孤立量子系における非平衡ダイナミクスと熱平衡化
東京大学大学院理学系研究科物理学専攻（上田研究室）

濱崎立資
共同研究者：藤本和也、上田正仁「熱場の量子論とその応用」2018/8/28-30

Outline
1. イントロダクション
2. 孤立量子系の熱平衡化の基礎理論
・熱平衡化の定義と目標
・Eigenstate thermalization hypothesis (ETH)

3. 非可積分系と熱平衡化
・非可積分性とETHや熱平衡化との対応
RH and M. Ueda, PRL,

・典型性の仮説はETHを説明するか？ 120, 080603 (2018).

4. 熱平衡化までの非平衡過程
・量子カオスとダイナミクス
・不可逆性と非可換性の等価性
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

Outline
1. イントロダクション
2. 孤立量子系の熱平衡化の基礎理論
・熱平衡化の定義と目標
・Eigenstate thermalization hypothesis (ETH)

3. 非可積分系と熱平衡化
・非可積分性とETHや熱平衡化との対応
RH and M. Ueda, PRL,

・典型性の仮説はETHを説明するか？ 120, 080603 (2018).

4. 熱平衡化までの非平衡過程
・量子カオスとダイナミクス
・不可逆性と非可換性の等価性
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

孤立量子系の統計力学の目標
ミクロな動力学から平衡統計力学を正当化したい（2, 3章の内容）?
ユニタリー時間発展

ミクロカノニカル分布（量子力学）（平衡統計力学）

1929年のJ. von Neumannに始まる問題
2000年代後半から急速に発展

非平衡過程での普遍的な物理を明らかにしたい（4章の内容）孤立量子系の実現
冷却原子系
・mK以下に冷却した原子集団を真空中に閉じ込め
超高真空、極低温 → ほとんど孤立した量子系

・高い制御性（相互作用、格子の形、次元など）
・観測技術の発展（運動量空間・実空間の粒子分布）

イオントラップ
長距離相互作用系などもモデル化

超伝導キュービット
ダイヤモンドのNV中心

J. F. Sherson et al., Nature (2010)

Outline
1. イントロダクション
2. 孤立量子系の熱平衡化の基礎理論
・熱平衡化の定義と目標
・Eigenstate thermalization hypothesis (ETH)

3. 非可積分系のクエンチと熱平衡化
・非可積分性とETHや熱平衡化との対応
RH and M. Ueda, PRL,

・典型性の仮説はETHを説明するか？ 120, 080603 (2018).

4. 熱平衡化までの非平衡過程
・量子カオスとダイナミクス
・不可逆性と非可換性の等価性
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

純粋状態による熱平衡状態の記述
初期状態を純粋状態に取る<latexit sha1_base64="mkWDxvKbqn1+gaWW91FnvpwgRT0=">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</latexit>→時間発展後も純粋状態のまま<latexit sha1_base64="ZctgZwXbjBeueODrdJnQi50lG/Q=">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</latexit>

状態のレベルではミクロカノニカル分布とは異なる<latexit sha1_base64="X2+WIkLrDUGrjwFfYsmMt4CHLYU=">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</latexit>

統計力学で興味のある物理量<latexit sha1_base64="gIZ2oq7km0kR50LSr3ufQbK5r40=">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</latexit><latexit sha1_base64="9H+rIixCgi+yYnj4KnjalPVb8yI=">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</latexit>に着目すれば、が熱平衡値を与える場合がある<latexit sha1_base64="EOmfVr0nGS6Dv6Tjh8/+ZsuN9lI=">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</latexit>

熱平衡化とは？
ある緩和時間

が存在し

において
を要求？
<latexit sha1_base64="1618BJJMR76gcnAbZPSZjXsG25k=">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</latexit>

再帰現象のため上式は成り立たない
任意の

に対しある

が存在し

熱平衡化とは？
ほとんどの時刻で本発表での熱平衡化の定義：
ほとんどの時刻での時、「 が熱平衡化する」と呼ぶことにする(注：文献によって定義は異なる)

熱平衡化の条件
ほとんどの時間でとなるには？

① 定常状態へ緩和するかほとんどの時間で←J. von Neumann,
Zeit. fur. Phys. (1929)

を要求
が十分小さいM. Srednicki, PRE (1994)

時間揺らぎ② 長時間平均が熱平衡分布で記述されるか②①

小量子系での熱平衡化
初期状態の時間発展

エネルギー固有値についての仮定
（非縮退）
（非共鳴）
→時間平均が実行可能

量子系での熱平衡化
時間平均を実行① ある定常状態へ緩和するかが小さい② 長時間平均が熱平衡分布で記述されるか
一般には初期状態と行列要素に依存←行列要素に関してETHを仮定すると、初期状態の詳細に依存せず条件を達成

Eigenstate thermalization hypothesis (ETH)
物理量の行列要素に着目
・ についての非対角項に関するETH：（興味あるエネルギー領域内の）

全ての

に対し

→任意の初期状態について（熱力学極限）の定常状態への緩和を正当化
J. von Neumann, Zeit. fur. Phys. (1929);
M. Srednicki, PRE (1994)

ETH

Eigenstate thermalization hypothesis (ETH)・ についての（対角項に関する）ETH
系のサイズに対しsubextensive

エネルギーシェル<latexit sha1_base64="fzziOix/4gje0EvlAYoUFnaILYU=">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</latexit>

内の全ての
に対し
熱力学極限で一定値へ漸近<latexit sha1_base64="lkRy4Pz52sZN/R5ih3VxwArBhMA=">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</latexit>（subextensiveな誤差は除く）→エネルギー揺らぎの小さいすべての初期状態に対しを正当化

M. Rigol et al., Nature (2008)

ゆらぎ

規格化

ETH

E0

E↵

Outline
1. イントロダクション
2. 孤立量子系の熱平衡化の基礎理論
・熱平衡化の定義と目標
・Eigenstate thermalization hypothesis (ETH)

3. 非可積分系と熱平衡化
・非可積分性とETHや熱平衡化との対応
RH and M. Ueda, PRL,

・典型性の仮説はETHを説明するか？ 120, 080603 (2018).

