NJL 模型

下地寛武 松尾大和 稲垣知宏
広島大学大学院理学研究科

有限境界条件 効果Introduction• 境界 効果 真空期待値 変化
•

境界

生
効果 関

現象 １

Casimir 効果

研究 数多

1/3

Introduction• 境界 効果 真空期待値 変化
•

境界

生

現象 １

効果 関

Casimir 効果

研究 数多Minkowski 時空 MD (2 ≤ D < 4)
作用
∫
]
λ0 [
S = dD x L,
L = ψ̄iγ µ ∂µ ψ +(ψ̄ψ)2 + (ψ̄iγ 5 ψ)2
2N

今回

簡単

模型

4

相互作用模型

用

効果

見

1/3

Introduction• 境界 効果 真空期待値 変化
•

境界

生

現象 １

効果 関

Casimir 効果

研究 数多Minkowski 時空 MD (2 ≤ D < 4)
作用
∫
]
λ0 [
S = dD x L,
L = ψ̄iγ µ ∂µ ψ +(ψ̄ψ)2 + (ψ̄iγ 5 ψ)2
2N

今回

簡単

模型

4

相互作用模型用効果

見MD =⇒ MD−1 × S 1
1/3

Analysis
S1

新長• 解析方法
1. 補助場 σ, π
2. 1/N 展開

導入

L

境界条件

MD−1

位相 δ

{x0 , . . . , xD−2 }
長

L

• 仮定
1. 真空期待値 持σ2. 真空期待値 σ 座標 依存
3. 境界条件 ψ(xD−1 + L) = eiπδ ψ(xD−1 )

S1
xD−1

2/3

Result
δ=0
0.00

0.06

δ = 0.5

0.04
δ=1

0.02

D = 2.5, Lm0 = 10

V/m0 D

V/m0 D- 0.01- 0.02- 0.030.00

δ = 0.5- 0.02- 0.04- 0.04

δ=1- 0.06- 0.050.0

D = 2.5, Lm0 = 2.5

δ=0

0.2

0.4

0.6

0.8
σ/m0

1.0

1.2

1.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

σ/m0

δ = 0: 周期的境界条件、δ = 1: 反周期的境界条件
T. Inagaki, T. Muta and S. D. Odintsov, Prog. Theor. Phys. Suppl. 127 (1997) 93 .
A. Flachi, M. Nitta, S. Takada and R. Yoshii, Phys. Rev. Lett. 119 (2017) no.3, 031601 .

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Persistent Homologyによる閉じ込め・非閉じ込め相の解析
開田丈寛、柏浩司1、河野宏明2、管野淳平、高橋純一3、八尋正信
九大院理、福工大情1、佐賀大理工2、気象庁福岡管区気象台3• Persistent Homology (PH) … 位相的データ解析の一種（図：SiO2のガラス相(上)と液相(下)の原子配列と解析結果）

https://www.jst.go.jp/pr/announce/20160614/index.html

Persistent Homologyによる閉じ込め・非閉じ込め相の解析
開田丈寛、柏浩司1、河野宏明2、管野淳平、高橋純一3、八尋正信
九大院理、福工大情1、佐賀大理工2、気象庁福岡管区気象台3• PHによる解析を閉じ込め・非閉じ込め相転移に適用（Eﬀective Polyakov-line modelを用いて解析）[J. Greensite, Phys. Rev. D 90, 114507 (2014)]

25

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10

5

5

5

0

0

0

0

5

10
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20
Sector 1

250

5

10

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25
0

5

10
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20
Sector 2

250

5

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25
0

5

10
15
20
Sector 3

250

5

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15

20

25

Persistent Homologyによる閉じ込め・非閉じ込め相の解析
開田丈寛、柏浩司1、河野宏明2、管野淳平、高橋純一3、八尋正信
九大院理、福工大情1、佐賀大理工2、気象庁福岡管区気象台3• PHによる解析を閉じ込め・非閉じ込め相転移に適用
M/T=10, κ=0.120
10Polyakov-line
modelを用いて解析）
Sector（Eﬀective
1
Sector2
Sector 3

9[J. Greensite, Phys. Rev. D 90, 114507 (2014)]

8
Death time

7
25
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15
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0

