38SSIf8BDf4J4s4IXD777gW0aEp1lZ973mfd55nl3x/RtKxSEnChj6qXLV66OX9Ou37h5a6IwObUVelHAeIN5thc0TRpy23J5Q1jC5k0/4NQxbb5tHjxI9rd7PAgtz30ojny+69Cua3UsRgVCrcKP1ZIh+KFIlWTA27E0xD4XNC4vV40wclryUVyiybwnjTbtdnkQV9O8rBmCRnvyMMasYnQCyqQey3qc1Vf0c4xKjpS1v6IZ+XhuyPGJaLVPVK/W6ZliIpHZ+A+hPnetQpHUSDpmBgM9D4qQjzVvUlHAgDZ4wCACBzi4IDC2gUKIzw7oQMBHbBckYgFGVrrPIQYNuRFWcaygiB7g3MVsJ0ddzBPNMGUzPMXGN0DmDMySr+QtOSVfyDvyjfweqiVTjcTLEa5mxuV+a+LZ7c1f/2Q5uArYP2ON9CygA4upVwu9+ymSdMEyfu/4xenm0sasvEdek+/o/xU5IZ+wA7f3k71Z5xsvR/gxsW+KOUfN4V/uou+5ge6HMwXW+BjpiDojK8//CQ00vDX6xTsyGGzVa/r9Wn19vriymN+fcbgDd6GEJy7ACqzCGjSAKU3lsfJEeao+V9+rH9SPWemYknOmoW+on/8A7BYOOA==</latexit>

ただし

(a†j

X

†
x
aj )⌧j+1 (aj+1 + aj+1 )2

jz,✓i ✓2 ⌧jx z
i ✓2 ⌧jx⌧j = e⌧j ez,✓
2a†j aj )⌧j+1

⌧jz,✓1 (12j2．このとき H(0) = H(⇡) .
<latexit sha1_base64="gVRcHWHY88yW65nYOgyJgMtoX6M=">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</latexit><latexit sha1_base64="PcQWA1hArg7cCqTw8i13IOoEVgQ=">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</latexit>

この模型は⾃明相に属することが確認できる．
対応する周期的なfMPSは
✓
◆
✓
◆
✓
◆
C ",• (✓) =<latexit sha1_base64="YfSLoB3/ROClgMsc2ucR56EcAVE=">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</latexit>

i sin ( ✓2 )cos ( ✓2 )✓◆, C #,• (✓) =✓✓cos ( ✓2 )✓cos ( ✓2 )i sin ( ✓2 ),◆u(✓) =

1

1<latexit sha1_base64="J64pI/SYepKkOR2HIkcc4UFXTts=">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</latexit>

sha1_base64="FIrxYIu8LOxuDc7YXQK9X0Gav64=">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</latexit>✓

i sin ( ✓2 ), C #, (✓) =𝝅 rot.
<latexit

C ", (✓) =i sin ( ✓2 )"cos ( ✓2 )

.
✓

✓

✓

を⽤いて C̃ i,j (✓) = e i 2 ei 2 x C i,j (✓)e i 2 x , ũ(✓) = e i 2 ei 2 x u(✓)e i 2 x で与えられる．
<latexit sha1_base64="C6uvGihf9Y98XiDjHQs9RMSeomo=">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</latexit>

tr(ũ(✓)2 ) =<latexit sha1_base64="YqyqzSRKYkQ3/bxheq0JvGfM+d0=">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</latexit>

2ei✓ なので，ntop. = 1 2 Z/2Z よりpump不変量は⾮⾃明．
<latexit sha1_base64="3PKxt7wUos491QIFkvnMYnnyp0Y=">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</latexit>

11

/92

Pump不変量の計算例①例) ガチャガチャ模型
XH(✓) =<latexit sha1_base64="qUhQTllpMpTphoLCWFkzB1543Ns=">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</latexit>

ただし

(a†j

X

†
x
aj )⌧j+1 (aj+1 + aj+1 )2

jz,✓i ✓2 ⌧jx z
i ✓2 ⌧jx⌧j = e⌧j ez,✓
2a†j aj )⌧j+1

⌧jz,✓1 (12j2．このとき H(0) = H(⇡) .
<latexit sha1_base64="gVRcHWHY88yW65nYOgyJgMtoX6M=">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</latexit><latexit sha1_base64="PcQWA1hArg7cCqTw8i13IOoEVgQ=">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</latexit>

この模型は⾃明相に属することが確認できる．
対応する周期的なfMPSは
✓
◆
✓
◆
✓
◆
C ",• (✓) =<latexit sha1_base64="YfSLoB3/ROClgMsc2ucR56EcAVE=">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</latexit>

i sin ( ✓2 )cos ( ✓2 )✓◆, C #,• (✓) =✓✓cos ( ✓2 )✓cos ( ✓2 )i sin ( ✓2 ),◆u(✓) =

1

1<latexit sha1_base64="J64pI/SYepKkOR2HIkcc4UFXTts=">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</latexit>

sha1_base64="FIrxYIu8LOxuDc7YXQK9X0Gav64=">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</latexit>✓

i sin ( ✓2 ), C #, (✓) =𝝅 rot.
<latexit

C ", (✓) =i sin ( ✓2 )"cos ( ✓2 )

.
✓

✓

✓

を⽤いて C̃ i,j (✓) = e i 2 ei 2 x C i,j (✓)e i 2 x , ũ(✓) = e i 2 ei 2 x u(✓)e i 2 x で与えられる．
<latexit sha1_base64="C6uvGihf9Y98XiDjHQs9RMSeomo=">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</latexit>

tr(ũ(✓)2 ) =<latexit sha1_base64="YqyqzSRKYkQ3/bxheq0JvGfM+d0=">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</latexit>

2ei✓ なので，ntop. = 1 2 Z/2Z よりpump不変量は⾮⾃明．
<latexit sha1_base64="3PKxt7wUos491QIFkvnMYnnyp0Y=">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</latexit>

interacting fermionic SPTの⾃明相におけるpumpを確認した．

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Pump不変量の計算例②例)⾮⾃明相のKitaev chain with parameter ✓H(✓) =<latexit sha1_base64="hC4a+pAmSWjIQchtbuP8+XFNCZI=">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</latexit>