4. 熱平衡化までの非平衡過程
・量子カオスとダイナミクス
・不可逆性と非可換性の等価性
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

量子多体系の分類
可積分系

F. H. L. Essler and M. Fagotti, J. Stat. Mech (2016).

・自由粒子系、横磁場Ising模型、XY模型など（準粒子の二次形式にマップできる）
・XXZ模型、Lieb-Liniger模型など（Bethe ansatz）
Nandkishore and D. A. Huse,
Many-body localization (MBL) R.
Ann. Review of Cond. Mat.

・強い乱れにより状態が局在

Phys. (2015).

局所保存量により熱平衡化が起こらない

非可積分系…可積分点・MBLから十分遠い系→ ETHによって熱平衡化する（いくつかの例外あり）

熱平衡化の冷却原子実験
非可積分系の熱平衡化

S. Trotzky et al., Nat. Phys. (2012).

1D Bose-Hubbard model 1

0.5

奇数サイトの粒子密度0time (ms)
0

1

2

3

8

12

16

可積分系やMBL下の非熱平衡化（可積分系）T. Kinoshita et al., Nature (2016).; T. Langen et al., Science (2015).
(MBL) M. Schreiber et al., Science (2016).; J. Choi et al., Science (2016).

20非可積分性と熱平衡化
M. Rigol, Phys. Rev. Lett. 103, 100403 (2009).

可積分性を調整できるハードコアボゾン系（一次元、PBC）：可積分系：非可積分系<latexit sha1_base64="knqZFjGNWQPqgfCnZneQ6JcrkAA=">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</latexit>

パラメーターを時刻0で変化させる：
固定
初期状態：初期ハミルトニアンの励起固有状態

非可積分性と熱平衡化
M. Rigol, Phys. Rev. Lett. 103, 100403 (2009).

物理量
定常状態のミクロ
カノニカル分布の精度

可積分点を離れると差は小さくなる→熱平衡化！

可積分点

非可積分領域

非可積分性とETH
W. Beugeling et al., Phys. Rev. E 89 (2014).

可積分性を変化できる格子スピン系
ETHのサイズ依存性などを検証:あるスピンのz成分

可積分系

非可積分系

非可積分性とETH
W. Beugeling et al., Phys. Rev. E 89 (2014).

可積分性を変化できる格子スピン系
ETHのサイズ依存性などを検証:あるスピンのz成分

可積分系

非可積分系

非可積分性とETH
W. Beugeling et al., Phys. Rev. E 89 (2014).

ETHは多くの非可積分系で数値的証拠（例）spinless & spinful fermions,interacting spin chains, Bose-Hubbard models, …
ただし、例外も構成できる

RH, T. N. Ikeda, M. Ueda, PRE (2016).
N. Shiraishi and T. Mori, PRL (2017).

ただし、数学的な証明は存在しない
可積分系やMBLの系ではETHは破れる

ETHはなぜ多くの系で成り立つか？
ETHは特殊な理由のない一般的な系で期待される
J. von Neumann (1929);←典型性の仮説による説明の試み P. Reimann (2015)
キーワード：
・典型性（数学的定理）
・典型性の仮説（物理的仮説）
これが正しければ対角項に関するETHを説明できる
・我々の結果…ほとんどの物理量についての非典型性（典型性の仮説を否定）

行列要素の別表現
・ミクロカノニカルシェル

内で、 を対角化<latexit 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sha1_base64="TsiVoVr6wr8apI8MmNOSMrvBseY=">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</latexit>→固有状態に対する行列要素はユニタリー行列を用いて表される<latexit 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sha1_base64="p8F5a8Tqi2wQQPmXd9gMj5wuH3w=">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</latexit><latexit 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sha1_base64="TENAaG7vsFWPN6723ALz5KNp6+M=">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</latexit><latexit 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典型性は非常に複雑

→確率的議論を用いるはユニタリー行列→ もっとも単純な測度としてハール測度を用いる( の正規直交する列ベクトルをランダムベクトルに取る)<latexit sha1_base64="gkCCEHyP2oT3xwixOQVrMcYoJ9U=">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</latexit>

典型性（数学的定理）（ハール測度で）ランダムなに対し、確率1で空間<latexit sha1_base64="zzw/gzMIcykpVtLN2CVE5/AQQLo=">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</latexit><latexit sha1_base64="aUSyofp9hgtBORuRxt1hZ6A4fos=">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</latexit>

典型的が系のサイズに対し指数的に減少
非典型的典型性の仮説
現実に興味のあるハミルトニアンと物理量から得られるは典型的か？
<latexit sha1_base64="zzw/gzMIcykpVtLN2CVE5/AQQLo=">AAAIh3icjZVLb9NAEMenLQ+3PNrCBYlLRVQEEmo3LYKKU4ELJ9QHaSs1oYqddbCyfsh2Q0vULwBXEAdOIHFAfAHuXPgCHPoREMciceHAfydO0sSJE1eNZ+fxm9kZr20GyoliIY7HxifOnD133picunDx0uXpmdkrW5G/H1qyYPnKD3fMciSV48lC7MRK7gShLLumkttm7bG2b9dlGDm+9yw+DGTJLVc9x3ascgzVemFvJicWBF9zaSGfCDlKrjV/duIbFalCPlm0Ty5J8iiGrKhMEf52KU+CAuhK1IAuhOSwXdIRTSF2H14SHmVoa/itYrWbaD2sNTPiaAtZFP5DRM7RvPgpvogT8UN8Fb/Ev4GsBjN0LYe4m81YGexNv762+XdolIt7TC/oDiJDyDXY4g4jI96G5EFziEifDjJ228plw093L8r0rXIdAWrSfczm6o657alkVxBh7bI2ey4x6lzheTigBqyRvFPuSv3V+5PNBxvzjZvik/iNGX0Ux+I7puTV/1if1+XGB9A130PUy3aFulcVMIuQ6iDp56bEax9riT0rzqdrzcF2lFQ5iOJyPR1Ka44W7mpkRtiXEUIalWDyE5dmaP3oDHsAw+5iZPdVoe6oi6PPoA3SLcxNU+agC9HlKtgx3R5Sl+I+9OMVsepHLA7dq+I592Pu9iWWkn0PJtp8WiysNHUpoba06alW+CluZjrKsC+xPXs3ndxPOfvySNmft2fayTOoDu253BMzalVBRk+C9ptan5bufqRtnV5kn2sFuTNb7WvCt8rnusF2PVePSTqj9pBJfLd1WK4g6dbgbC0P/Swd9M2Y9hiW1Wm/6Vtd7c3q8LvSPeWn2K85uynu5ekaev25Bnyj871f5LSwtbSQFwv59bu51UfJ19qg63QDpz1P92mVntAaFbALSW/oLb0zJo1F456x0nQdH0tirlLXZTz8D6uVsQY=</latexit>→ 論理的には決定できない空間<latexit sha1_base64="gkCCEHyP2oT3xwixOQVrMcYoJ9U=">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</latexit><latexit sha1_base64="zzw/gzMIcykpVtLN2CVE5/AQQLo=">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</latexit>

