スピンネットワークに基づく
ハミルトニアン格子ヤン-ミルズ理論
の定式化
早田智也
慶應義塾大学
共同研究者:日高義将 (KEK)
文献: TH-Hidaka, arXiv:2305.05950 [hep-lat],arXiv:2306.12324 [hep-lat]
早田智也

熱場の量子論とその応用, 2023年8月29日

目標

i@t | i = H| i<latexit sha1_base64="rOm7Qv6E2aPDnfkBQLngkHHehd4=">AAAFKXicjZPNbtNAEMcnxUAJH03hgsQlIipCFYo2VQUICSkSlx7TtGkr1SWynU2yqr02600gmLwAL8CBE0gcUB+DCxw5IChXTohDD60ESBwYTxwaGtXpWrZn/ju/mf20A1eEmrHdzNQp4/SZs9PnsucvXLw0k5u9vBb6HeXwmuO7vtqwrZC7QvKaFtrlG4Hilme7fN3efhD3r3e5CoUvV3Uv4Fue1ZKiKRxLo1TPOabNW0JG/JG0lLJ68/1sooRYXvezwgwspYXl1nX+mVkJhaks2XL5/aVRL2ty2RgiZI8krOcKrMio5ceNUmIUIGkVfzbzGUxogA8OdMADDhI02i5YEOKzCSVgEKC2BRFqCi1B/Rz6kEW2g1EcIyxUt/HbQm8zUSX6cc6QaAeruPgqJPMwxz6xt2yfvWc77Dv7c2yuiHLEY+nh34Zb+HqDDDyozzy/uvJzIuvhX0P7kEoduYYm3KURC5xBQEo8F2fAd5++2F+5V52LbrDX7AfO4hXbZe9wHrJ74LxZ5tWXlF0i85jm7FF9iascod7FjG3M3E+8eE2HSjpXGeMqJ+D0sfUewvwEdpWo2FKo58lPJ54kdeI9ishLjz8cWZv2KCI/nWlgTBOfATfcW0UVGxNpQadnnJxcV9GZH12PKinpVLwS6h9nE6dORPEjFNXCG146ep/HjbWFYul2cXF5oVAuJ3d9Gq7BdbiJ9/kOlGEJz04N636APfgFv40d46Pxxfg6CJ3KJMwV+K8Z3/4Cjikv/A==</latexit>

QCD(YM)の時間発展を解きたい!

早田智也

熱場の量子論とその応用, 2023年8月29日

格子ゲージ理論○ 格子化＋経路積分形式＋モンテカルロ

1
hOiQCD =ZZdU e SYM det (D + m)

Nf

O(U )

1
hOiQCD =Z

Z

dU eiSYM det (iD

Nf

O(U )

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m)

符号問題

○ 格子化＋ハミルトニアン形式＋量子計算/TN
早田智也

熱場の量子論とその応用, 2023年8月29日

量子計算の形式的/技術的な困難○ゲージ場のヒルベルト空間(ゲージ群)を有限自由度で近似する○ ユニタリ性の問題<latexit sha1_base64="3fhOlwBFXRdrshi3TpgAhuP3h04=">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</latexit>

・リンク変数（演算子）のユニタリ性

Ua ⇠ Ŝ
Ua Ua† 6= 1
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implies the existence of local basis chanwhich will become essential for SNAQs.
SN state has the form |ji = ⌦0` |j` i wit
resentation label j` 2 {0, 21 , 1, . . . } assigof the resulting lattice. The rules oftum addition lead to an additional “tri|j1 j2 |  j3  j1 + j2 , together withwhich has to be satisfied by all triples othat meet at a vertex, which we indica
product. One can show that the collectistates forms an orthonormal basis of thHilbert space (see [51] and SM).
We propose to regularize the KS mo
the corresponding defining Lie algebra
example, we proceed by replacing the d
the representation theory of SU(2) wi
FIG. 1. (a) In a 2+1D Kogut-Susskind LGT [see Eq. (1)]pression for the quantum group SU(2)gauge fields live on links of a spatial 2D square lattice, which
and the SM). More precisely, we defin
contains elementary plaquettes (blue) and four-vertices (red).
states with j` 2 {0, 12 , 1, . . . , k2 }, which(b) For the gauge-invariant SN basis (see main text) every
four-vertex is split into two three-vertices, resulting in an adcal Hilbert space dimension that physic
ditional link (dashed). (c) The elementary plaquette operator
to a maximum electric flux jmax = k2 .
on the point-split lattice acts on elementary hexagons accordtriangle constraint for triples (j1 , j2 , j3 )ing to Eq. (2). (d) A key feature preserved by our proposed
SU(2)k fusion rule: j1 + j2
j3 and j
q-deformed regularization are local unitary transformations
Toremainclose
totheoriginal
KS mod(“F -moves”) that eﬀect a basis transformation between inelectric energy operator E`2 analogousl
equivalent ways of point-splitting [see Eq. (6)].
熱場の量子論とその応用,
2023年8月29日
cate it to admissible
states. That is, E・ヒルベルト空間のユニタリ同値性<latexit sha1_base64="848qSUGUKGu4JEMylf61+4w1A5o=">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</latexit>