X<latexit sha1_base64="Ns6jXVzKT/11xGO2LV6WS37FPdU=">AAADMXichVK7ThtRED3eEB4mARMKkNJYWCCKyLoLkYIQBVIaShswIGFk7S4DXrEv7V5bwMo/wA9QpEoQBUqo+ASa/ACFO0SHXIJEQ8F4vRIPy2ZWu3fm3Dlzz9wd3bPMQApRTygfej729vUPJAc/fR4aTo18WQvcim9QwXAt19/QtYAs06GCNKVFG55Pmq1btK7v/Wzur1fJD0zXWZUHHm3Z2q5j7piGJhkqpcaKkvZlVCf0absWFmWZpFYrpTIiKyJLtztq7GQQW84dSSRQxDZcGKjABsGBZN+ChoCfTagQ8BjbQsiYz54Z7RNqSDK3wlnEGRqje/zd5WgzRh2OmzWDiG3wKRa/PjPTmBRX4kzcif/ir7gVjx1rhVGNppYDXvUWl7zS8NH4ysO7LJtXifIzq6tmiR3MRVpN1u5FSLMLo8WvHh7frcwvT4ZT4o9osP7foi4uuQOnem+c5mn5Vxc9OvetcUxcs/PNvdX9ra37zkzJOR57KqN218yXfyKJJE+N+nZG2p21maw6m53Jf88sLsTz04+vmMA0n/gDi1hCDoXork7wD+fKhVJXrpWbVqqSiDmjeGVK4wnHRrmi</latexit>a†j+1 aj+1

a†j+1 aj + ei✓ a†j a†j+1 + e i✓ aj+1 aj

j

11

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Pump不変量の計算例②例)⾮⾃明相のKitaev chain with parameter ✓X<latexit sha1_base64="Ns6jXVzKT/11xGO2LV6WS37FPdU=">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</latexit>a†j+1 aj+1 a†j+1 aj + ei✓ a†j a†j+1 + e i✓ aj+1 aj
j ✓
◆
✓
◆
✓
◆
1i
1
1
0i i
1
i ✓2
i ✓2
i
i
A
(✓)
=
A
(✓)
=
e
u(✓)
=
e
H(✓)}✓2[0,2⇡] ! {AA(✓),
u(✓)}
(0) =
A (2⇡),
u(0) = u(2⇡)
A (0) =1 A (2⇡), u(0) = u(2⇡) 1
1✓2[0,2⇡]
✓
◆
✓
◆
✓
◆
1
1
i
i 1
0 ii(✓), u(✓)}1
H(✓)}✓2[0,2⇡] ! {A(0)== Ai (2⇡),
u(0)
=
A
u(2⇡)
(0)
=
A
(2⇡),
u(0)
=
u(2⇡)
ÃA
(✓)
Ã
(✓)
=
ũ(✓)
=
1✓2[0,2⇡]ei✓
ei✓
H(✓) =<latexit sha1_base64="hC4a+pAmSWjIQchtbuP8+XFNCZI=">AAAD1HichVLPSxtBFH7ramtTrbFeCr2EBotiDbO2UBEEoVC8GbWJgqthdvOMU/cXu5OgrnMqvfTQaw89VfAg/ge9evEf6ME/oXhU6KWHvmwW1MSks+zOe9+875vvzY4VOCKSjF1ofXr/wIOHg48yj4eGn4xkR5+WI78e2liyfccP1y0eoSM8LEkhHVwPQuSu5eCatfuuub7WwDASvvdB7ge46fKaJ7aFzSVBlezPxQlT4p5MlOIQqyo25Q5KribnzajuVuKPappvxWaV12oYKsqnDMVb0/0LaqpdEbdikaqquxTVoXAfefqGfWubSjbPCiwZuc7ASIM8pKPoj2oamFAFH2yogwsIHkiKHeAQ0bMBBjAICNuEmLCQIpGsIyjIELdOVUgVnNBd+tYo20hRj/KmZpSwbdrFoTckZg7G2S92wq7YOTtlv9nfrlpxotH0sk+z1eJiUBn58mz1z39ZLs0Sdm5YPT1L2IbZxKsg70GCNLuwW/zGwber1bmV8fglO2KX5P8Hu2Bn1IHXuLaPl3Hlew8/FvXNKUfS7H5y7b5fdXTfnSmpJqDIINTtWXn7T2QgQ7fGaL8jnUF5pmC8Lswsv8kvzKb3ZxCewwuYoB3fwgIsQhFKYGs57b22pBX1sn6of9I/t0r7tJQzBneG/vUfG3D3eA==</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit 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11

/92

Pump不変量の計算例②例)⾮⾃明相のKitaev chain with parameter ✓X<latexit sha1_base64="Ns6jXVzKT/11xGO2LV6WS37FPdU=">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</latexit>a†j+1 aj+1 a†j+1 aj + ei✓ a†j a†j+1 + e i✓ aj+1 aj
j ✓
◆
✓
◆
✓
◆
1i
1
1
0i i
1
i ✓2
i ✓2
i
i
A
(✓)
=
A
(✓)
=
e
u(✓)
=
e
H(✓)}✓2[0,2⇡] ! {AA(✓),
u(✓)}
(0) =
A (2⇡),
u(0) = u(2⇡)
A (0) =1 A (2⇡), u(0) = u(2⇡) 1
1✓2[0,2⇡]
✓
◆
✓
◆
✓
◆
1
1
i
i 1
0 ii(✓), u(✓)}1
H(✓)}✓2[0,2⇡] ! {A(0)== Ai (2⇡),
u(0)
=
A
u(2⇡)
(0)
=
A
(2⇡),
u(0)
=
u(2⇡)
ÃA
(✓)
Ã
(✓)
=
ũ(✓)
=
1✓2[0,2⇡]ei✓
ei✓
H(✓) =<latexit sha1_base64="hC4a+pAmSWjIQchtbuP8+XFNCZI=">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</latexit><latexit 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sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="xr5evtGdzp8SdtdG9izkR7b6IOg=">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</latexit>

tr(ũ(✓)2 ) =<latexit sha1_base64="YqyqzSRKYkQ3/bxheq0JvGfM+d0=">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</latexit><latexit sha1_base64="RCLzO/8yFV3it9pr2xBSHrjcrfw=">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</latexit><latexit sha1_base64="V98scwraBhzvxsItTvz61y7pSOE=">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</latexit><latexit sha1_base64="5SDuM1TfnHzA+5yZf2tEF4chz+4=">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</latexit>

2ei✓ なので，ntop. = 1 2 Z/2Z よりpump不変量は⾮⾃明．
<latexit sha1_base64="3PKxt7wUos491QIFkvnMYnnyp0Y=">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</latexit>