典型的？
<latexit sha1_base64="gkCCEHyP2oT3xwixOQVrMcYoJ9U=">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</latexit>

非典型的典型性の仮説
典型性の仮説（物理的仮説）
実際の興味あるに対しても（特別な理由がない限り）<latexit sha1_base64="aUSyofp9hgtBORuRxt1hZ6A4fos=">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</latexit>

が系のサイズに対し指数的に減少→

内の（対角項についての）ETHを正当化<latexit sha1_base64="ARy1UJpJ/lrSF29NWfwjdr/FmTg=">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</latexit>

J. von Neumann (1929); P. Reimann (2015)

空間<latexit sha1_base64="gkCCEHyP2oT3xwixOQVrMcYoJ9U=">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</latexit><latexit sha1_base64="zzw/gzMIcykpVtLN2CVE5/AQQLo=">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</latexit>

典型性の仮説
典型的
非典型的ほとんどのfew-body物理量の非典型性
RH and M. Ueda, Phys. Rev. Lett. 120, 080603 (2018).

真に興味があるのはfew-bodyのセッティング
・ハミルトニアンをfew-bodyに取る（任意）・ほとんどのfew-body物理量に対し行列要素は非典型的
は指数的に小さくならない<latexit sha1_base64="aUSyofp9hgtBORuRxt1hZ6A4fos=">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</latexit>

ほとんどの few-bodyの<latexit sha1_base64="QJkJACXAeLIf30gKSxP+XQcEunY=">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</latexit>(few-body)空間<latexit sha1_base64="59QSTMB4Sp17dg73L3AfFtM2DHQ=">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</latexit><latexit sha1_base64="zzw/gzMIcykpVtLN2CVE5/AQQLo=">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</latexit>

典型的
非典型的

ほとんどのfew-body物理量の非典型性
RH and M. Ueda, Phys. Rev. Lett. 120, 080603 (2018).