0

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10
05 2
15
20
Sector 1
250

20

25
0

4

6

5

15

10
10
8 015 20
2 4 5 6
Sector 2
250

Birth time

20

25
0

8

0

5

10
10
2 415 68 5
20
Sector 3
250

15

20

25

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i tH(r, t)⌘1

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æ! r ö
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Q( r , t ) = q ç R + , t ÷ - q ç R - , t ÷
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1 2
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r2r /M + V (r)i{D(0)t0 ) abØ +=>7%():61

Ø +=>7%():61
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QQ̄ (r, t) = H⇠ (r, t)H⇠ (r, t) '<latexit sha1_base64="X3v1hpSNAA447uvssN5dTDhjtIM=">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</latexit>

r2r /M

r2r /M

QQ̄ (r, t)

iCF D(0) + [V (r) + iD(r)](ta ⌦ t

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[Magazzú et al., Nat. Commun. (2018).]
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QGP粘性係数導出に向けた
Nf=2+1 QCDエネルギー運動量テンソル相関関数の研究
谷口裕介
forWHOT QCD collaboration

馬場惇, 江尻信司, 金谷和至, 北沢正清, 下条昂礼,
鈴木遊, 鈴木博, Y.T, 梅田貴士

QGPの粘性係数が面白い
ずれ粘性: shear viscosity (Wikipedia)
速度 Uh
剪断応力 F
面積 S

12
1 2
hT
i
=
⌘@
u
計算すべき量しかし熱平衡系では hT

12

i =0

非平衡効果を実時間形式の外場として取り入れる
しかし実時間形式の応答関数は格子上では計算できない
スペクトル関数を使ってEuclid空間の相関関数に変換する

格子計算の問題点
スペクトル関数Z

d3 xh ˆ( i⌧, ~x) ˆ(0, ~0)i =

Z 1

⇣

2
dk0 cosh k0 ⌧⇢(k0 , ~0)2⇡sinh k0 2

ρ(k0, 0 )⃗
η = lim2k0
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τは離散化された有限個の点⌘

病的な逆解き問題

スペクトル関数のモデル化
Breit-Wigner ansatz
⇢(!)
F
=
!
1 + b2 (!

F
+
2
!0 )
1 + b2 (! + !0 )2

hard thermal loop ansatz
⌘
⇡

⇢(!)
=
+ ✓(!
2
2
!
1+b !

A! 3
!0 )
!
tanh 4T

Whatsʼ new?
Gradient Flowを使って格子上のエネルギー
運動量テンソルを定義する
格子上の非摂動論的な繰り込みを与える
Nf=2+1 QCDへの応用
本邦初公開
m⇡
⇠
0.6
a〜0.07
[fm],
heavy
ud
quark
•
m⇢

• T=174ー464 MeV

Shear viscosity
Breit-Wigner ansatz

T=232 MeV (Nt=12)⟨ΔTij(x4)ΔTij(0)⟩

0.15

η
s

1

linear window

0.5

0.1

T=232 MeV (Nt=12)
linear fit

0

0.05
-0.5

0
-1
0

-0.05
0

2

4

flow time=0.5
flow time=1.0

6
x4

8

10

flow time=1.5
flow time=2.0

12

0.5

1
flow timeη= 0.145(51)
s

1.5

2

Viscosity as a function of temperature
Shear viscosity
0.8

Bulk viscosity
2

Shear viscosity
Breit-Wigner
Hard thermal loop
ADS/CFT

0.6

1.5
1zeta/s

eta/s

0.4
0.2

0.5

0

0

-0.2

-0.5

-0.4

Bulk viscosity
Breit-Wigner
Hard thermal loop

0

100

200

300

400

T [MeV]

Breit-Wigner ﬁt ansatz

500

-1
150

200

250

300

350

400

450

T [MeV]

Hard thermal loop ﬁt ansatz

500

.

.