11

/92

Pump不変量の計算例②例)⾮⾃明相のKitaev chain with parameter ✓X<latexit sha1_base64="Ns6jXVzKT/11xGO2LV6WS37FPdU=">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</latexit>a†j+1 aj+1 a†j+1 aj + ei✓ a†j a†j+1 + e i✓ aj+1 aj
j ✓
◆
✓
◆
✓
◆
1i
1
1
0i i
1
i ✓2
i ✓2
i
i
A
(✓)
=
A
(✓)
=
e
u(✓)
=
e
H(✓)}✓2[0,2⇡] ! {AA(✓),
u(✓)}
(0) =
A (2⇡),
u(0) = u(2⇡)
A (0) =1 A (2⇡), u(0) = u(2⇡) 1
1✓2[0,2⇡]
✓
◆
✓
◆
✓
◆
1
1
i
i 1
0 ii(✓), u(✓)}1
H(✓)}✓2[0,2⇡] ! {A(0)== Ai (2⇡),
u(0)
=
A
u(2⇡)
(0)
=
A
(2⇡),
u(0)
=
u(2⇡)
ÃA
(✓)
Ã
(✓)
=
ũ(✓)
=
1✓2[0,2⇡]ei✓
ei✓
H(✓) =<latexit sha1_base64="hC4a+pAmSWjIQchtbuP8+XFNCZI=">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</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="04PLFq9DZQtHmm+Arf+tV7llpvI=">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</latexit><latexit sha1_base64="EaQklouobvLJ7HZ8V+0zQklHJys=">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</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">AAADPXicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQsoO8ZVZ9ZqGGjaQhhGMQWZmjox1qUgoVIIF6jKQM8ADBQwGYZQhjIDFATkCzIyMsQwpDDkMyQzlDLkMqQy5DGUANk5DIkMxUAYzWDIYMBQABSLZagGihUBWZlg+VSGWgYuoN5SoKpUoIpEoGg2kEwH8qKhonlAPsjMYrDuZKAtOUBcBNSpwKBqcNVgpcFngxMGqw1eGvzBaVY12AyQWyqBdBJEb2pBPH+XRPB3grpygXQJQwZCF143lzCkMViA3ZoJdHsBWATki2SI/rKq6Z+DrYJUq9UMFhm8Brp/ocFNg8NAH+SVfUleGpgaNBuPe5KA/k4E8lOBZuIOOXR362D4HrfOEqCaAiDLECiai1clckxwMXABU40hehrBZIQZ6Rka6xkFmig7WEDTDweDNIMSgwbQRnMGBwYPhgCGUKDZnQzrGXYw7GTaz/SU6Q3TO4hSJkaoHmEGFMD0HQDe+7rS</latexit><latexit sha1_base64="xr5evtGdzp8SdtdG9izkR7b6IOg=">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</latexit>

tr(ũ(✓)2 ) =<latexit sha1_base64="YqyqzSRKYkQ3/bxheq0JvGfM+d0=">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</latexit><latexit sha1_base64="RCLzO/8yFV3it9pr2xBSHrjcrfw=">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</latexit><latexit sha1_base64="V98scwraBhzvxsItTvz61y7pSOE=">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</latexit><latexit sha1_base64="5SDuM1TfnHzA+5yZf2tEF4chz+4=">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</latexit>

2ei✓ なので，ntop. = 1 2 Z/2Z よりpump不変量は⾮⾃明．
<latexit sha1_base64="3PKxt7wUos491QIFkvnMYnnyp0Y=">AAADQXichVI9SyNRFD0ZXVfjqok2gk3YoGuxZN9EwSAIARtLv6KiCWFm9qmPzBczLwEd8gcEawsrBQuxtN2tbPwDFv4E2UZU2GK32JvJgGtC4h1m3nnn3XPfuW+e7prCl4zdx5Se3g99H/sH4oOfhoZHEsnRDd+pegYvGI7peFu65nNT2LwghTT5lutxzdJNvqlXFhvrmzXu+cKx1+WBy0uWtmeLXWFokqhy4otdDoqelZKOm6kvqEVhFy1N7ut6sF3/ln3F5USaZVgYqXagRiCNKJadZCyGIr7DgYEqLHDYkIRNaPDp2YEKBpe4EgLiPEIiXOeoI07aKmVxytCIrdB3j2Y7EWvTvFHTD9UG7WLS65EyhUl2xy7ZM7tlV+yB/e1YKwhrNLwc0Kg3tdwtjxyNr/1+V2XRKLH/qurqWWIXudCrIO9uyDS6MJr62uHJ89r86mQwxc7ZL/J/xu7ZDXVg116MixW+etrFj059azTnVLPzybX6/trWfWelpByXkEqs1TXz/z8RR5xujdp6R9rBRjajzmSyK7PpfC66P/2YwGdM045zyGMJyyhQ7WNc4wd+KjfKg/KoPDVTlVikGcObUP78A1yJvzw=</latexit>

Kitaev chainのgap関数の位相を回すことでf.p.がpumpされることは，
free fermionの範囲で知られている．[Kitaev,Teo-Kane]
11/92

Pump不変量の計算例②例)⾮⾃明相のKitaev chain with parameter ✓X<latexit sha1_base64="Ns6jXVzKT/11xGO2LV6WS37FPdU=">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</latexit>a†j+1 aj+1 a†j+1 aj + ei✓ a†j a†j+1 + e i✓ aj+1 aj
j ✓
◆
✓
◆
✓
◆
1i
1
1
0i i
1
i ✓2
i ✓2
i
i
A
(✓)
=
A
(✓)
=
e
u(✓)
=
e
H(✓)}✓2[0,2⇡] ! {AA(✓),
u(✓)}
(0) =
A (2⇡),
u(0) = u(2⇡)
A (0) =1 A (2⇡), u(0) = u(2⇡) 1
1✓2[0,2⇡]
✓
◆
✓
◆
✓
◆
1
1
i
i 1
0 ii(✓), u(✓)}1
H(✓)}✓2[0,2⇡] ! {A(0)== Ai (2⇡),
u(0)
=
A
u(2⇡)
(0)
=
A
(2⇡),
u(0)
=
u(2⇡)
ÃA
(✓)
Ã
(✓)
=
ũ(✓)
=
1✓2[0,2⇡]ei✓
ei✓
H(✓) =<latexit sha1_base64="hC4a+pAmSWjIQchtbuP8+XFNCZI=">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</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">AAADPXicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQsoO8ZVZ9ZqGGjaQhhGMQWZmjox1qUgoVIIF6jKQM8ADBQwGYZQhjIDFATkCzIyMsQwpDDkMyQzlDLkMqQy5DGUANk5DIkMxUAYzWDIYMBQABSLZagGihUBWZlg+VSGWgYuoN5SoKpUoIpEoGg2kEwH8qKhonlAPsjMYrDuZKAtOUBcBNSpwKBqcNVgpcFngxMGqw1eGvzBaVY12AyQWyqBdBJEb2pBPH+XRPB3grpygXQJQwZCF143lzCkMViA3ZoJdHsBWATki2SI/rKq6Z+DrYJUq9UMFhm8Brp/ocFNg8NAH+SVfUleGpgaNBuPe5KA/k4E8lOBZuIOOXR362D4HrfOEqCaAiDLECiai1clckxwMXABU40hehrBZIQZ6Rka6xkFmig7WEDTDweDNIMSgwbQRnMGBwYPhgCGUKDZnQzrGXYw7GTaz/SU6Q3TO4hSJkaoHmEGFMD0HQDe+7rS</latexit><latexit 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sha1_base64="EaQklouobvLJ7HZ8V+0zQklHJys=">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</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="xr5evtGdzp8SdtdG9izkR7b6IOg=">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</latexit>