真に興味があるのはfew-bodyのセッティング
・ハミルトニアンをfew-bodyに取る（任意）・ほとんどのfew-body物理量に対し行列要素は非典型的
現実の系はミクロカノニカルシェル内で非典型的は指数的に小さくならない
ETHの正当化には従来の典型性以外の説明が必要
ほとんどのfew-bodyの<latexit sha1_base64="aUSyofp9hgtBORuRxt1hZ6A4fos=">AAAJDXicjZU7b9NQFIBPWh6mPNrCgugSERWBVKqbFgnEVIGQOkEf9CHVIbKd69TK9UO2G1pcjywsjAxMIDEgZiQGxMIAf4ChPwF1LBILA+eeOE8nTlw1Pvc8vnPuOb627gkrCBk7yo2Nnzp95qxybuL8hYuXJqemL28G7p5v8A3DFa6/rWsBF5bDN0IrFHzb87lm64Jv6bWH0r5V535guc7T8MDjJVurOpZpGVqIqvLUS9XW9suRWuNh9KisasLb1eK55lLnoRarlpNXbSNajsuq7Ue2ZcTxoar7WmfMobqrhZF0e4LWFul2h5+Epdwow2F5qsDmGV35tFBMhAIk14o7Pf4FVKiACwbsgQ0cHAhRFqBBgH87UAQGHupKEKHOR8kiO4cYJjB2D704emioreFvFVc7idbBtWQGFG1gFoH/PkbmYZb9Yh/ZCfvJPrHf7N9AVkQMWcsB3vVGLPfKk6+urv8dGmXjPYRdmMNIH+Ua2sI2IyPeRMlBzQFGurCfsdtmLhP9ZPeCTN8q1eFhTbKP2VzZMbs1lewKAlzbpM2eS4h13qN5WEj1SMNpp9SV+os3J+v312ajG+w9O8YZvWNH7DtOyan/MT6s8rW3SJd8B6OetyqUvaogU0WpjiT53JRo7eKa454F5ZO1FtAWJ1UOothUT5vSnKOBdzEyw+/L8FEalaDTE5dmSP3oDHMAw+xiZPdVYN1BF0eeQRNJN3FukpJHnY9driI7hFtD6hLUh348FVf9iOrQvQqacz/mTl9iKdn3YKJJp8XAlaQuJNSmNj3VCj3FjUxxhn2B7Nm7aed+TNkXR8r+rDXTdp5BdUjPxZ6YUavyMnritd7U8rR09yNta/ci+1wLlNuzlb46+lbpXEdkl3N1iCQzSg+exHdbh+Xykm4Nztb0kM/Sft+MaY9hWa3Wm77Z1d6sFr0r7Q4/QX6N2U1QLztr6PWnGvAbXez9IqeFzYX5Ipsvrt4pLD1IvtYKzMB1PO1FuAtLsAwrsIG7OM5N567lZpTXymflq/Kt4TqWS2KuQNel/PgPe4roiQ==</latexit><latexit sha1_base64="QJkJACXAeLIf30gKSxP+XQcEunY=">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</latexit>(few-body)空間<latexit sha1_base64="59QSTMB4Sp17dg73L3AfFtM2DHQ=">AAAIh3icjZU7b9NQFMdPWx5uebSFBYklIioCCYWbtGpTpgILE+qDtJWaUMXOdbDql2w3tET9ArCCGJhAYkB8AXYWvgBDPwJiLBILA/974iRNEztx1fjc8/idc8/xtXXftsJIiOOx8Ylz5y9c1CanLl2+cnV6ZvbaZujtB4YsGZ7tBdt6NZS25cpSZEW23PYDWXV0W27pe4+Vfashg9Dy3GfRoS8rTrXuWqZlVCOo1kq7M1mRWyguLC8XMyInFvMFITJKM19YWsrkc4KvLMXXqjc78Y3KVCOPDNonhyS5FEG2qUoh/nYoT4J86CrUhC6AZLFd0hFNIXYfXhIeVWj38FvHaifWulgrZsjRBrLY+A8QmaE58VN8ESfih/gqfol/iawmM1Qth7jrrVjp706/vrHxd2iUg3tEL+geIgPIe7BFXUZKvAnJheYQkR4dpOy2ncuEn+pemOpb5zp81KT6mM5VHXM6U0mvIMTaYW36XCLUWeR5WKD6rJG8U+5K49X7k40H63PN2+KT+I0ZfRTH4jum5Db+GJ/X5PoH0BXfRdTLToWqVzUwy5AaIKnnpsJrD2uJPducT9Wahe0orjKJ4nA9XUp7jgbu9siMYCAjgDQqQecnrp+h9KMzzASG2cNI76uNusMejjqDJkh3MDdFyUAXoMt1sCO6O6Qum/swiFfGahCxPHSvNs95EHNnILES7zuZaPJpMbBS1EJMbWv7p1rjp7iV6SjFXmB7+m66uZ9y9vmRsj/vzLSbJ6kO5Tl/JmbUqvyUnvidN7U6Lb396Ld1e5F+rm3I3dkqXx2+dT7XTbarubpMUhmVh4zje63Dcvlxt5KztT3Us3QwMGO/x7CsVudN3+7q2awWvyudU342+7VmN8W9PF3DWX+uAd/o9oc4kyxsFnJ5kcuvLWRXHsVfa41u0i2c9jwt0Qo9oVUqYReS3tBbeqdNave1Ra3Ych0fi2OuU8+lPfwPQbWxWA==</latexit><latexit sha1_base64="zzw/gzMIcykpVtLN2CVE5/AQQLo=">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</latexit>

典型的
非典型的

Outline
1. イントロダクション
2. 孤立量子系の熱平衡化の基礎理論
・熱平衡化の定義と目標
・Eigenstate thermalization hypothesis (ETH)

3. 非可積分系と熱平衡化
・非可積分性とETHや熱平衡化との対応
RH and M. Ueda, PRL,

・典型性の仮説はETHを説明するか？ 120, 080603 (2018).

4. 熱平衡化までの非平衡過程
・量子カオスとダイナミクス
・不可逆性と非可換性の等価性
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

古典カオスと量子カオス
古典カオス：摂動に対する軌道の不安定性: Lyapunov指数

量子的なユニタリー時間発展は線形→ 波動関数のレベルでは初期状態鋭敏性はない

量子カオスの特徴づけ I: 不可逆性
Loschmidtの議論

J. Loschmidt, Wiener Ber. (1876).

系が平衡に達したあとに全ての粒子の速度を反転すると、一定時間後に元の状態へと戻る→ 実際には、このような完全な反転操作は不可能
W. Thomson (1874)

摂動>tixetal/<==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>"=wv/6o/AhWQK1HFl1VkfELP/XH8H"=46esab_1ahs tixetal<

時間反転テスト<latexit 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時間発展
と

時間反転

はカオス系では大きく異なると期待される<latexit sha1_base64="h7mQnUo3lB1B4hxlOVbPRHV0M+k=">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</latexit><latexit sha1_base64="+aFdcxgq1PWmSWLlYJ2gabGyTZ0=">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</latexit>

時間反転テストによる不可逆性
・Loschmidt echo
パラメータをうまく選ぶと「Lyapunov減衰」

RA. Jalabert and HM. Pastawski, PRL (2001).
<latexit sha1_base64="z43pYEF+xBHSgYKEyxmPgK4rqmI=">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</latexit><latexit sha1_base64="YL4qw8+RoZd2+lpxZqGpB/i/Dtc=">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</latexit>

時間発展<latexit sha1_base64="+HonW+8Vn9f49zIojmNoEgdBaI0=">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</latexit>

時間反転 + 摂動<latexit sha1_base64="hfJd2lx6hqXpxOKoJ/2VzlWWOU4=">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</latexit><latexit sha1_base64="LlGr61O0b66xleBq5jxpXtHU31g=">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</latexit>

・物理量に着目<latexit sha1_base64="OPtMP6vORHaL7Nit/zT4z4NoNTY=">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</latexit>

S. Adachi, M. Toda, and K. Ikeda, PRL (1988).;
H. S. Yamada and K. S. Ikeda, PRE (2010).

例）<latexit 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sha1_base64="YL4qw8+RoZd2+lpxZqGpB/i/Dtc=">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</latexit><latexit 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時間発展

摂動

時間反転
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観測量子カオスの特徴づけ II: 非可換性
異時刻の演算子の交換子の二乗
量子的な非可換性の増大の指標

半古典極限…カオス的指数増大！

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Larkin and Ovchinnikov (1969); Kitaev (2015)

短時間

カオス系の場合: Wigner関数

非可換性に関する先行研究
指数増大の現れる量子多体系

Kitaev’s talk at KITP (2015).