Statistical Fluctuation in Stick-Slip motion:
Continuous-Time Approach vs Step-Flow
Method一ノ瀬 祥一ichinose@u-shizuoka-ken.ac.jp
静岡県立大学

理研シンポジウム： 熱場の量子論とその応用 2018.08.28 - 30

一ノ瀬 祥一 (Univ. of Shizuoka)

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

1. Introduction

Sec 1. Introduction: a.Dissipative Model.
Figure: 1 The spring-block model

Spring Forcek(V t-x(t)-l )Fric. Force
Fri
x(t)
-mg
Grav. Force
一ノ瀬 祥一 (Univ. of Shizuoka)

Vt

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

1. Introduction

Sec 1. Introduction c.Burridge-Knopoﬀ Model

Figure: 2 Burridge-Knopoﬀ Model, 1967

一ノ瀬 祥一 (Univ. of Shizuoka)

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

1. Introduction

Sec 1. Introduction: d.Friction Forces.
Figure: 3 Friction Forces [left] rain dropFric[right] stick slip model.
Fric

κ
1/2α
x

-1/2α
−κ

rain drop

一ノ瀬 祥一 (Univ. of Shizuoka)x

m=1

stick slip

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

2. Spring-Block Model

Sec 2. Spring-Block Model b.Variat. Principle
Minimal Energy Priciple δKn (x)/δx|x=xn = 0.
k1(xn + ℓ̄ − nhV̄ ) + 2 (xn − 2xn−1 + xn−2 ) +mh√κ sgn(xn−1 − xn−2 )k=0, ω≡,
1 + 2α|xn−1 − xn−2 |/hm(1)

where n = 2, 3, 4, · · · , N − 1, N.
Recursion relation among n-th, (n-1)-th and (n-2)-th positions
Parameters: ℓ̄ = 1, V̄ = 0.1, ω = 1.0, κ = 1.0, α = 2.5
1 Step Interval: h = 2.5 × 10−3 , Total Step Number: N = 2 × 104(h · N = 50 Total Step Length(’Time’))Initial condition: x0 = −ℓ̄, (x1 − x0 )/h = 0.
See 3.a,b Movement, 3.c,d Velocity
一ノ瀬 祥一 (Univ. of Shizuoka)

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

Sec 3. Movement and Velocity

Sec 3. a. 位置 x(n), N1=100000,
Figure: 5 SB Model, Movement, xn The step-wise solution (1) correctly
reproduces the continuous-time solution: stick-slip motion O(10−1 )ht= 0.00250,om= 1.00,al= 2.500,be= 1.000,size= 1.000Vel= 0.100,N1=100000
25’outSBd.dat’

20

x

15

10

5

0

-5
0

一ノ瀬 祥一 (Univ. of Shizuoka)

50

100

150

200

250

理研シンポジウム： 熱場の量子論とその応用 20nhStatistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

Sec 7. Stick-Slip 運動の周期の分布 (揺らぎ)Sec 7. a. 加速度 ac(n), N1=100000
Figure: 27 10 万ステップでの加速度分布 0 ≤ t ≤ 250

ht= 0.0025, om= 1.0, al= 2.5, be=1.0, size= 1.0, Vel= 0.1, N1= 100000

5’accel.dat’

2

5

1

5

0

5

1

5

2
0

50

100

150

200

250

nh

一ノ瀬 祥一 (Univ. of Shizuoka)

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

Sec 7. Stick-Slip 運動の周期の分布 (揺らぎ)Sec 7. b. 加速度 (詳細) ac(n), N1=100000
Figure: 27 10 万ステップでの加速度分布 (詳細) 0 ≤ t ≤ 50

ht= 0.0025, om= 1.0, al= 2.5, be=1.0, size= 1.0, Vel= 0.1, N1= 100000

5’accel.dat’

2

5

1

5

0

5

1

5

2
0

10

20

30

40

50

nh

一ノ瀬 祥一 (Univ. of Shizuoka)

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

Sec 7. Stick-Slip 運動の周期の分布 (揺らぎ)Sec 7. c.63 個の周期分布 N1=500000

DissipSysSF ht= 0.00250, om= 1.00, al= 2.500, be= 1.000, size= 1.000, Vel= 0.100, N1=500000, Nst=

63

’Periods.dat’
20

period18

16

14

Figure: 27 63 個の周期分布120

一ノ瀬 祥一 (Univ. of Shizuoka)

10

20

30
40
n-th StickState

50

60

70

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

Sec 7. Stick-Slip 運動の周期の分布 (揺らぎ)Sec 7. d.62 個の周期分布 (詳細) N1=500000

DissipSysSF ht= 0.00250, om= 1.00, al= 2.500, be= 1.000, size= 1.000, Vel= 0.100, N1=500000, Nst=
21
’Periods.dat’

63

period

20.5

20

19.5

Figure: 27 62 個の周期分布190

一ノ瀬 祥一 (Univ. of Shizuoka)