tr(ũ(✓)2 ) =<latexit sha1_base64="YqyqzSRKYkQ3/bxheq0JvGfM+d0=">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</latexit><latexit sha1_base64="RCLzO/8yFV3it9pr2xBSHrjcrfw=">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</latexit><latexit sha1_base64="V98scwraBhzvxsItTvz61y7pSOE=">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</latexit><latexit sha1_base64="5SDuM1TfnHzA+5yZf2tEF4chz+4=">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</latexit>

2ei✓ なので，ntop. = 1 2 Z/2Z よりpump不変量は⾮⾃明．
<latexit sha1_base64="3PKxt7wUos491QIFkvnMYnnyp0Y=">AAADQXichVI9SyNRFD0ZXVfjqok2gk3YoGuxZN9EwSAIARtLv6KiCWFm9qmPzBczLwEd8gcEawsrBQuxtN2tbPwDFv4E2UZU2GK32JvJgGtC4h1m3nnn3XPfuW+e7prCl4zdx5Se3g99H/sH4oOfhoZHEsnRDd+pegYvGI7peFu65nNT2LwghTT5lutxzdJNvqlXFhvrmzXu+cKx1+WBy0uWtmeLXWFokqhy4otdDoqelZKOm6kvqEVhFy1N7ut6sF3/ln3F5USaZVgYqXagRiCNKJadZCyGIr7DgYEqLHDYkIRNaPDp2YEKBpe4EgLiPEIiXOeoI07aKmVxytCIrdB3j2Y7EWvTvFHTD9UG7WLS65EyhUl2xy7ZM7tlV+yB/e1YKwhrNLwc0Kg3tdwtjxyNr/1+V2XRKLH/qurqWWIXudCrIO9uyDS6MJr62uHJ89r86mQwxc7ZL/J/xu7ZDXVg116MixW+etrFj059azTnVLPzybX6/trWfWelpByXkEqs1TXz/z8RR5xujdp6R9rBRjajzmSyK7PpfC66P/2YwGdM045zyGMJyyhQ7WNc4wd+KjfKg/KoPDVTlVikGcObUP78A1yJvzw=</latexit>

Kitaev chainのgap関数の位相を回すことでf.p.がpumpされることは，
free fermionの範囲で知られている．[Kitaev,Teo-Kane]

我々の結果はこのpumpが相互作⽤に対して安定であることを意味している．
12/92

[H.Katsura,et.al., ” Exact ground states and topological order in interacting Kitaev/Majorana chains”]

Pump不変量の計算例③
例)⾮⾃明相のinteracting Kitaev chain with parameter ✓ [Katsura et.al.]H(✓) =<latexit sha1_base64="7SG0H4XLmp1oJLux36KSDcfHFck=">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</latexit>

X<latexit sha1_base64="Ns6jXVzKT/11xGO2LV6WS37FPdU=">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</latexit>ta†j ajj<latexit sha1_base64="OhHQY7tLA8UZLmHIEma6qepi1fI=">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</latexit>

taj+1 a†j + | | ei✓ aj aj+1 + | | e i✓ a†j+1 a†jµ(2a†j aj21) + U (2a†j aj

1)(2a†j+1 aj+1

1)

12

/92

[H.Katsura,et.al., ” Exact ground states and topological order in interacting Kitaev/Majorana chains”]

Pump不変量の計算例③
例)⾮⾃明相のinteracting Kitaev chain with parameter ✓ [Katsura et.al.]H(✓) =

X<latexit sha1_base64="Ns6jXVzKT/11xGO2LV6WS37FPdU=">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</latexit>ta†j ajj<latexit sha1_base64="7SG0H4XLmp1oJLux36KSDcfHFck=">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</latexit>r<latexit sha1_base64="OhHQY7tLA8UZLmHIEma6qepi1fI=">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</latexit>µ(2a†j aj2

特に µ = 4 U 2 + tU +<latexit sha1_base64="dheRnvh9slKqhXauAQWnuk7a4kI=">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</latexit>

taj+1 a†j + | | ei✓ aj aj+1 + | | e i✓ a†j+1 a†j

t2
4

| |

1) + U (2a†j aj

1)(2a†j+1 aj+1

1)のところではG.S.が求まる．

12

/92

[H.Katsura,et.al., ” Exact ground states and topological order in interacting Kitaev/Majorana chains”]

Pump不変量の計算例③
例)⾮⾃明相のinteracting Kitaev chain with parameter ✓ [Katsura et.al.]H(✓) =

X<latexit sha1_base64="Ns6jXVzKT/11xGO2LV6WS37FPdU=">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</latexit>ta†j ajj<latexit sha1_base64="7SG0H4XLmp1oJLux36KSDcfHFck=">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</latexit>r<latexit sha1_base64="OhHQY7tLA8UZLmHIEma6qepi1fI=">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</latexit>µ(2a†j aj2

特に µ = 4 U 2 + tU +<latexit sha1_base64="dheRnvh9slKqhXauAQWnuk7a4kI=">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</latexit>

taj+1 a†j + | | ei✓ aj aj+1 + | | e i✓ a†j+1 a†j

t2
4

| |

1) + U (2a†j aj

1)(2a†j+1 aj+1のところではG.S.が求まる．

このとき周期的なfMPSは以下で与えられる：
✓

◆

1)

✓
✓
◆
1i
0i
1
i
i 1
ÃA(✓)=ũ(✓) == u(2⇡)
Ã
(✓)
=
(|
|
,
µ)
(0) = A (2⇡),
u(0)
=
u(2⇡)
A
(0)
=
A
(2⇡),
u(0)
i✓
1
ei✓
e
<latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="xr5evtGdzp8SdtdG9izkR7b6IOg=">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</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="5SDuM1TfnHzA+5yZf2tEF4chz+4=">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</latexit><latexit sha1_base64="oPlIifBAjKEvUd0E1hiiFvh5hDs=">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</latexit>

◆
1

12

/92

[H.Katsura,et.al., ” Exact ground states and topological order in interacting Kitaev/Majorana chains”]