・Sachdev-Ye-Kitaev (SYK) モデル @ 有限温度
Majorana fermionsと<latexit sha1_base64="+GtZQXydRQdtfglJaK1KKDS10ew=">AAAIjXicjZVLb9NAEMenLQ8THm3hgsSlIipqJVRtWhAIIVTBAU6oD9JWakIVO+tgZf2Q7YSWqF+AGyckegKJA+ILcOfCF+DQj4A4FokLB/47cZLm5cRV49l5/GZ2xmubgXKiWIjjicmpM2fPnTcuZC5eunxlemb26lbk10JL5i1f+eGOWYqkcjyZj51YyZ0glCXXVHLbrD7R9u26DCPH917EB4EsuqWK59iOVYqh2i4EkbMQL+7NZMWS4GuuX8glQpaSa82fnfpGBSqTTxbVyCVJHsWQFZUowt8u5UhQAF2RGtCFkBy2SzqkDGJr8JLwKEFbxW8Fq91E62GtmRFHW8ii8B8ico7mxU/xRZyIH+Kr+CX+DWU1mKFrOcDdbMbKYG/67fXNvyOjXNxjekW3ERlCrsIWdxgp8TYkD5oDRPq0n7LbVi4bfrp7UapvhesIUJPuYzpXd8xtTyW9gghrl7Xpc4lR532ehwNqwBrJO+Wu1N+8P9l8sDHfuCU+id+Y0UdxLL5jSl79j/V5XW4cga75HqJetyvUvSqDWYBUB0k/N0Ve+1hL7FlxPl1rFrbDpMphFJfr6VBac7RwV2MzwoGMENK4BJOfuH6G1o/PsIcw7C5Gel8V6o66OPoM2iAtYG6aMgddiC5XwI5pcURdivswiFfAahCxMHKviuc8iLk7kFhM9j2caPNpsbDS1OWE2tL2T7XMT3Ez02GKfZnt6bvp5H7O2VfGyv6yPdNOnmF1aM+VnphxqwpSehK039T6tHT3o9/W6UX6uVaQO7PVviZ8K3yuG2zXc/WYpDNqD5nEd1tH5QqSbg3P1vLQz9L+wIz9HqOyOu03faurvVkdfle6p/wU+zVnl+Fenq6h159rwDc61/tF7he2lpdyYim3fie7+jj5Wht0g27itOfoHq3SM1qjPL/b39EHOjKmjbvGQ+NR03VyIom5Rl2X8fQ/wz6zWg==</latexit>の反交換子の二乗平均<latexit sha1_base64="opo/8vbzBHuqOlvb5Wxjhernds8=">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</latexit><latexit sha1_base64="CDJ7R/jsOpV6K6fHVGUK3uCSGGA=">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</latexit>（フェルミオンのため）・有限温度状態でカオスを想起？

の時、Maldacena et al., JHEP (2016).
<latexit sha1_base64="4eVAlshgs0SuqPySHrK/wfyqtKg=">AAAIn3icjZVJb9NQEMenLYspS1s4gMQlEBVaCVUvLRKIU0UPcKq60AU1aWU7L8Hq8yLbDS1RvgBfgAMnQBwQcOfOhS/AoR8BcSwSFw7838RJms2Jq8bzZvnNvBk/2wqUE8VCHI+Mjp05e+68cWH84qXLVyYmp65uRv5BaMsN21d+uG2ZkVSOJzdiJ1ZyOwil6VpKbln7S9q+VZFh5Pjes/gokAXXLHtOybHNGKq9yetLM/FsPnLcjNyt5hUCi+bdTFzbm8yKOcFXplvIJUKWkmvFnxr7Rnkqkk82HZBLkjyKISsyKcLfDuVIUABdgarQhZActkuq0ThiD+Al4WFCu4/fMlY7idbDWjMjjraRReE/RGSGpsVP8UmciB/is/gl/vVlVZmhaznC3arHymBv4vWN9b8Do1zcY3pB9xAZQt6HLW4xUuJLkDxojhDp02HKbhu5SvDT3YtSfctcR4CadB/TubpjbnMq6RVEWLusTZ9LjDof8jwcUAPWSN4pd6Xy6s3J+qO16eod8V78xozeiWPxHVPyKn/sj6ty7S3omu8h6mWzQt2rIph5SBWQ9HNT4LWPtcSeFefTtWZhqyVV9qO4XE+L0pijjbsamhH2ZISQhiVY/MR1M7R+eEapD6PUxkjvq0LdURtHn8ESSDOYm6ZkoAvR5TLYMc0OqEtxH3rx8lj1IuYH7lXxnHsxd3oSC8m++xNLfFpsrDR1PqE2tN1TLfJTXM9US7HPsz19N63cy5x9Yajsu82ZtvL0q0N7LnTEDFtVkNKToPmm1qelvR/dtlYv0s+1gtyarfa14Fvmc11lu56rxySdUXvIJL7dOihXkHSrf7aGh36WDntm7PYYlNVpvukbXe3M6vC70j3lp9ivPrtx7uXpGjr9uQZ8o3OdX+RuYXN+Lifmcqv3s4uPk6+1QTfpNk57jh7QIj2lFdrgt+QH+kJfjVvGE2PZWKm7jo4kMdeo7TKe/wduCbm+</latexit>“a bound on chaos”
conjecture

・SYKなど、いくつかの系でこのバウンドが達成される

モチベーション
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

量子カオスとして多くの指標が提案されている・物理量の不可逆性

・物理量の非可換性

・Loschmidt echo

・固有値の準位統計・固有状態のエンタングルメントエントロピー
・Ehrenfest時間での量子古典対応の破れ

etc.