6=○ プラケット相互作用の実装の問題

量子群に基づく定式化を提案する！
早田智也

SU(N)kとグラフの変形ルールI
da 量子次元<latexit sha1_base64="ClZ5CIgKid7B59kjfxJphEF1Zdk=">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</latexit>○ Wilson line

c0
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c a =
da U

a

<latexit sha1_base64="1o+Ss9Bkic02d/GZE+68eflPRYo=">AAACZHichVFNSwJBGH7cvswsLQmCICQxOskYUtFJ6tLRj/wAE9ndRltcd5fdVTDpD9S16NCpICL6GV36Ax38A0F0NOjSodd1IUqqd5iZZ555n3eemZEMVbFsxroeYWR0bHzCO+mb8k/PBIKzc3lLb5oyz8m6qptFSbS4qmg8Zyu2youGycWGpPKCVN/p7xda3LQUXduz2wYvN8SaplQVWbSJSsuVYITFmBPhYRB3QQRupPTgLfZxAB0ymmiAQ4NNWIUIi1oJcTAYxJXRIc4kpDj7HMfwkbZJWZwyRGLrNNZoVXJZjdb9mpajlukUlbpJyjCi7IndsR57ZPfshX38Wqvj1Oh7adMsDbTcqAROFrLv/6oaNNs4/FL96dlGFZuOV4W8Gw7Tv4U80LeOLnrZrUy0s8Ku2Sv5v2Jd9kA30Fpv8k2aZy7how+I/3zuYZBfi8XXY4l0IpLcdr/Ci0UsY5XeewNJ7CKFHJ3LcYoznHueBb8QEuYHqYLH1YTwLYSlT886iew=</latexit><latexit sha1_base64="IjxlY4nZadXdoUaSiSPRKJ75jfQ=">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</latexit>a = (pa , qa , · · · ) Dynkin index<latexit sha1_base64="naKRkbkVpWRP9rM5WChQ0gTqt3c=">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</latexit>○ SU(2)k, SU(3)k,…

<latexit sha1_base64="wK7cjGsp/iPhut1NduMiCY7vdsY=">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</latexit>(0, k)

c<latexit sha1_base64="1o+Ss9Bkic02d/GZE+68eflPRYo=">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</latexit>ab<latexit sha1_base64="71+j3Bl8C6hB4k6eN2M4nHKlmVk=">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</latexit><latexit sha1_base64="IjxlY4nZadXdoUaSiSPRKJ75jfQ=">AAACZHichVFNSwJBGH7cvswsLQmCICQxOskYUtFJ6tLRj/wAE9ndRltcd5fdVTDpD9S16NCpICL6GV36Ax38A0F0NOjSodd1IUqqd5iZZ555n3eemZEMVbFsxroeYWR0bHzCO+mb8k/PBIKzc3lLb5oyz8m6qptFSbS4qmg8Zyu2youGycWGpPKCVN/p7xda3LQUXduz2wYvN8SaplQVWbSJSouVYITFmBPhYRB3QQRupPTgLfZxAB0ymmiAQ4NNWIUIi1oJcTAYxJXRIc4kpDj7HMfwkbZJWZwyRGLrNNZoVXJZjdb9mpajlukUlbpJyjCi7IndsR57ZPfshX38Wqvj1Oh7adMsDbTcqAROFrLv/6oaNNs4/FL96dlGFZuOV4W8Gw7Tv4U80LeOLnrZrUy0s8Ku2Sv5v2Jd9kA30Fpv8k2aZy7how+I/3zuYZBfi8XXY4l0IpLcdr/Ci0UsY5XeewNJ7CKFHJ3LcYoznHueBb8QEuYHqYLH1YTwLYSlT8s6ieo=</latexit>

01<latexit sha1_base64="SK5SNZT8GsNP26VxnZ4Uk49Jh8c=">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</latexit><latexit sha1_base64="G6QFf9+wvcnO/IDWl4oDsvXCDnw=">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</latexit>k<latexit sha1_base64="W35bY+QFNWjW/PCI2wjBOWRFwZE=">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</latexit><latexit sha1_base64="MRnvPRyJvPpmdXItbqoUauy4LKw=">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</latexit>(0, 1)<latexit sha1_base64="ZGtVJgEyONa89E1nJ0dC1v5U7JQ=">AAAFYXicjZS/b9NAFMdfQ0Pb8KNJWZC6RESFUlXRpUItYorEwpimTVupKZHtXJJT7bM5OynByj+AxMzABBIDYmSFiYV/gKEzE2IsEgsD716cNjSq3bNsv/e993nvfulMzxZ+wNjxVOrKdPrqzOxc5tr1Gzfns7mFHd/tKovXLNd21Z5p+NwWktcCEdh8z1PccEyb75qHj3X/bo8rX7hyO+h7/MAx2lK0hGUEKDWy9+ombwsZ8mfSUMrorwwyy2yV3c/UuWyOqY1sgRUZtfykUYqMAkSt4uZSOahDE1ywoAsOcJAQoG2DAT4++1ACBh5qBxCiptAS1M9hABlkuxjFMcJA9RC/bfT2I1Wir3P6RFtYxcZXIZmHJfadfWAn7Bv7yH6yvxfmCimHHksf/yas4usMM3CvMf/y9tafRNbBfwCdMyp25AG04CGNWOAMPFL0XKwh33vx+mTrUXUpvMvesV84i7fsmH3Fecjeb+v9Jq++oewSmSOas0P1Ja5yiHoPM3Yw8yDy9JqOlHiuMsFVLsEFF9Z7CisJ7DZR2lKo58mPJ55HdfQeheTFx5+NrEN7FJIfzzQxpoXPkBvtraKKzURa0OmZJJPrKjrz4+tRJSWe0iuhTjmTOHUpip+jkmt1cWcHp5beaYUxR4nrqWNkRI68MRpvldL5O2TS2FkrltaLDzbXCuVydL/MwiLcgWW8QzagDE/wvNaw/iv4BJ/hy/SP9Fw6m14YhqamIuYW/NfSi/8AFt8qzA==</latexit>(0, 0)
早田智也<latexit sha1_base64="SS1OvqRNojcwhs3h3PVY9TDWg7A=">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</latexit>(1, 0)<latexit sha1_base64="Lyu1T5KOO3bRSs0gEW4nTh122jY=">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</latexit>(k, 0)
熱場の量子論とその応用, 2023年8月29日