10

20

30
40
n-th StickState

50

60

70

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

Sec 7. Stick-Slip 運動の周期の分布 (揺らぎ)Sec 7. e.50 万回ステップでの 62 個の周期の統計
Figure: 28 62 個の周期の統計, Python3.6(Spyder)

一ノ瀬 祥一 (Univ. of Shizuoka)

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

Sec 8. Conclusion

Sec 8. Conclusion a.Step-Flow(DMF) vsRunge-Kutta

双方とも摩擦力がギャップを持つ形であれば、揺らぎの統計変化は生ずる。DMF(ステップ) 法の結果のほうが、数値グラフが幾何学的に美しい。
揺らぎがこのメカニズムで一般的に生ずるならば、面白い。

Burridge-Knopoﬀ model はステップフロー法で解析し直すべき。
.

一ノ瀬 祥一 (Univ. of Shizuoka)

理研シンポジウム： 熱場の量子論とその応用 20
Statistical Fluctuation in Stick-Slip motion: Continuous-Time Approach vs Step-Flow
/ 12
Meth

18/8/28 @ 熱場2018

Equation of state in Nf = 2+1 QCD
using gradient flow-- lattice spacing effects
Kazuyuki Kanaya

Tomonaga Center for the History of the Universe
Univ. Tsukuba

in collaboration with
S.Ejiri, R.Iwami, M.Kitazawa, T. Shimojo, M. Shirogane, H.Suzuki, Y.Taniguchi, T.Umeda(WHOT-QCD Collaboration)

QCD thermodynamics with gradient flow
Gradient flow

Narayanan-Neuberger ('06), Lüscher ('09)

tBµ(t,x)Flow ≈ smearing over a physical range of √(8t).
Flowed operators have no UV divergences nor shortdistance singularities
Lüscher-Weisz ('11)=> New method to calculate EMT etc. on the lattice
H.Suzuki ('13)

t=0

Aµ(x)

Our project: Application of Suzuki's idea to (2+1)-flavor QCD
Iwasaki gauge + nonperturbatively O(a) improved Wilson quarks

As the first systematic application to full QCD thermodynamics

1st step: heavy QCD on a fine lattice

PRD96, 014509 (’17); D95, 054502 (’17)=> Looks fine. The method works well.

2nd step (a): QCD @ physical point on a less fine lattice=> Looks similar to the heavy QCD case.

2nd step (b): heavy QCD on a less fine latticefor continuum extrapolation.

EPJ Conf. 175, 07023 (’18)
熱場2017

PRD96, 014509 (’17); D95, 054502 (’17)

1st step:
➤

EoS
30

Nf=2+1 heavy QCD thermodynamics with GF• Heavy ud quarks (mPS/mV ≈0.63) with ≈physical s quark.
• Fine lattice a≈0.07fm (ß=2.05), T≈174-697 MeV (Tpc~190 MeV).
Errors include statical as well as systematical ones due to fit ansatz etc.
12
gradient flow
10
T-integration

gradient flow
T-integration25

8(e-3p)/T4(e+p)/T4

20
15
10

6
4
2
0

5

-2

0

-4
0

100

200

300

400

500

600

0

T (MeV)

➤

100

200

300 400
T (MeV)

500

600

EoS by GF agrees withconventional method atT≤300 MeV. Suggests oura≈0.07fm lattice close to thecont. limit.
Disagreement at T≥350 MeV
due to O((aT)2 =1/Nt2)
artifacts at Nt ≤ 8.

Chiral suscept. (disc.) / topological suscept.
Crossover at Tpc~190 MeV.
Gluonic and fermonic def.'s of
top. charge and 𝜒top agree.

MS scheme at µ=2 GeVPower-low behavior of 𝜒top
consistent with DIGA withexponent –8.

Wilson quarks have guaranteed cont. limit, but violates the chiral sym. explicitly.

GF powerful to extract physical properties even with Wilson quarks!