これらの指標の関係性は重要な問題だが、一般に非自明不可逆性と非可換性の関係は？

不可逆性と非可換性の関係
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

我々のConjecture: 不可逆性と非可換性の等価性の時、局在した初期状態に対し<latexit sha1_base64="UTJSsRjm1MvPXx7pjuRt3Jbbnv0=">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</latexit>

二つの指標は等価！
<latexit sha1_base64="m90PDWfaOh2Swh5ebc9E61wuPs4=">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</latexit>

不可逆性の指標：<latexit sha1_base64="13x6y3WvCczj7y9izghRXVtvKoA=">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</latexit>

非可換性の指標：<latexit sha1_base64="cijbXqzAMoMJYkT1k8SaNe1HGHg=">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</latexit>

上の予想を、量子kicked rotorモデル、
相互作用する量子多体系を用いて正当化⑴B̂ † B̂

Forward time evolution

具体例：量子kicked rotor⑵

RH, K. Fujimoto and M. Ueda. arXiv:1807.02360
ρ̂・ハミルトニアン

Protocol (a): DAB (t) = Tr[ρ̂′t Â† Â]Tr[ρ̂B̂ † B̂]Protocol (b): IAB (t) = Tr[ρ̂t Â† Â]Tr[ρ̃ˆ t B̂ † B̂]Perturbed initial state

Initial state⑴⑸⑹

Forward time evolution

・局在した波束に対応するWigner関数はB̂ † B̂⑵ρ̂ρ̂
時間反転テストに対し不可逆に広がる
′
t=0

⑸

→

⑶

⑹

⑺

ρ̂′t

Â† Â
⑺

時間発展Back to t = 0

局在波束
⑻
p
ρ̃ˆ t

B̂ † B̂

Wigner関数

xÂ† Â

Backward time evolution<latexit sha1_base64="sHkXWOciQB1gTigqQHsNSaQdnBg=">AAAIvnicjZXLbtNQEIanLVATLm1hg8SmIipqJVSdtAgQElIFG9igXuhFakJku8epFd9ku6HFygvwAixgAxILxAuwhg0SbFn0ERDLIrFhwX8mTtLcnLhqPGcu38yZ8bGNwLGjWIjjsfGJM2fPTWrncxcuXro8NT1zZSvyD0JTbpq+44c7hh5Jx/bkZmzHjtwJQqm7hiO3jeojZd+uyTCyfe9ZfBTIkqtXPNuyTT2Gqjx950k5CYL6fLzwoGiEelXGSXFfjxNWpVLnDfp6rjydF4uCr9leoZAKeUqvVX9m4jMVaY98MumAXJLkUQzZIZ0i/O1SgQQF0JUogS6EZLNdUp1yiD2Al4SHDm0VvxWsdlOth7ViRhxtIouD/xCRszQnfoqP4kR8E5/EL/FvICthhqrlCHejESuD8tSraxt/h0a5uMe0T7cQGUKuwha3GRnxFiQPmiNE+nSYsdtmLgt+qntRpm+F6whQk+pjNld1zG1NJbuCCGuXtdlziVHnPZ6HDWrAGsk75a7UXr4+2bi/PpfcFO/Fb8zonTgWXzElr/bH/LAm19+Arvgeol60KlS92gOzCKkGknpuSrz2sZbYs8P5VK152OpplYMoLtfTpjTnaOLujMwI+zJCSKMSDH7iehlKPzrDGsCwOhjZfXVQd9TBUWfQAmkec1OUWehCdLkCdkwLQ+pyuA/9eEWs+hGLQ/fq8Jz7MXf7EkvpvgcTLT4tJlaKupRSm9reqe7xU9zIVM+wL7E9ezft3E85+/JI2Z+3ZtrOM6gO5bncFTNqVUFGT4LWm1qdls5+9Nravcg+1w7k9myVrwHfCp/rhO1qrh6TVEblIdP4TuuwXEHarcHZmh7qWTrsm7HXY1hWu/Wmb3a1O6vN70r3lJ/Dfo3Z5biXp2vo9uca8I0udH+Re4WtpcWCWCys3c6vPEy/1hpdpxs47QW6Syv0mFZpE7t4S1/oO/3QVjRLczW/4To+lsZcpY5LO/wPlyvH2w==</latexit>ρ̂t
Backward time evolutionの増加⑷

Â† Â

ρ̂t

摂動＋Back to t = 0⑻<latexit sha1_base64="6rZk0QtXCz5UDht0HVUaa6kyLfc=">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</latexit>

時間反転

p
ρ̃ˆ t

<latexit sha1_base64="cJTd+E6Ltuc1oLh/MP5MVgDK6r0=">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</latexit>B̂ † B̂具体例：量子kicked rotor
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

我々の主張<latexit sha1_base64="sHkXWOciQB1gTigqQHsNSaQdnBg=">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</latexit>の増加→の増加と等価<latexit sha1_base64="2A9rOOo6X+xEElm7n6UvBDvkuks=">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</latexit>

主張を確認！
ただし<latexit sha1_base64="ZXuArCcVeK52o3pJ3rjpUEHHcX8=">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</latexit><latexit sha1_base64="i4xF7qUP+aFSWXF+NsWnCQiUMZQ=">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</latexit>非可換性はなぜ増大するか？
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

交換子の二乗を展開<latexit sha1_base64="3pojSyLeKXxEOXg8pV9uRYtDmYw=">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</latexit><latexit sha1_base64="HnZ1Ej1R3jP8LB8Z2FauJLjyZMM=">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</latexit><latexit sha1_base64="iG9dgrKFUQjBhkqvnYESu81Jkf8=">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</latexit>

熱平衡状態（多くの先行研究）<latexit 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非可換性はなぜ増大するか？
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

交換子の二乗を展開<latexit sha1_base64="3pojSyLeKXxEOXg8pV9uRYtDmYw=">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</latexit><latexit sha1_base64="HnZ1Ej1R3jP8LB8Z2FauJLjyZMM=">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</latexit><latexit sha1_base64="iG9dgrKFUQjBhkqvnYESu81Jkf8=">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</latexit>