SU(N)kとグラフの変形ルールII
da 量子次元<latexit sha1_base64="ClZ5CIgKid7B59kjfxJphEF1Zdk=">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</latexit>○ Wilson line

c0
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c a =
da U
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2

implies the existence of local basis changes [see Fig. 1d]
a = (pa , qa , · · · ) Dynkin index
which will become essential for SNAQs. A general SU(2)<latexit sha1_base64="naKRkbkVpWRP9rM5WChQ0gTqt3c=">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</latexit>

SN state has the form |ji = ⌦0` |j` i with one SU(2) representation label j` 2 {0, 21 , 1, . . . } assigned to every linkof the resulting lattice. The rules of angular momentum addition lead to an additional “triangle” constraint
6|j1 j2 |  j3  j1 + j2 , together
with j1 + j2 + j3 2 N,
which has to be satisfied by all triples of spins (j1 , j2 , j3 )that meet at a vertex, which we indicateii by thei primed
product. One can show that the collection of all such SN
i=1states forms an orthonormal basis of the gauge-invariantc4
Hilbert space (see [51] and SM).
We propose to regularize the KS model by deforming
c3
a04
a4 In thea3 present
the corresponding defining Lie algebra.
c5µ4c5
example, we proceed by replacing the µdataarisingµ05µ3 from
50aaathe representation theory of SU(2) 5with analogous
52 exµ06
pression for the quantum group SU(2)µk6 (see,µ2e.g., [49]µ1c2
and the SM). More precisely, we cdefineSN
a6generalizeda1
c6 a06
6states with j` 2 {0, 12 , 1, . . . , k2 }, which truncates the lo1
cal Hilbert space dimension that physically ccorrespondskto a maximum electric flux jmax = 2 . Additionally, thetriangle constraint for triples (j1 , j2 , j3 ) is replaced by theSU(2)k fusion rule: j1 + j2
j3 and j1 + j2 + j3  k.
To remain close to the original KS model, we define theelectric energy operator E`2 analogously and only truncate it to admissible
states. That is, E`2 is diagonal
and
熱場の量子論とその応用,
2023年8月29日○ 状態のグラフ表現 (2+1次元のYang-Mills理論)

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FIG. 1. (a) In a 2+1D Kogut-Susskind LGT [see Eq. (1)]gauge fields live on links of a spatial 2D square lattice, whichcontains elementary plaquettes (blue) and four-vertices (red).
(b) For the gauge-invariant SN basis (see main text) every
four-vertex is split into two three-vertices, resulting in an additional link (dashed). (c) The elementary plaquette operator
on the point-split lattice acts on elementary hexagons according to Eq. (2). (d) A key feature preserved by our proposed
q-deformed regularization are local unitary transformations(“F -moves”) that eﬀect a basis transformation between inequivalent ways of point-splitting [see Eq. (6)].

・3点頂点にGauss law (singlet条件)を課す
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SU(N)kとグラフの変形ルールIII
c0○ Wilson line

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c a =
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熱場の量子論とその応用, 2023年8月29日<latexit sha1_base64="1o+Ss9Bkic02d/GZE+68eflPRYo=">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</latexit>

SU(N)kとグラフの変形ルールIV○ ユニタリ性[(Fdabc ) 1 ](e,µ,⌫)(d,⇢, ) = [(Fdabc )† ](e,µ,⌫)(d,⇢, ) = [Fdabc ]⇤(d,⇢, )(e,µ,⌫)<latexit sha1_base64="9bQGYIF02RW8To2zfzE6dmzIETc=">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</latexit>○ pentagon方程式
X<latexit sha1_base64="vFiu90vLqpP5cvB5LyBU88Op40Q=">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</latexit>[Fef cd ](g, , )(l,⌫, ) [Feabl ](f,↵, )(k,µ, )=X[Fgabc ](f,↵, )(h, , ) [Feahd ](g, , )(k,⇢, ) [Fkbcd ](h, ,⇢)(l,⌫,µ)

h, , ,⇢

・‘格子間’のユニタリ同値性を保証する
早田智也

熱場の量子論とその応用, 2023年8月29日

量子群に基づく格子ゲージ理論の定式化
K = 1/g 4○ Kogut-Susskindハミルトニアン

X1

H=

2
x,µ

K X
(trUp + trUp† )2

E a2 (x, µ)

p2P

0
0
○ 状態への作用(ハニカム格子を採用)<latexit sha1_base64="rV72XRgremWFiGzCZ/1S3+/ZTEE=">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</latexit>