Pump不変量の計算例③
例)⾮⾃明相のinteracting Kitaev chain with parameter ✓ [Katsura et.al.]H(✓) =

X<latexit sha1_base64="Ns6jXVzKT/11xGO2LV6WS37FPdU=">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</latexit>ta†j ajj<latexit sha1_base64="7SG0H4XLmp1oJLux36KSDcfHFck=">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</latexit>r<latexit sha1_base64="OhHQY7tLA8UZLmHIEma6qepi1fI=">AAADeXichVLNTttAEP4Sl7+0hQAXpF6AiCotEK0DEogTEpce+QsgEWqtnU3q4j/Zm0hg+QV4AQ6cWolD1ceoKvUFOPAIiCOgHtpDx44lIFHSsbw78818s9/Yq3uWGUjGrjNZ5cXA4NDwSO7lq9ejY/nxib3AbfqGqBiu5foHOg+EZTqiIk1piQPPF9zWLbGvH2/E+f2W8APTdXbliSeObN5wzLppcEmQltcWq3WfG2HVbkZhOSqW+cewWuONhvAjLfwc8XhZVN/NV3qmOhPzapKijZJavsBKLLHpbkdNnQJS23THMxlUUYMLA03YEHAgybfAEdBzCBUMHmFHCAnzyTOTvECEHHGbVCWoghN6TGuDosMUdSiOewYJ26BTLHp9Yk5jjl2xb+yO/WLf2Q3727NXmPSItZzQrre5wtPGzqZ2fv+XZdMu8emR1VezRB2riVaTtHsJEk9htPmt0/O7nbXtufAt+8puSf8Xds1+0ARO69643BLbF3306DQ3p1hQz95frlP3Qtf0vZmSajzyVELtvpVP/0QOObo1aucd6Xb2yiV1qVTeWi6sr6b3ZxhvMIsinbiCdXzAJirU+yducY+H7B9lRikq79ul2UzKmcQzU5b+AQTc0RU=</latexit>µ(2a†j aj2

特に µ = 4 U 2 + tU +<latexit sha1_base64="dheRnvh9slKqhXauAQWnuk7a4kI=">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</latexit>

taj+1 a†j + | | ei✓ aj aj+1 + | | e i✓ a†j+1 a†jt2
41) + U (2a†j aj| |1)(2a†j+1 aj+1のところではG.S.が求まる．

このとき周期的なfMPSは以下で与えられる：
✓

◆

1)

✓
✓
◆
1i
0i
1
i
i 1
ÃA(✓)=ũ(✓) == u(2⇡)
Ã
(✓)
=
(|
|
,
µ)
(0) = A (2⇡),
u(0)
=
u(2⇡)
A
(0)
=
A
(2⇡),
u(0)
i✓
1
ei✓
e
<latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="xr5evtGdzp8SdtdG9izkR7b6IOg=">AAADZnichVJLaxNRFP6S8VFHbVOLKLgJhoYKEu60BUtBqLhx4aIP0xY6NdyZnKaXzouZm9A6zB/wD7hwpeBC3PkX3PgHBLvUleKyghsXnkwGHw2JZ5i553z3fOd+585xIk8lWojjUtk4c/bc+YkL5sVLlyenKtNXNpOwG7vUdEMvjLcdmZCnAmpqpT3ajmKSvuPRlnNwv7+/1aM4UWHwSB9FtOvLTqD2lCs1Q63KQ1srr03pvexxKrI5W9OhzqumMbWz1Nb7pGV2667tUEcFaeRLHavDzLTqtm3WLdOmoP0bbVVqoiFyqw47VuHUUNhqOF0qwUYbIVx04YMQQLPvQSLhZwcWBCLGdpEyFrOn8n1CBpO5Xc4izpCMHvC3w9FOgQYc92smOdvlUzx+Y2ZWMSs+iNfiRLwXb8RX8XNkrTSv0ddyxKsz4FLUmnp6fePHf1k+rxr7f1hjNWvsYSnXqlh7lCP9LtwBv/fk2cnG8vpsWhcvxTfW/0Ici3fcQdD77r5ao/XnY/Q43LfkmLjm6Js7rfv2UPejmZpzIvYsRv2xmX//CRMmT411ekaGnc35hrXQmF9brK0sFfMzgRu4iTk+8Q5W8ACraHLtt/iIT/hc/mJMGleNa4PUcqngzOAfM6q/ADXUyuE=</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="5SDuM1TfnHzA+5yZf2tEF4chz+4=">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</latexit><latexit sha1_base64="oPlIifBAjKEvUd0E1hiiFvh5hDs=">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</latexit>◆
1

ntop. = 1 2 Z/2Zよりpump不変量は⾮⾃明．
従ってfreeの場合と同様に，<latexit sha1_base64="3PKxt7wUos491QIFkvnMYnnyp0Y=">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</latexit>

12

/92

[H.Katsura,et.al., ” Exact ground states and topological order in interacting Kitaev/Majorana chains”]

Pump不変量の計算例③
例)⾮⾃明相のinteracting Kitaev chain with parameter ✓ [Katsura et.al.]H(✓) =

X<latexit sha1_base64="Ns6jXVzKT/11xGO2LV6WS37FPdU=">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</latexit>ta†j ajj<latexit sha1_base64="7SG0H4XLmp1oJLux36KSDcfHFck=">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</latexit>r<latexit sha1_base64="OhHQY7tLA8UZLmHIEma6qepi1fI=">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</latexit>µ(2a†j aj2

特に µ = 4 U 2 + tU +<latexit sha1_base64="dheRnvh9slKqhXauAQWnuk7a4kI=">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</latexit>

taj+1 a†j + | | ei✓ aj aj+1 + | | e i✓ a†j+1 a†jt2
41) + U (2a†j aj| |1)(2a†j+1 aj+1のところではG.S.が求まる．

このとき周期的なfMPSは以下で与えられる：
✓

◆

1)

✓
✓
◆
1i
0i
1
i
i 1
ÃA(✓)=ũ(✓) == u(2⇡)
Ã
(✓)
=
(|
|
,
µ)
(0) = A (2⇡),
u(0)
=
u(2⇡)
A
(0)
=
A
(2⇡),
u(0)
i✓
1
ei✓
e
<latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="xr5evtGdzp8SdtdG9izkR7b6IOg=">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</latexit><latexit sha1_base64="3squkNWfeKS1E6i1NCDw+chKomg=">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</latexit><latexit sha1_base64="5SDuM1TfnHzA+5yZf2tEF4chz+4=">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</latexit><latexit sha1_base64="oPlIifBAjKEvUd0E1hiiFvh5hDs=">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</latexit>◆
1

ntop. = 1 2 Z/2Zよりpump不変量は⾮⾃明．
従ってfreeの場合と同様に，<latexit sha1_base64="3PKxt7wUos491QIFkvnMYnnyp0Y=">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</latexit>

interacting fermionic SPTの⾮⾃明相におけるpumpを確認した． 12/92

Appendix

12/92

1-parameter familyとpump

12

/92

[A.Kitaev, ”Unpaired Majorana fermions in quantum wires” Phys.-Usp. 44 131 (2000).]