局在した非平衡状態（本研究）<latexit sha1_base64="QGeg8pVH+hbxj8WfFOdm0Q/s8co=">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</latexit><latexit sha1_base64="ObPLdxZtKi/izaF2OmUDa0hgNbA=">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</latexit>の非自明な増大がの増大に寄与する

まとめ
実験技術の発展により、孤立量子系のダイナミクス・
熱平衡化の問題が注目を集めている
ETHは熱平衡化の十分条件を与える
ETHは多くの非可積分系で成り立つことが期待されるが、その証明はいまだ存在しない（従来の典型性の議論でもうまくいかない）
非可換性など、量子カオスのダイナミクスが注目されている（局在した初期状態に対しては、これは不可逆性と等価）補足

（純粋）状態による熱平衡状態の記述と典型性状態はミクロカノニカル分布と一般に異なる
物理量を限定すれば、 も熱平衡状態を記述する<latexit sha1_base64="h7mQnUo3lB1B4hxlOVbPRHV0M+k=">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</latexit>: few-bodyの物理量

典型的な純粋状態は熱平衡状態

S. Popescu et al.,Nat. Phys. (2018).; etc.

エネルギーシェル内のほぼ全ての状態に対し非典型的状態

典型的状態→熱平衡<latexit sha1_base64="2X9aNyj9Uxa7wqhVLjxygrLfeC0=">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</latexit>

ランダム緩和時間の問題
熱力学極限では必ず熱平衡化が見られるか？
非可積分なIsing模型（無限系）
M. C. Banuls et al.,PRL (2011).

初期状態：全てのスピンが+x
物理量の量子期待値とカノニカル分布の予言を比較→の熱平衡化は見られない

緩和時間に達していない可能性

緩和時間の問題（無限に待てば緩和するが）長い緩和時間
・系が可積分系に近い場合など… “Prethermalization”
M. Kollar, F. Alexander, and M. Eckstein, PRB (2011).

・時間発展に非自明な拘束のある系…ガラスに近い状態
C. J. Turner et al., Nat. Phys. (2018).; N. Shiraishi and T. Mori, PRE (2017).

古典ダイナミクスでは遅い緩和は自然・Coarsening dynamics （秩序化過程）
対称性の保たれた相→破れた相へのクエンチ→多くのトポロジカル欠陥→欠陥の緩和は一般に非常に遅い

spin down

spin up緩和時間の問題
半古典レベルでは遅い緩和ダイナミクスは自然
例：超流動相にあるspinor Bose-Hubbard模型を常磁性相から強磁性相へクエンチ→ 磁化間のドメインの遅い秩序化過程 (coarsening)(Experiment) J. Guzman et al., PRA (2011).
(2D, Gross-Pitaevskii) L. A. Williamson and P. B. Blakie, PRL (2016).
(1D, Truncated Wigner approximation) K. Fujimoto, RH and M. Ueda, PRL (2018).

近可積分系での実験
T Kinoshita et al. Nature 440.7086 (2006): 900-903.

一次元ボーズ気体：Lieb-Liniger模型（可積分）：トラップポテンシャル

定常状態の運動量分布
初期状態の記憶→非熱的分布運動量

Many-body localizationの実験
J. Choi et al., Science 352.6293 (2016)

二次元Bose-Hubbard模型

MBL転移を観測量子カオスの特徴づけの例
半古典極限“quantum chaology”
MV. Berry, Phys. Scr. (1989)

ダイナミクスによる特徴づけ
・Wigner関数のダイナミクス W. H. Zurek, RMP (2003).
Ehrenfest
古典的運動方程式の成立限界
時間統計性による特徴づけ
・固有値・固有状態の統計
カオス系 ランダム行列
O. Bohigas, MJ. Giannoni, and C. Schmit (1984); MV. Berry (1977)

半古典極限から離れた量子多体系
非可積分性による特徴付けが有効と期待されている

非可換性に関する先行研究
量子多体系のいくつかの性質の特徴づけ（ただし、カオスとの関係は不明）
D. J. Luitz and Y. B. Lev, PRB (2017).

・情報伝播の特徴づけ
非可積分系における光円錐状の<latexit sha1_base64="1Is7dLqZC3I8XHtot8GvenFKdPk=">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</latexit>

情報の伝播 (Lieb-Robinson限界)を可視化<latexit sha1_base64="j1kmfUbSkszWJEhzBg9qEJGT1M8=">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</latexit><latexit sha1_base64="5pLfI7C2UdMCOwUS6KtgSuiMs1s=">AAAIvXicjVXLbtNAFL1teZjyaAsbJDYVUVErlWqSgkBIiNJuWKG+WylJI9sdp1bHD9lu+rDyA/wAi65AYoH4AbaIDYI1i34CYlkkNiw4M3GS5uXEUez7PPfMvTO24Qs7jBg7GxoeuXT5ylXt2uj1GzdvjY1P3N4MvYPA5BumJ7xg29BDLmyXb0R2JPi2H3DdMQTfMvaXpH+rwoPQ9tz16NjnRUcvu7Zlm3oEU2n88VIpfrlYnY5mnj8sOEG8HlTz+cKeHsVr1ZK9cwLHbF1lUNlMcSdXLI1n2BxT12SnkE2EDCXXsjcx8pkKtEsemXRADnFyKYIsSKcQvzxliZEPW5Fi2AJItvJzqtIocg8QxRGhw7qPexlaPrG60CVmqLJNVBH4B8icpCn2k31k5+wb+8R+sX89sWKFIbkc42nUcrlfGntzd+1v3ywHz4j2aBaZAeR9+KImRkq+BcmF5RiZHh2lrLZey0Kc7F6YGltWPHxwkn1Mx5UdcxpTSWcQQneUNX0uEXg+VfOwgeorC1crVV2pnLw9X3u2OhU/YO/Zb8zoHTtjXzElt/LH/LDCV0+BLvFdZB02GMpe7QKzAKkCJLlvikr3oHOsWah6kmsGvmrCsheKo/g0UepzNPEUA2MEXTECSIMiGGrHdWJI++AYVg8MqwUjva8CvMMWHHkGLSBNY24SZRK2AF0uAzuimT68hOpDN7wCtG6Ihb5rFWrO3TDzXRGLybp7I1rqtJjQJGouQa1bO6e6q3ZxrVI1xZ9T/vTVNGu/VtXnB6q+05hps04vHjJyvi1nUFZ+Sk/8xptanpbWfnT6mr1IP9cCcnO2MtZAbFmd61j55VxdhSQrygie5Ld6+9Xyk271rlaPkHvpqGvFzoh+Ve3Gm77e1faqtnpXOhfihIqrzW5U9fIih/Z4xQHf6Gz7F7lT2MzNZdlcduVRZmEx+VprdI/u47Rn6Qkt0Ctapg2s4pS+0Hf6ob3QuCY0txY6PJTk3KGWSzv8D82fxXA=</latexit>