TH-Hidaka, arXiv:2306.12324 [hep-lat]<latexit sha1_base64="rV72XRgremWFiGzCZ/1S3+/ZTEE=">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</latexit>cc2ca (E a (x,b µ)) ac = C2 (a)<latexit sha1_base64="IFA5DHzIUNUYIt90FhliWhLniTw=">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</latexit>ab<latexit sha1_base64="71+j3Bl8C6hB4k6eN2M4nHKlmVk=">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</latexit><latexit sha1_base64="IjxlY4nZadXdoUaSiSPRKJ75jfQ=">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</latexit><latexit sha1_base64="9Xq4G1qpNNKXix1uOXP9xaEU/6s=">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</latexit><latexit sha1_base64="1o+Ss9Bkic02d/GZE+68eflPRYo=">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</latexit><latexit sha1_base64="IjxlY4nZadXdoUaSiSPRKJ75jfQ=">AAACZHichVFNSwJBGH7cvswsLQmCICQxOskYUtFJ6tLRj/wAE9ndRltcd5fdVTDpD9S16NCpICL6GV36Ax38A0F0NOjSodd1IUqqd5iZZ555n3eemZEMVbFsxroeYWR0bHzCO+mb8k/PBIKzc3lLb5oyz8m6qptFSbS4qmg8Zyu2youGycWGpPKCVN/p7xda3LQUXduz2wYvN8SaplQVWbSJSouVYITFmBPhYRB3QQRupPTgLfZxAB0ymmiAQ4NNWIUIi1oJcTAYxJXRIc4kpDj7HMfwkbZJWZwyRGLrNNZoVXJZjdb9mpajlukUlbpJyjCi7IndsR57ZPfshX38Wqvj1Oh7adMsDbTcqAROFrLv/6oaNNs4/FL96dlGFZuOV4W8Gw7Tv4U80LeOLnrZrUy0s8Ku2Sv5v2Jd9kA30Fpv8k2aZy7how+I/3zuYZBfi8XXY4l0IpLcdr/Ci0UsY5XeewNJ7CKFHJ3LcYoznHueBb8QEuYHqYLH1YTwLYSlT8s6ieo=</latexit><latexit sha1_base64="IjxlY4nZadXdoUaSiSPRKJ75jfQ=">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</latexit>

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早田智也

c1

c6

a6

µ4

c3

a3
µ3

µ1

µ2
a01
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a2
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2023年8月29日<latexit sha1_base64="1o+Ss9Bkic02d/GZE+68eflPRYo=">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</latexit>

平均場近似TH-Hidaka, arXiv:2306.12324 [hep-lat]○ 変分波動関数µ<latexit sha1_base64="5TYLSQkjmU0VaygoCCg+nJ3iyxA=">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</latexit><latexit sha1_base64="IHsVBbFe0QzzvYOLG0ia95y7ams=">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</latexit>| ci =<latexit sha1_base64="wsQ8dz10u97oUgXpP+XUS7M8xhU=">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</latexit>a<latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">AAADL3ichVK7ThtBFL3e8Ip52UkTiQZhGaWyxghIlAqFhtKPGJAcy9pdj2HEvrQ7duSs+IGkJaKgIlIKRMU30PADEaKJ0iJKkNKkyJnxQpQgzIx25u6Ze+7j6FqBIyLJ2EXKeDI0PDI69jQ9PjE5NZ3JPluP/E5o85rtO364aZkRd4THa1JIh28GITddy+Eb1s6qet/o8jASvvdO9gLecM0tT7SFbUpAZbOZybEC02v2vlFMjBwlq+RnUxP0nlrkk00dcomTRxK2QyZF2HUqEqMAWINiYCEsod857VIa3A68ODxMoDs4t/BXT1AP/ypmpNk2sjj4QjBnKc++syN2zc7YMbtkvx+MFesYqpYebqvP5UFz+tOL6q9HWS5uSdt/WQMYFrwH9SQQyUQHqv8Wzja26itA/B4YCo8GRIjvqpFQQWkxyFtpKJHhtc4hoGWgEaWq3e+n+3H/uvqmko/n2Vd2BT0P2QU7haJe98b+VuaVAx3dA+eDVt/VFXioPgYeIuqutm7rCmHFGkVdmKXi/5Nz31hfKBSXC0vlxdzK22SqxmiG5uglJucVrdAalaiG7Jw+0x59MU6Mc+OH8bPvaqQSznP6ZxmXfwD4/rUf</latexit>b<latexit sha1_base64="DxfVtzs4sFRdFRSHU8J7zJPuJl8=">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</latexit>a<latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">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</latexit>

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sha1_base64="DxfVtzs4sFRdFRSHU8J7zJPuJl8=">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</latexit><latexit sha1_base64="tnq1ct4/nOMkvVfrRGzjdmyGSaU=">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</latexit><latexit sha1_base64="5TYLSQkjmU0VaygoCCg+nJ3iyxA=">AAADMXichVK7ThtBFL3eQDCPhFeDRIOwQFTWGPESlUWalDbGBgkQ2l0GZ8S+tDs2Mit+IBJtKKiCRIEo+QQafoDCHUoXuQSJhoIz44UooDgz2pm7Z+65j6NrBY6IJGPNlPGhq/tjT7q3r3/g0+fBoeGRSuTXQpuXbd/xww3LjLgjPF6WQjp8Iwi56VoOX7f2v6j39ToPI+F7a7IR8G3XrHpiT9imBFTacms7QxmWZXpNvDdyiZGhZBX84dQAbdEu+WRTjVzi5JGE7ZBJEfYm5YhRAGybYmAhLKHfOR1RH7g1eHF4mED3cVbxt5mgHv5VzEizbWRx8IVgTtAUu2UX7J7dsEv2mz39M1asY6haGritNpcHO4Pfx0qP/2W5uCV9+8PqwLDg3akngUgmOlD97+Lcw1Z9BYjfAEPhUYcI8Ws1EiooLTp5Kw0lMizpHAJaBhpRqtrtfuqHJ/el5dWpeJqdsRb0/Mma7BqKevUH+7zIV091dA+cA62+qyvwUH0MPETUI2291BXCijWKujBLubeT896ozGZzC9n54lwmv5JMVZrGaZJmMDmLlKevVKAyslfpmH7QiXFlNI0741fb1UglnFH6axmtZ6/nthA=</latexit>