periodic Kitaev chain●periodic Kitaev chain(時空1+1次元の超伝導の模型)
X X
H = H = a†j+1 aj+1
a†j+1 aja†j+1+ aa†j ja†j+1
aj+1+ajaj+1
a†j+1 aj+1
+ a†j+a†j+1( aj ：complex fermion)<latexit sha1_base64="aQodVYA8SB7AVEu9nyH8oTSGdOo=">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</latexit>Xj<latexit sha1_base64="aQodVYA8SB7AVEu9nyH8oTSGdOo=">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</latexit>jhopping

gap 関数H はgappedかつuniqueな基底状態を持つ．
=
a†j+1 aj+1 a†j+1 aj + a†j a†j+1 + aj+1 aj<latexit sha1_base64="aQodVYA8SB7AVEu9nyH8oTSGdOo=">AAADgnichVLLSiNBFD2xfcZX1M2AG5mgiNFQrYIyOCC4cekrKhgN1W0ZW/tFdyXgNP0D/oALVwouxM9w48qdCz9BZqngxhm86TT4iInVdNe9p865dW51aa5p+JKx+0ST0tzS2tbekezs6u7pTfX1r/tOydNFTndMx9vUuC9MwxY5aUhTbLqe4JZmig3tcKGyvlEWnm849po8csW2xYu2sWfoXBJUSO0v/s77JasQHIQTfCfI7/JiUXgh5Rk15NXp64Uw8xEOa1gZ/o5dSKVZlkVjqDZQ4yCNeCw5fYkE8tiFAx0lWBCwISk2weHTswUVDC5h2wgI8ygyonWBEEnSlogliMEJPaRvkbKtGLUpr9T0I7VOu5j0eqQcwjC7Y5fskd2wK/bAXurWCqIaFS9HNGtVrXALvcc/Vp+/VVk0S+y/qRp6ltjDbOTVIO9uhFS60Kv68p+Tx9VfK8PBCDtnf8n/Gbtn19SBXX7SL5bFymkDPxr1zSkXVLP+yX32PV7TfX2lJI5LkUqo1ZD5/k8kkaRbo36+I7XB+mRWncpOLk+n52fj+9OOQfzEKO04g3ksYgk5qn2LJ/zDf6VZGVNUZapKbUrEmgF8GMrcK4AI1eM=</latexit>j●任意のd次元の閉多様体の上でgappedかつuniqueな基底状態として得られる状態を時空d+1次元Short-Range Entangled(SRE)状態と呼ぶ．
SRE状態はtensor積に対する逆元が存在する：| i ⌦ | ¯i ⇠ |0i ⌦ |0i<latexit sha1_base64="vt6feN+p8fRyVKt3UXgOmaoJ80s=">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</latexit>

ここで“〜”はgapを保った連続変形．
12/92[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:

12

/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:

d次元⾃明状態

13/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:

d+1次元⽬⽅向

13/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:

⾃明相 M

d+1

<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>⇠

d+1次元⽬⽅向

13/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:d次元SRE状態| id<latexit sha1_base64="LXKm3oYA8wmAGDOQiUMKV1iLOWY=">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</latexit>

⾃明相 M

d+1

<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>⇠

d+1次元⽬⽅向

13/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:d次元SRE状態| id<latexit sha1_base64="LXKm3oYA8wmAGDOQiUMKV1iLOWY=">AAADInichVI7SwNBEP5yvuMjPhrBJhgUCwl7UVCsRBtLX4miEbk7V7PkXtxtAhr8A9aChSgoWog/w0bsRCwE/4BYKthYOLkc+AiJc9ztzLfzzX6zN7prCl8y9hRRGhqbmlta26LtHZ1dse6e3ozvFDyDpw3HdLxVXfO5KWyelkKafNX1uGbpJl/R87Pl/ZUi93zh2Mty1+UblrZji21haJKgtWyey1LWyIn9ze4ES7LA4tWOGjoJhDbv9EQiyGILDgwUYIHDhiTfhAafnnWoYHAJ20CJMI88Eexz7CNK3AJlccrQCM3Td4ei9RC1KS7X9AO2QaeY9HrEjGOIPbIr9sZu2TV7YZ81a5WCGmUtu7TqFS53N2MH/Usf/7IsWiVy36y6miW2MRloFaTdDZByF0aFX9w7eluaWhwqDbNz9kr6z9gTu6EO7OK7cbnAF4/r6NGpb41iTjVr39xf3aNV3ddmSspxyVMJtepm/vwTUURpatS/M1LtZFJJdSyZWhhPTM+E89OKAQxihE6cwDTmMI801bZxiBOcKhfKnXKvPFRSlUjI6cMvU56/AG0ms0A=</latexit>

⾃明相 M

d+1

<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>⇠

d+1次元⽬⽅向

13/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:d次元SRE状態| id<latexit sha1_base64="LXKm3oYA8wmAGDOQiUMKV1iLOWY=">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</latexit>

パスの⽅向

⾃明相 M

d+1

<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>⇠

d+1次元⽬⽅向

13/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:d次元SRE状態| id<latexit sha1_base64="LXKm3oYA8wmAGDOQiUMKV1iLOWY=">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</latexit>

パスの⽅向

⾃明相 M

d+1

<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>⇠

d+1次元⽬⽅向

13/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:d次元SRE状態| id<latexit sha1_base64="LXKm3oYA8wmAGDOQiUMKV1iLOWY=">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</latexit>

ループ⽅向

⾃明相 M

d+1

<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>⇠

d+1次元⽬⽅向

13/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:d次元SRE状態

d+1次元GTP

| id

pump

<latexit sha1_base64="LXKm3oYA8wmAGDOQiUMKV1iLOWY=">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</latexit>

ループ⽅向

⾃明相 M

d+1

<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>⇠

d+1次元⽬⽅向

13/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:d次元SRE状態

d+1次元GTP

| id

pump

<latexit sha1_base64="LXKm3oYA8wmAGDOQiUMKV1iLOWY=">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</latexit>

ループ⽅向

⾃明相 M

d+1

<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>⇠

d+1次元⽬⽅向

要点① Md+1 のループはd+1次元のGTPを与える.
⇠<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>

13

/92

[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:d次元SRE状態| id 7!

d+1次元GTP
pump<latexit sha1_base64="rcQF4ZPnHRI8quH5n9t/8iAyVsA=">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</latexit><latexit sha1_base64="LXKm3oYA8wmAGDOQiUMKV1iLOWY=">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</latexit>

ループ⽅向

⾃明相 M

d+1

<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>⇠

d+1次元⽬⽅向

要点① Md+1 のループはd+1次元のGTPを与える.
⇠<latexit sha1_base64="irb0SJJ97vG0tO0nZ42jiz8MdRQ=">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</latexit>

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[ A. Kitaev, ”On the classification of short-range entangled states. ”Talk at Simons Center (2013).]