・MBL相の特徴付け

etc…

<latexit sha1_base64="Lt3dFnnz0VBZ3VJxFnJCIjQNPVQ=">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</latexit><latexit sha1_base64="Lt3dFnnz0VBZ3VJxFnJCIjQNPVQ=">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</latexit>

R. Fan et al., Science Bulletin (2016).

実験もなされている<latexit sha1_base64="eTiSvOpt0sUjIz503mq+PxQLqVg=">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</latexit>

・イオンやNMRでの測定、その他多くの理論提案
M. Gärttner et al., Nat. Phys. (2017).; J. Li et al., PRX (2017).; …

Out-of-ordered time correlators (OTOC)
交換子の二乗を展開→3種類の異時刻相関関数
out-of-time-orderedcorrelator (OTOC)

はの非自明な振る舞いに寄与すると期待され、盛んに研究されている
例：有限温度-はで0に減衰- 他の相関関数ははるかに早い時間
で一定値にOTOCの例：高エネルギー物理
・Sachdev-Ye-Kitaev (SYK) モデル @ 有限温度
Majorana fermions

OTOCは

で

他の相関関数ははるかに早い時間
で一定値に・普遍的なバウンド
Maldacena et al., JHEP (2016).

・Blackholeはこのバウンドを達成
・SYKもこのバウンドを達成(holographic duality?)のように振る舞うカオスを想起？
“a bound on chaos”

OTOCの例: 物性物理
スピン系、ボゾン系でも多くの研究
量子多体系のいくつかの性質の特徴づけ（ただし、カオスとの関係は不明）
・情報伝播の特徴づけ
非可積分系における光円錐状の情報の伝播 (Lieb-Robinson限界)を可視化

・MBL相の特徴付け

etc…

A. Bohrdt et al., NJP (2017).

OTOCの例: 実験・イオントラップ
M. Gärttner et al.,Nat. Phys. (2017).

measure

だけ回転

・NMR

J. Li et al., PRX (2017).

・冷却原子などを用いた実験の理論的提案

非時間順序相関関数(OTOC)への示唆
RH, K. Fujimoto and M. Ueda. arXiv:1807.02360

交換子の二乗を展開<latexit 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sha1_base64="DuQEFBaQGkneVmaJNDkSVM5m1AI=">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</latexit><latexit sha1_base64="HnZ1Ej1R3jP8LB8Z2FauJLjyZMM=">AAAIx3icjZU7b9NQFMdPWx6hPJrCgsQSERW1EqquU0RbJEQbGGBBfdCHVKeR7VynVvyS7YaWKAMrX4CBCaQOiC/AzoLESod+BMRYJBYG/vfGSZomceKq8bnn8TvnnuNr675thRFjJyOjYxcuXrqcujJ+9dr1GxPpyZubobcfGHzD8Gwv2Na1kNuWyzciK7L5th9wzdFtvqVXngr7VpUHoeW5r6JDnxccrexapmVoEVTF9JNnxdpyvj4dzTxW9UCr8Kim7mlRLV/fVUtaWcrLDRk+8bIp5ev1YjrLZhljiqJkhKDMP2QQFhcXcspCRhEmXFmKrxVvcuwrqVQijwzaJ4c4uRRBtkmjEH87pBAjH7oC1aALIFnSzqlO44jdhxeHhwZtBb9lrHZirYu1YIYy2kAWG/8BIjM0xY7ZZ3bKvrMv7Bf715dVkwxRyyHueiOW+8WJd7fX/w6McnCPaI/uIzKAXIEtajMS4k1ILjSHiPToIGG3zVwm/ET3wkTfsqzDR02ij8lc0TGnNZXkCkKsHalNnkuEOhfkPCxQfanhcqeyK9U370/XH61N1e6xT+w3ZvSRnbBvmJJb/WMcrfK1D6ALvouo160KRa9KYKqQqiCJ56Yg1x7WHHu2ZT5Raxa2elxlP4oj62lTmnM0cLeHZgQ9GQGkYQm6fOK6GUI/PMPswzA7GMl9tVF32MERZ9AEaRpzE5QMdAG6XAY7opkBddmyD714Kla9iOrAvdpyzr2YOz2JhXjf/YmmPC0GVoKai6lNbfdUS/IpbmSqJ9hz0p68m3bulzL73FDZd1szbefpV4fwnDsXM2xVfkJP/NabWpyWzn5029q9SD7XNuT2bIWvDt+yPNc1aRdzdSVJZBQePI7vtA7K5cfd6p+t6SGepYOeGbs9BmW1Wm/6ZlfPZ7Xku9I542dLv8bsxmUvz9Zw3l/WgG9080Oc6S9s5mYVNqusPsgu5eOvdYru0F2cdoXmaYme0wptYBdH9IN+0nHqRcpLVVMHDdfRkTjmFnVcqbf/AQntyxE=</latexit>→ 時間相関→ならばOTOC<latexit sha1_base64="d631udMyzooQU++UpnLHva2n04w=">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</latexit>→ 3-OTOC<latexit sha1_base64="3pojSyLeKXxEOXg8pV9uRYtDmYw=">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</latexit>→ OTOC→ 4-OTOC