0i =<latexit sha1_base64="fDUMQx09KrfD1yGj5cLRgU95zBE=">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</latexit>bf○ グラフ(fusion rulesだけ)を使って期待値を計算できる
早田智也

熱場の量子論とその応用, 2023年8月29日

平均場近似TH-Hidaka, arXiv:2306.12324 [hep-lat]○有効ハミルトニアン<latexit sha1_base64="O89iHGxJ/w3Tu4nGuslY18xCXAE=">AAADDnichVHLahRBFL3dvuL4yEQ3gpvGITKjcagZNBFBGM0mIJE8nCSQzjRVNTU9TfpFdc1A7NQPuHLnwpWCgmTvTly4yQ+4yCeIywhuXHj7gaJBc5uue+6pe26fqmax7yWKkAPDPHHy1OkzE2cr585fuDhZnbq0lkQjyUWXR34kNxhNhO+Foqs85YuNWAoaMF+ss+35bH99LGTiReETtROLrYC6oTfwOFVIOdU9O6BqyKmfLuj7lmXZyShwUjrDZri25p12nTce91KundRmVKZUM20PJOVp30FWZ4lqXJjW1q4dJ16dNnZ77QKyDFZuWYXikU7b+td8nXX0bmC77YuBqi8iy/TNxZ7dp64rZF7a0nOHqlEOc6o10iR5WEdBqwQ1KGMpqr4FG/oQAYcRBCAgBIXYBwoJPpswCzEyW5AiIxF5+a4ADRVUjrBHYAdFdhtXF6vNkg2xziYmuZrjN3x8JSotmCafyTtySPbJHvlCfvxzVprPyJzsYGaFVsTO5LMrq9+PVQWYFQx/q/7rWcEA7uZePfQe50x2Cl7ox09fHK7eW5lOr5PX5Cv6f0UOyCc8QTj+xt8si5WXUMHrb/192UfBWrvZmm3eWb5d6zwsf8QEXIVrUIcWzEEHFmAJusCNKWPO6BgPzOfme/OD+bFoNY1Scxn+CHP/J6oDyhw=</latexit>

H=

X

a,b,c

c dc
C2 (c)Nāb| (a)|2 | (b)|2
da db

KX
2

⇤

(a)

a,b

○ 時間依存Ginzburg-Landau方程式<latexit sha1_base64="vu6zWOR41cmbbYEDBl6a5JMG888=">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</latexit>

X⇣†
Mab + Mab

⌘

(b)

a
Mab = N(1,0)b<latexit sha1_base64="2gMP9YihotBM8UDE0nzmVgSpdgQ=">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</latexit>dc
c dc| (⌧, b)|2 (⌧, a) + C2 (c)Nb̄a| (⌧, b)|2 )dā dbda db̄
b,c
!
⇣
⌘
X
X
K
†
+
Mab + Mab(⌧, b) ⇤| (⌧, b)|2 1(⌧, a)
2@⌧ (⌧, a) =c
(C2 (c)Nāba,bb○ 2+1次元のSU(3)kの基底状態を調べた
早田智也

熱場の量子論とその応用, 2023年8月29日

Cutoff依存性の検証
100

0.8

80

htrU(1,0)i/d(1,0)

1.0htrU(1,0)i 1/g 2 = 8/3

0.6

60

2

hE 2i

htrU(1,0)i 1/g = 40/3
htrU(1,0)i 1/g 2 = 80/3

0.4

40hE 2i 1/g 2 = 8/3

hE 2i 1/g 2 = 40/3

0.2

2

20

2

hE i 1/g = 80/3
0.0

0
0

10

20

30

40

50

k

・cutoffを臨界値より十分大きくとって外挿するが必要ある
早田智也

熱場の量子論とその応用, 2023年8月29日

相図
50<latexit sha1_base64="7wo/0y5midvqnsgrispajLzCB/4=">AAACj3ichVE9S8NAGH6M3/Gr6iK4FIsiDuXSQcVBCi66+VUVrJQkvdajySUkaUFD/4CTm4OTgoi4uyq4+Acc/AniqODi4Ns0IirqG3L33PO+z3vP3RmuJfyAsYcWpbWtvaOzq1vt6e3rH0gMDm34TtUzec50LMfbMnSfW0LyXCACi2+5Htdtw+KbRmWhkd+scc8XjlwP9ly+Y+tlKUrC1AOiCompvMHLQoa6Jcpyqq6Gec9OLjiyRP2KdTXPZfEjV0ikWJpFkfwJtBikEMeykzhHHkU4MFGFDQ6JgLAFHT5929DA4BK3g5A4j5CI8hx1qKStUhWnCp3YCo1lWm3HrKR1o6cfqU3axaLfI2US4+yeXbBndscu2SN7+7VXGPVoeNmj2WhquVsYOBhZe/1XZdMcYPdT9afnACXMRl4FeXcjpnEKs6mv7R89r82tjocT7JQ9kf8T9sBu6QSy9mKerfDVY6j0ANr36/4JNjJpbTqtrWRS2Uz8FF0YxRgm6b5nkMUilpGjfQ9xhWvcKIPKjDKvZJulSkusGcaXUJbeAWbGmhs=</latexit>