自明相におけるGTP : Kitaevの予想
d+1次元の⾃明相におけるGTPを構成する:d次元SRE状態| id 7!

d+1次元GTP
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ループ⽅向

⾃明相 M

d+1

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d+1次元⽬⽅向

要点① Md+1 のループはd+1次元のGTPを与える.
⇠
要点②d+1次元のGTPはd次元のSRE状態で分類される.(Kitaevの予想)
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bosonic MPS with sym. G

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bosonic SRE with symmetry G
y H に対するGの表現を固定する：Oû(g)Û y H ：unitaryかつlinear s.t. Û (g) =jinjective MPSに対するGの作⽤は以下の通り：<latexit sha1_base64="C+PLqYGoEH9/d1tOh9phyYZdJFI=">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</latexit><latexit sha1_base64="1xezwUYZ+eeukn5syAw4sEAcX30=">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</latexit>Û (g) |{A }i =

X

=

X

i

<latexit sha1_base64="4oMXFdIL3n5AT+XLAoru3ssMQcY=">AAADeXichVLLbtNAFD2JgZbwaFo2ldgEoqKAImvcIlEhVSqwYcGir7SV6tay3WlqxS/Zk4gymh/gB1iwAokF4jMQEj/Aop+AumwRi7Lg+iEBjRLGsufec++5c+74OrHvpYKx40pVu3T5ysTk1dq16zduTtWnZzbTqJ+4vONGfpRsO3bKfS/kHeEJn2/HCbcDx+dbTu9ZFt8a8CT1onBDHMV8N7C7oXfgubYgyKpb5qEtZEe1uvfNHhfSlE/2pKdMpZbMtB+Ygr8UUiSqlcGWNJQy3f1IpI3Cf6FUwctje7Kr2rqut/NI5imr3mQ6y1dj2DBKo4lyrUTTlQpM7COCiz4CcIQQZPuwkdKzAwMMMWG7kIQlZHl5nEOhRtw+ZXHKsAnt0bdL3k6JhuRnNdOc7dIpPr0JMRuYY9/YR3bKvrJP7Dv7NbKWzGtkWo5odwouj62p17PrP//LCmgXOPzDGqtZ4ACLuVaPtMc5knXhFvzBqzen64/X5uQ99p6dkP537Jh9pg7CwZn7YZWvvR2jx6G+bfI51Rx9cxd1t4e6H80UlBOTZRAajM38+0/UUKOpMS7OyLCxOa8bC/r86sPm8tNyfiZxG3fRohMfYRnPsYIO1f6CE5zhR/Vcu6O1tAdFarVScm7hn6Ut/Aa5r9N4</latexit><latexit sha1_base64="y0NJh0KtseGS1wVwI741q55ovZo=">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</latexit>

tr(Ai1 · · · AiL ) |ig1 , ..., igL iXû(g) |ii =|ji [u(g)]j,ij|ig i := û(g) |iiXj(g · A) =[u(g)]j,i Ai<latexit sha1_base64="f32zTmkJb1Ia6wnoOgdYJCA76CI=">AAADOHicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvIxWSnllRnxlWn19Za2cZkJJZUl9ZqpGtChGvjBZQN9AzAQAGTYQhlKDNAQUC+ICMjQwxDCkM+QzJDKUMuQypDHkMJkJ3DkMhQDITRDIYMBgwFQLFYhmqgWBGQlQmWT2WoZeAC6i0FqkoFqkgEimYDyXQgLxoqmgfkg8wsButOBtqSA8RFQJ0KDKoGVw1WGnw2OGGw2uClwR+cZlWDzQC5pRJIJ0H0phbE83dJBH8nqCsXSJcwZCB04XVzCUMagwXYrZlAtxeARUC+SIboL6ua/jnYKki1Ws1gkcFroPsXGtw0OAz0QV7Zl+SlgalBs/G4Jwno70QgPxVoJu6QQ3e3DobvcessAaopALIMgaK5eFUixwQXAxcw1RiipxFMRpiRnqGxnlGgibKDEzT9cDBIMygxaABtNGdwYPBgCGAIBZrdwrCKYTPDFqZdTA+YnjO9hChlYoTqEWZAAUyfAMCfvAU=</latexit>jtr((g · A) · · · (g · A) ) |i1 , ..., iL i = |{(g · A)i }i<latexit sha1_base64="qvZ2Tc761dh3v0PCoko5i9t/taE=">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</latexit>

i1

iL<latexit sha1_base64="1/Wdwk+25Z6EWaDy6zn7/+X8524=">AAADQnichVI9T9tQFD0xpdCUlkCXSl0ioqAgoegZkEBISKFdOpLQQKQkRLZ5hAf+kv0cKVj5Ax26dujUSh0Qa9d2aQf+QAd+QtupSiUWBm4cS3xESa9lv/POu+e+c5+f7prCl4xdJJSxB+MPJyYfJR9PPXk6nZqZ3fGdwDN42XBMx6voms9NYfOyFNLkFdfjmqWbfFc/ftVb321xzxeO/Ua2XV63tKYtDoShSaIaqVyuWTP2HZneXNgLjzobNT+wGgSqQa65UCe0KDqbe6HoNFIZlmdRpAeBGoMM4thyZhIJ1LAPBwYCWOCwIQmb0ODTU4UKBpe4OkLiPEIiWufoIEnagLI4ZWjEHtO3SbNqzNo079X0I7VBu5j0eqRMI8t+slPWZefsjP1iV0NrhVGNnpc2jXpfy93G9Nvn25f/VVk0ShzeqEZ6ljjAWuRVkHc3YnpdGH196+R9d3u9lA3n2Sf2h/x/ZBfsO3Vgt/4Zn4u89GGEH5361mjOqebwk7vve3Gg++FKSTkuIZVYa2Tm7T+RRJJujXr/jgyCnaW8upxfKq5kCi/j+zOJF5hDjnZcRQGvsYUy1X6HL/iKb8oP5bfyV+n2U5VErHmGO6FcXQNWC78+</latexit><latexit sha1_base64="Io8isy2cmyQUE9YM7HjW5LYXYqg=">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</latexit>