EPJ Conf. 175, 07023 (’18)

Nf=2+1 phys.pt. QCD• ud and s quarks with physical mass
• a≈0.09fm (ß=1.9), T≈157-549 MeV

EoS

➤
35

40

linear fit

30(e-3p)/T4(e+p)/T

4

25
20
15
10

25
20
15
10
5

5

0

0
0

100

200

300 400
T (MeV)

500

-5

600

0

100

200

300 400
T (MeV)

500

600

0

u quark

-0.05
-0.1
-0.15
-0.2

0

50

100 150 200 250 300 350
T (MeV)

Figure 12. The entropy density normalized by T 3 as a function of the temperature on Nt = 6, 8
Figure 15. Continuum estimate for the trace anomaly normalized by T 4 together with theand 10 lattices. The Stefan-Boltzmann limit sSB = 4pSB /T is indicated by an arrow.
parametrization of equation (3.1) using the nf = 2 + 1 parameters from table 2.

Borsany et al., JHEP 1011, 077 (2010), stout, cont. limit

e+p consistent with KS
e-3p may be larger than KS– 18 –

Chiral condensate
subtracted chiral condensate

subtracted chiral condensate

linear fit

35

30

➤

Figure 11. The energy density normalized by T 4 as a function of the temperature on Nt = 6, 8
and 10 lattices. The Stefan-Boltzmann limit ϵSB = 3pSB is indicated by an arrow.

JHEP11(2010)077

2-a step:0s quark-0.05

Definite comparison possible only
after continuum extrapolation.

-0.1
Figure 16. The normalized trace anomaly for two diﬀerent values of the light quark masses on
Nt = 8 lattices: the physical mud = mphys
and the three degenerate flavor mud = mphys
case.
s
ud

-0.15
-0.2

Sharper chiral trans. with u,dTpcphys ≤ 169 MeV : consistent with KS– 21 –

050

100 150 200 250 300 350
T (MeV)

Similar to the heavy QCD case. The method seems to work.
Linear window for t→0 extrapolation narrower <= larger a and/or smaller mq
Need more statistics, lower T, more study of the fits => on-goingb=1.9 linear fit

30

b=1.9 linear fit
10

20

2-b step:
15
10(e-3p)/T4(e+p)/T425

Nf=2+1 heavy QCD
5preliminary!

0• ud and s quarks with physical mass
0-5
• 200a≈0.097fm(ß=1.9),MeV
0 50 100 150
250 300 350 400 450
0 T≈169-254
50 100 150 200 250 300
350 400 450
5

35

30
0.004
(e+p)/T4

ud quark

0.003525

T [MeV]T [MeV]15

b=2.05b=1.9 linear fit
b=1.9 non-linear
b=2.05fit

0.004
100.0035b=1.9 linear fit

0.003
20

0.0025150.002
100.001550.001
00.000500➤(e-3p)/T4

EoS

s quark➤b=2.05b=1.9 linear fit
b=2.05fit
b=1.9 non-linear

b=1.9 linear fit

0.003

5
0.00250.002
00.00150.001-550 100 150 200 250 300 350 400 450

0.00050

T [MeV]00 50 100 150 200 250 300 350 400 450

ß=1.9 data at T ≥ 254 MeV may becontaminated by O((aT)2 =1/Nt2)
lattice artifacts at Nt ≤ 8.

Chiral condensate

50 100 150 200 250 300 350 400 450
T [MeV]

Linear fit => our central values
Non-linear fits => systematic erros

0 50 100 150 200 250 300 350 400 450

a-dep. looks small so far.

T [MeV]T [MeV]11. The entropy density (ϵ + p)/T 4 as a function
ofFIG.
temperature(MeV).
Black circles
are 4 as a function
14. The trace
anomaly(ϵ − 3p)/Tof temperature (MeV). Black circles are

s at β = 2.05 including
all the systematic errors. Redresults
up andatblue
traiangles
aretheresults0.004
β =down
2.05 includingallsystematic errors. Red up and blue down traiangles are results
0.004

0.0025s quark

ud quark

b=2.05b=2.050.0035b=1.9
linear
= 1.90 with liner fit and non-linear fit error is the statistical
only.with liner fit 0.0035
fit only.
at βfit= 1.90
and non-linear fit errorb=1.9
is the linear
statisticalb=1.9 non-linear fit
b=1.9 non-linear fit
0.003
0.003
0.002
0.00150.00250.002
0.0015

0.001

0.001

0.00050.000500 50 100 150 200 250 300 350 400 450
T [MeV]00 50 100 150 200 250 300 350 400 450
T [MeV]

Look similar to the fine lattice case so far.