Confined

40

k

30

20

<latexit sha1_base64="gfTWNXM4SnUqGdiQKabERp2SG28=">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</latexit>

10

Topological<latexit sha1_base64="uyNn8X+qHHdbjYe+Fk9yC/EpCmU=">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</latexit>(Stringnet condensation)
0
0

5

10

15

20

25

30

35

40

1/g 2

・ゲージ群の有限自由度による近似に伴う人工的な相転移・外挿時のcutoff依存性に気を付ける必要がある
早田智也

熱場の量子論とその応用, 2023年8月29日

モンテカルロ計算との比較
1.0

htrU(1,0)i/d(1,0)

0.8

0.6

0.4

Monte Carlok = 50k = 20k = 10

0.2

P. Bialas et. al., Nucl. Phys. B 807 (2009) 547

0.0
0

5

10

15

20

1/g

25

30

35

40

2

・cutoffを大きくとればモンテカルロの結果に近づく・平均場が定量的にも使えることが分かった
早田智也

熱場の量子論とその応用, 2023年8月29日

張力の結合定数依存性
2.0pæ(1,0)

1.5

1.0

Monte Carlok = 50k = 20k = 10

0.5

0.0
0

5

10

15

20

1/g

25

30

35

40

2

・面積則を示せたが定量的には不十分
・平均場の限界 (Wilson loop同士が相関していない)
早田智也

熱場の量子論とその応用, 2023年8月29日

まとめ○ 量子群に基づく定式化を整備した・Wilson loop = ユニタリ演算子

・Yang-Mills理論の効率的な実装→ F moveの効率的な実装
・平均場近似を用いたcutoff依存性の解析○ 今後の展望・平均場近似の改善

・量子計算機/テンソルネットワークを用いた計算の実装

・時間依存平均場近似による非平衡状態の計算
・3+1次元への拡張

・量子群ゲージ理論のダイナミクス

・フェルミオン
早田智也

熱場の量子論とその応用, 2023年8月29日

expressed by a loop encircling the defect,

Wilson loopの計算 I

oop encircling the defect,

tr Ud
tr Ud

=

=

.

(3.9)

.

Using the deformation rules in eqs (2.5)-(2.7):

mation rules in eqs (2.5)-(2.7):
eq. (2.5)

eq. (2.7)=)

eq. (2.5)=)

eq. (2.7)=)=)

eq. (2.7)

eq. (2.7)=)

eq. (2.7)=)=)

eq. (2.7)

eq. (2.7)

=)

=)

早田智也

eq. (2.6)

=)

(3

=)

(3.10)=)

eq. (2.7)

eq. (2.7)

eq. (2.7)=)

eq. (2.6)

=)

,

,

we obtain eq. (3.7), by tracing the changes of indices.
熱場の量子論とその応用, 2023年8月29日

4.1

Expectation values of observables

where we introduce a shorthand notation,
Let us evaluate the expectation value of an operator hOi := h |O| i. Asµana example,
X we
YX
=
.
onsider the expectation value of a Wilson loop, O = tr Ud (@S),@S represents
path
a wherecbb the
f 2F af{a}af the Wilson loop, which is the boundary of an area S. The expectation value is given asµ
We repeatedly used the fact that all Wilson loops commute with e
htr Ud (@S)i
µ
0
1
c
b a c
Y X
⇤
tr Ubf (f ) | 0 i = bf,
= h | tr U (@S)@(a ) (b ) tr U (f ) tr U (f )A µ
| i

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0

d

f

0

X

f 2F af ,bf

āf

f

1

Y

bf

0

(4.2)

Y of the Wilson loops in eqs. (2.13) and (2.14)
and the fusion rulef0⇤
and@then apply
on cthestate.
atrsmalllatticeboundaries for illustr
A h 0 | We=
(af 0 )h (b0f|0 )N
tr theUdconsider
(@S)
UThe
| 0 i , withcf (f )nontrivialāf 0 bf 0
obtainsecond line.
part of the calculation is no
the computation,and the state (4.6) is representedas
f 2F
f 0 2F{a},{b},{c}h0 | tr Ud (@S)

Y

tr Uc (f ) |0i .

fwhere we introduce a shorthand notation,
f 2F
a
X YX
µ
To this=. end, let us consider the state
(4.3)
a b
c
b {a} f 2F af
Y
a
tr
U
(@S)
tr Ucf (f ) | 0 i ,dµf 2FWe repeatedly used the fact that all Wilson
Y loops commute with each other. We also used
Ud (@S)
tr Ucf (f ) | 0 i =
,
µ
c
f 2F
b a c
µ
tr Ubf (f ) | 0 i = bf,
(4.4)
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(2.13) and (2.14) on the same plaquette to
0 と内積をとる
btain the second line. The nontrivial part of the calculation is now the evaluation of|<latexit sha1_base64="3GRdXnQZXQ15RwExl61Pztm9pIk=">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</latexit>iwhere the red line is the pathshaded by light gray is S. By a
Yof @S, and the inside region
熱場の量子論とその応用, 2023年8月29日

早田智也

h

| tr U (@S)

tr U (f ) |

i.