従ってG対称性を持つinjective MPSは以下の条件を満たす：|{Ai }i / |{(g · A)i }i ⇔ ei (g) 2 U(1), U (g) 2 U(n), s.t. (g · A)i = ei (g) U (g)Ai U (g)† .
<latexit sha1_base64="ppiDM02xlcN1lLbJVkZp2dCT/Xc=">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</latexit><latexit sha1_base64="Jon+kVz6MEO6PrUwutpxoZ9dnN4=">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</latexit><latexit sha1_base64="DLJwUrNvc68QsjYMKu/egADl5cQ=">AAADT3ichVLLbtNAFD1xCrTh0RQ2SN1EREWpVEXjAmrFqoJNl33gtlJTIttMk1H9kj2JVKz8AD/AoiuQWFR8BeoG0RULFv0ExLJI3SDEiWupjyjhWvacOXPPnXPH40SeSrQQJwWjOHbj5q3xidLtO3fvTZan7m8kYSd2peWGXhhvOXYiPRVISyvtya0olrbveHLT2XvZX9/syjhRYfBK70dyx7dbgdpVrq1JNcvP5OtUNaK2qrVmew0VVBq+rduxn1q9mjk7V7HIk77EBmSb5aqoiywqg8DMQRV5rIRThQIaeIMQLjrwIRFAE3uwkfDZhgmBiNwOUnIxkcrWJXooUdthlmSGTXaP3xZn2zkbcN6vmWRql7t4fGMqK5gRP8ShOBVfxWfxU/wZWivNavS97HN0zrUyak6+e7h+9l+Vz1GjfaEa6VljF4uZV0XvUcb0u3DP9d2370/Xn6/NpI/FR/GL/j+IE3HEDoLub/fTqlw7GOHHYd8255I1h5/cdd9zA90PV2rmREQmWX9k5uU/UUKJt8a8fkcGwcZ83XxSn199Wl16kd+fcUzjEWrccQFLWMYKLNY+wBd8w7Hx3Tgz/hbzVKOQgwe4EsWJf3kXwUI=</latexit>

i (g)
(g)
i (g)
U(1),
U (g)
2
2 U(1), U (g) はGの射影表現．
2 U(n),ei e(g)
22
U(1),
U (g)
2 はup
U(n),
eU(n),
U (g) e2i U(n),は⼀意，
to2 U(1), phaseで⼀意で，<latexit sha1_base64="DLJwUrNvc68QsjYMKu/egADl5cQ=">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</latexit><latexit sha1_base64="DLJwUrNvc68QsjYMKu/egADl5cQ=">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</latexit><latexit sha1_base64="DLJwUrNvc68QsjYMKu/egADl5cQ=">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</latexit><latexit sha1_base64="DLJwUrNvc68QsjYMKu/egADl5cQ=">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</latexit>(•)}, ei上の条件を満たす, U (•)) | s.t.} をG-sym. inj. matricesと呼ぶ{({Ai }, ei (•) , U {({A(•)) |is.t.}<latexit sha1_base64="l0NcyAubshBr3pwSr+uU5ixoqro=">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</latexit><latexit sha1_base64="l0NcyAubshBr3pwSr+uU5ixoqro=">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</latexit>

14

/92

不変量の構成のアイデア

i!
{A(✓), u(✓)}✓2[0,2⇡] に対して不変量を構成したい．
]

0+1次元系におけるBerry位相：

1+1次元系におけるPump不変量：

C ：状態の位相#L
全空間#
X ：parameter空間
低空間
Berry接続/曲率 = L のtopology

ファイバー

ファイバー

<latexit sha1_base64="+/csAw7twFUG78CcVMfyF/ASVfc=">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</latexit>

A ：Z/2-gr. CSA# G-sym. CSAA#
X ：parameter空間<latexit sha1_base64="8FZL1cvikCziplX+4+LKl3rtA4w=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">AAADInichVK7SgNRED1ZXzE+o41gIwbFQsJdFRQr0cYyPmKCGsLu5qqL+2L3JkGDP2AtWIiCooX4GTZiJ2Ih+ANiqWBj4WSzoCYkzrJ7Z86dM/fM3VEdQ/cEY88hqam5pbUt3B7p6Ozq7umN9q15dt7VeFKzDdtNq4rHDd3iSaELg6cdlyumavCUurtQ3k8VuOvptrUq9hyeMZVtS9/SNUUQtL6Zs4uW4rp2MdsbY3Hm21CtIwdODIEl7GgohE3kYENDHiY4LAjyDSjw6NmADAaHsAxKhLnk6f4+xwEixM1TFqcMhdBd+m5TtBGgFsXlmp7P1ugUg16XmEMYYU/smr2zO3bDXtlX3Volv0ZZyx6taoXLnWzP4cDK578sk1aBnR9WQ80CW5jxteqk3fGRchdahV/YP35fmV0eKY2yC/ZG+s/ZM7ulDqzCh3a1xJdPGuhRqW+FYk41699cte7xmu7rMwXlOOTJhJoNM3//iQgiNDVy9YzUOmsTcXkyPrE0FZubD+YnjEEMY4xOnMYcFpFAkmpbOMIpzqRL6V56kB4rqVIo4PTjj0kv36ujs1c=</latexit><latexit sha1_base64="O3bTm2KaIGDpz3gEtK7P8HscJH8=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

全空間<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit><latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit><latexit sha1_base64="O3bTm2KaIGDpz3gEtK7P8HscJH8=">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</latexit><latexit sha1_base64="zU4Fcgah3eDcozCMeZ8378Zpu3Y=">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</latexit>

低空間<latexit sha1_base64="tVI2NC5pd3uVnTryZK5jVCnJOyU=">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</latexit>

Pump不変量 = A のtopology<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit>

0i
i
i
0i 0
{A
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{A
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,
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A の変換関数は何か？
の時と<latexit sha1_base64="VXBigEG5gwnxyatoC+CZIPzHpjI=">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</latexit><latexit sha1_base64="YEvL6GzZ9XXCWOWoq6cEWzIRL4o=">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</latexit><latexit sha1_base64="rJV2lAqxKXEdBrvvR7Mi0jMcMLI=">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</latexit><latexit sha1_base64="oxZk14Y3FLQIaA2A9cczqDAptgU=">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</latexit>|({Ai }, ei (•) , U (•))i / |({A0i }, ei<latexit sha1_base64="atr/zLwoxYcb18aBSROlwU4acnE=">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</latexit>0(•), U 0 (•))i

14

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14

/92

14

/92