!
""Chiral condensate {ψ̄u ψu } with VEV subtraction
a functionof the!{temperature.
FIG. 22. as
Chiralcondensateψ̄s ψs } with VEV subtraction as a function of the temperature.
14
17
es are results at β = 2.05 including all theBlacksystematic
errors.
Red
up
and
blue
down
circles are results at β = 2.05 including all the systematic errors. Red up and blue down

Linear window for t→0 extrapolation narrower than the fine lattice case.
s is in lattice unit.
axis is in latticeunit.
Need moreverticalstatistics,
more
T, more study of the fits => on-going

are results at β = 1.90 with liner fit and non-linear
error
is theatstatistical
The fit and non-linear fit error is the statistical only. The
traianglesfitareresults
β = 1.90 only.
with liner    
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(!. #.

Boltzmann∗ Keldysh2
Wigner()…

J. P. Blaizot, E. Iancu(2002), Y. Hidaka, S. Pu, and D. L. Yang(2017)

Dyson-Schwinger

ポスター番号:103体引力相互作用する
１次元ボソンの量子液滴とその普遍性
Y. Sekino & Y. Nishida, Phys. Rev. A 97, 011602(R) 2018

関野裕太

東工大

共同研究者 西田祐介 東工大
理研シンポジウム・iTHEMS研究会
熱場の量子論とその応用
Tuesday, August 28, 2018 @RIKEN

1/4

No. 10 関野裕太 2/4

Ultracold atoms

Highly controllable system
New quantum statesØ Spatial geometry

2D

1D

Latticehttps://www.uibk.ac.at/exphys/ultracold/projects/cs3/index1.html

No. 10 関野裕太 3/4

Ultracold atoms

Highly controllable system
New quantum statesØ Spatial geometry

Ø Interaction

Tuning of two-body interactions by a magnetic field

Three-body interaction in ultracold atoms
Various proposals
Büchler, Micheli, & Zoller, Nat. Phys. (2007)
Daley et al., PRL (2009), ...

6=<latexit sha1_base64="s2S7rdI0nKjdCXf+QJ9lEvQ4SLM=">AAACZ3ichVHLSsNAFD2Nr1ofrQpScKMWxVW5EUFxVXTjsg9bC1UkiaMG0yQmaUGLP+DCbQVXCiLiZ7jxB1z0E4rLCm5ceJMGREW9w8ycOXPPnTMzqm3orkfUikg9vX39A9HB2NDwyGg8MTZecq2ao4miZhmWU1YVVxi6KYqe7hmibDtCqaqG2FKP1v39rbpwXN0yN70TW+xUlQNT39c1xfOpbVMc7yZSlKYgpn8COQQphJG1EnfYxh4saKihCgETHmMDClxuFcgg2MztoMGcw0gP9gXOEGNtjbMEZyjMHvF4wKtKyJq89mu6gVrjUwzuDiunMUfPdE8deqIHatP7r7UaQQ3fywnPalcr7N34ebLw9q+qyrOHw0/Vn5497GMl8Kqzdztg/FtoXX39tNkprObnGvN0Qy/s/5pa9Mg3MOuv2m1O5K8Q4w+Qvz/3T1BaTMuUlnNLqcxa+BVRTGEWC/zey8hgA1kU+dxDXKCJy0hbikuTUrKbKkVCzQS+hDTzAbj0iz4=</latexit>

No. 10 関野裕太 4/4

Set up of our study

1D bosons w/ a resonant 3-body interaction
w/o a 2-body interaction

ü Quantum anomaly of scale invariance

Bright soliton stabilized
by quantum effect|�� (�)|Exotic many-body bound state

Running coupling�����

Asymptotic freedom

��
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Length scale

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理研シンポジウム・iTHEMS研究会
熱場の量子論とその応用

2018年 8月28日

熱場ダイナミクスを用いた
熱力学的距離の導入と非散逸カレントの寄与

橋爪洋一郎1 鈴木増雄2 中嶋宇史1,3 岡村総一郎1
東理大理1，理研2，JSTさきがけ3

熱場ダイナミクス(thermo field dynamics; TFD)
量子力学的期待値：

統計力学的期待値：

状態の空間を拡張して、表現形式を統一する。

ＴＦＤ状態ベクトル（平衡状態）：

２重ヒルベルト空間

U.Fano, Rev.Mod.Phys.A42(1957)74
I.Prigogine et al.,Chemica Scripta 4(1973)5
T.Takahashi and H.Umezawa, Collect Phenom.2(1975)55
c.f. 松原グリーン関数(1955)