(4.5)

a

a
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a<latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">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</latexit>a<latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">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</latexit>aa<latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">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</latexit><latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">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</latexit>

Wilson loopの計算 IIIa<latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">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</latexit>aa<latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">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</latexit><latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">AAADL3ichVK7ThtBFL3e8Ip52UkTiQZhGaWyxghIlAqFhtKPGJAcy9pdj2HEvrQ7duSs+IGkJaKgIlIKRMU30PADEaKJ0iJKkNKkyJnxQpQgzIx25u6Ze+7j6FqBIyLJ2EXKeDI0PDI69jQ9PjE5NZ3JPluP/E5o85rtO364aZkRd4THa1JIh28GITddy+Eb1s6qet/o8jASvvdO9gLecM0tT7SFbUpAZbOZybEC02v2vlFMjBwlq+RnUxP0nlrkk00dcomTRxK2QyZF2HUqEqMAWINiYCEsod857VIa3A68ODxMoDs4t/BXT1AP/ypmpNk2sjj4QjBnKc++syN2zc7YMbtkvx+MFesYqpYebqvP5UFz+tOL6q9HWS5uSdt/WQMYFrwH9SQQyUQHqv8Wzja26itA/B4YCo8GRIjvqpFQQWkxyFtpKJHhtc4hoGWgEaWq3e+n+3H/uvqmko/n2Vd2BT0P2QU7haJe98b+VuaVAx3dA+eDVt/VFXioPgYeIuqutm7rCmHFGkVdmKXi/5Nz31hfKBSXC0vlxdzK22SqxmiG5uglJucVrdAalaiG7Jw+0x59MU6Mc+OH8bPvaqQSznP6ZxmXfwD4/rUf</latexit>a a inute the partial inner product with h 0 | for three edges on the boundary
ates with trivial representation survive, since h0|ai = a0 . Thus, we can replace
Where the Wilson loops overlap, we can use eq. (2.6) to evaluate
erator by
Where the Wilson loops overlap, we can use eq. (2.6) to evaluate<latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">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</latexit><latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">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</latexit>µµ<latexit sha1_base64="5TYLSQkjmU0VaygoCCg+nJ3iyxA=">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</latexit><latexit sha1_base64="5TYLSQkjmU0VaygoCCg+nJ3iyxA=">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</latexit>p
cc ccµµ
Xp dc
cc cc
X
cc
p
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d
p cdd dc1 ccµµ(4.9)= c,µcc11
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! c1 d d
dd.

<latexit sha1_base64="WP/Vp+fFGalJ4FEVz86B6brUeLU=">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</latexit><latexit sha1_base64="WP/Vp+fFGalJ4FEVz86B6brUeLU=">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</latexit>ddHere we used the same reduction as eq. (4.9). Repeating this procedure until t
Here we used the same reduction as eq. (4.9). Repeating this procedure until theshrinks to a point, we obtainn of this plaquette to the expectation value (4.2) becomes trivial:
shrinks to a point, we obtain

X

⇤

c 0
(a) (b)Nāb
c =

a,b,c

X

⇤

(a) (b) ba = 1,

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a,b

dc1

(4.10)

c1<latexit sha1_base64="WXg5Hb3q++/GekQ6YR94c5iffZ8=">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</latexit><latexit sha1_base64="EEKD67W8LgvmTJaH6rVjCHp/J/g=">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</latexit>

0 = a . Repeating this procedure until we reach @S, eq. (4.7) becomes
here we employ Nābdb<latexit sha1_base64="WP/Vp+fFGalJ4FEVz86B6brUeLU=">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</latexit><latexit sha1_base64="StseTYhaTyd7nRugK2UpaTXfGho=">AAADMXichVK7ThtBFL1eILyDTRqkNCgWKJU1TniJCkFDCRjbSARZu8tgRuxLu2Mjs+IHkGhDQUWkFIgyn0DDD1C4Q3SIkkg0FJwZb0CAYma0M3fP3HMfR9cKHBFJxpopo6Oz60N3T29f/8Dgx6F0ZrgU+bXQ5kXbd/xwzTIj7giPF6WQDl8LQm66lsPL1s6Cei/XeRgJ31uVjYBvuGbVE1vCNiWggl35XklnWY7pNfrWyCdGlpK15GdSA/SDNsknm2rkEiePJGyHTIqw1ylPjAJgGxQDC2EJ/c5pn/rArcGLw8MEuoOzir/1BPXwr2JGmm0ji4MvBHOUxtglO2V37IKdsRv28N9YsY6hamngtlpcHlSGDkYK9++yXNyStp9ZbRgWvNv1JBDJRAeq/02cW9iqrwDxG2A<latexit sha1_base64="StseTYhaTyd7nRugK2UpaTXfGho=">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</latexit>

c3

c3<latexit sha1_base64="0sPxWgQ8pYtt22mInrn0UH/Tfn8=">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</latexit>

c4

c2

c4

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c2<latexit sha1_base64="GshLbhy5tbPZ2saWD77V+4wnUsI=">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</latexit>

c5

c5<latexit sha1_base64="WP/Vp+fFGalJ4FEVz86B6brUeLU=">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</latexit><latexit sha1_base64="WP/Vp+fFGalJ4FEVz86B6brUeLU=">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</latexit><latexit sha1_base64="6zAsdiEfALesqw0+FU0VhFbeB64=">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</latexit>

c6

c6<latexit sha1_base64="sUdw8zuRQBAexEjJqmMMY3gnC3E=">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</latexit>

c µ

c

!

d

c1

.µ

c

c7

c7

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dd

dd

0
1
0 Y 1d¯@ cd¯ c A!Y
dd
! @ f 2Si A
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d

d

f 2S

(4.11)c1

01
0 Y 1d¯@ cd¯ c A
= ddY
iA
@
dd
= dd
f 2S
dd

.

f 2S

– 11
Where the Wilson loops overlap, we
can–use eq. (2.6) to evaluate

dd

cc11

cc µµ
p
cc loopの数を数える
内側のWilson
X
d
cc

早田智也

p c
=
dd dc1 µµ
cµ

cc11

!

d¯
c1

dd

.

d

(4.12)

熱場の量子論とその応用, 2023年8月29日

.

4.1

Expectation values of observables

where we introduce a shorthand notation,
Let us evaluate the expectation value of an operator hOi := h |O| i. Asµana example,
X we
YX
=
.
onsider the expectation value of a Wilson loop, O = tr Ud (@S),@S represents
path
a wherecbb the
f 2F af{a}af the Wilson loop, which is the boundary of an area S. The expectation value is given asµ
We repeatedly used the fact that all Wilson loops commute with e
htr Ud (@S)i
µ
0
1
c
b a c
Y X
⇤
tr Ubf (f ) | 0 i = bf,
= h | tr U (@S)@(a ) (b ) tr U (f ) tr U (f )A µ
| i

Wilson loopの計算 IV<latexit sha1_base64="wPY8mtMgQBn4Trh+ix2KgbsIm3g=">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</latexit><latexit 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0

=

d

X

{a},{b},{c}f0@f 2F af ,bfY⇤(af 0 )

f 0 2F

āf

f

bf

0

By plugging the obtained numerical factor into(4.2)
eq. (4.2), the expec
loop is 1

Y of the Wilson loops in eqs. (2.13) and (2.14)
and the fusion rulecf 0
A(bf 0 )Nā 0 b 0 hobtain
Ud (@S)
tr UThe| 0 i , part of the calculation is no0 | tr thecf (f )nontrivial
second line.
f f
Y X
cf
f 2F
⇤

Y (af ) (bf )Nā b
htr Ud (@S)i = ddgging the obtained numerical factor into eq. (4.2), the expectation
value
the
Wilson
h of
|
tr
U
(@S)
tr Ucf (f ) | 0 i . f f
0 f 2S ad ,b ,cf f f
where
shorthand factor
notation,
gging wetheintroduce
obtainedanumericalinto eq. (4.2), the expectation value of the
Wilsonf 2FP✓aa⇤ (a) (b) ◆|S|
X YXµ
a,b Ndb
To thisthe= dstate=. end, let us consider
(4.3)
d¯
d
Y
X
dd
cf
c
a b
c
f
b
⇤
af ) (b )N
f 2F (a¯Y
htr Ud (@S)i = dd Y {a}
d
f
ā b
X
af
Ud (@S)|S| dtr),Ucf (f ) | 0 i ,cff f dcfd
= dtr⇤
d exp(µ dd f 2S af ,bf ,cf (af ) (bf )N
htr Ud (@S)i =āf bf eachf 2F
We repeatedly used the fact that all P
Wilson loops commute withdd other. We also used✓◆|S|
(4.14)⇤ (a) (b)f 2S af ,bN
f ,ca
f
µ
a,b dbwe employ N d¯ = N a in the second line, |S| represents the
where
P
✓
āb
db
(4.14)a ⇤ (a) (b) ◆|S|c
b a c = dd
N
d
d
a,b
db
=trddUbf (fplaquettesµ) | 0 i = bfinside the ,path @S, and d is the string(4.4) tension,= dd exp( |S| ddd),
dd= dd exp( |S| d ),
P
:
=
ln
d
a ⇤ (a) (b) .
¯daNa,b dbto
weNābrule= ofNdbtheinWilson
the second|S|(2.13)
represents
the numberof theplaquettehexagonalnd employthe fusion
loops line,
in eqs.
and (2.14)
on the samed¯patha inandttes
insidethestringbtain
the second
nontrivial
partof |S|
thetension,
calculation theis now
the evaluationof
we
employNāb=line.
N@S,
thesecondline,
represents
numberof the hexagonald is thedbThe
If d is of order unity, the Wilson loop exhibits the area law. A
Y tension,
ttes inside
the path @S, and d is thestring早田智也
熱場の量子論とその応用,
2023年8月29日
functionthe(fstringh | tr U (@S) ddintr U
) | i net
. condensed state is (a)(4.5)= da /D [42]. 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Casimir scalingの検証
500

350

1/g 2 = 1 k = 50
1/g 2 = 3 k = 50
1/g = 10 k = 50
1/g 2 = 30 k = 50

1/g 2 = 3 k = 10250

æa/æ(1,0)æa/æ(1,0)

1/g 2 = 3 k = 20

300

2

400

1/g 2 = 3 k = 50

300

200

200
150
100

100
50
0

0
0

50

100

150

200

250

0

50

C2(a)/C2(1, 0)・弱結合/強結合極限から離れた状態
・cutoff依存性が見えない表現
早田智也

100

150

200

250

C2(a)/C2(1, 0)<latexit sha1_base64="D0G1y7kqzaPT7WYLbgbK4nPwe8A=">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</latexit>a / C2 (a)熱場の量子論とその応用, 2023年8月29日

On going○ 量子クエンチ

cf. TH-Hidaka (KEK), PRD 103, 094502 (2021)

SU(2)

SU(3)

熱化？
早田智也

熱場の量子論とその応用, 2023年8月29日

On going II○ 量子クエンチ[SU(2)]早田智也

cf. TH-Hidaka (KEK), PRD 103, 094502 (2021)

熱場の量子論とその応用, 2023年8月29日