オリジナル空間

チルダ空間

ＴＦＤは有限温度の状態表現に適している
ＴＦＤによって状態変化をとらえることはできるだろうか？

目的
準静的な状態変化に対してTFD波動関数がどのように変わるのか
を調べる．

内容
１．一般論として，状態変化の特徴量を導入する
２．具体的な系での確認
３．一般の系での議論

幾何学距離との関係
quantum distanceFubini-Study metric （混合状態についてはBures metric）
として （TFDを導入して有限温度にする）

だったので
として，

・・・今回はこの特別な場合（温度のみの1変数）を考慮すると・・・Fisher metricＴＦＤ状態ベクトルの変化
エントロピー演算子

・・・系の制御変数（温度，磁場など）が含まれる．
・・・基底ベクトルの表現，（系の制御変数によらない）

制御変数 x を少し変化させたときのTFD状態ベクトルの変化

11/12一般の系での有限温度状態変化
制御変数を温度とする場合は互いに可換は比熱

一般の熱力学系で成り立つ！

多体系で相転移がある場合

Flux current がある場合について位相に起因するフラックスを導く．

準平衡の時に限っては密度行列の規格化などによってゼロとなる．

Thermodynamic Distance （有限温度の量子距離）
一般形

平衡系

カレントがあるとき

その他，具体例や散逸系への拡張について

非平衡物理の最前線 –素粒子・宇宙から物性まで-

P12

人工磁場クエンチによるゲージ依存性
清水啓太 1 、一瀬郁夫 1 、久野義人 2
名工大院 1 、京大理 2

熱場の量子論とその応用
２０１８年８月２８日

非平衡物理の最前線 –素粒子・宇宙から物性まで-

Introduction

Optical lattice2D➢ Clean➢ High controllability➢ Various dimensions
and configurations

3D
I. Bloch, J. Dalibard, W. Zwerger,
Rev. Mod. Phys. 80, 885 (2008)

Strong magnetic field
Experimental

TheoryQuantum Hall effect etc
D. Jaksch, P. Zoller,
New J. Phys. 5, 56 (2003)

M. Aidelsburger, et al.
Phys. Rev. Lett. 111, 185301 (2013)

非平衡物理の最前線 –素粒子・宇宙から物性まで-

Results

StaticB

B : magnetic field

Time independent

Gauge invariance非平衡物理の最前線 –素粒子・宇宙から物性まで-

Results

StaticB

B : magnetic field

Gauge invarianceTime independent

Dynamical

B

B

Time dependent

Breakdown ofdynamical gauge invariance !!

非平衡物理の最前線 –素粒子・宇宙から物性まで-

Results

Order parameter

Density

Landau gauge

Symmetric gauge

なぜゲージ不変性を破るのか？

非平衡物理の最前線 –素粒子・宇宙から物性まで-

Results

Order parameter

Density

Landau gauge

Symmetric gauge

なぜゲージ不変性を破るのか？

磁場が“人工的”であることが原因！！

回転する高密度
バリオン物質におけるカイラルソリトン格子
西村健太郎 (慶應義塾大学)
共同研究者 Xu-Guang Huang (Fudan Univ),山本直希(慶應義塾大学)
JHEP02,069 (2018)「熱場の量子論とその応用」２０１８年８月２８日

The STAR Collaboration, Nature (2017)22⌦ ' 10 s

1!

回転するQCD物質の低エネルギー有効理論• カイラル渦効果：回転軸の方向に粒子が流れる異常輸送
1 2
1 2
j5 =
µ
⌦
+
T
⌦
,
(µ
=
µ
=
µ)
R
L
2⇡ 2
6
• カイラルアノマリー由来の係数• 非摂動的に厳密でエネルギースケールに依存しない• カイラル渦効果を再現する低エネルギー有効理論µB µI
L=r⇡0 · ⌦
22⇡ f⇡
Huang, Nishimura and Yamamoto, JHEP (2018)

回転するバリオン物質にアノマリーがもたらす新奇効果• カイラルソリトン格子
パリティの破れ トポロジカルチャージ 周期構造8⇡m⇡ f⇡2
以上で回転するQCD物質の基底状態⌦CSL =µB |µI |• 異常項が導く新奇物理現象
エネルギー流のホール効果：新奇なトポロジカル輸送現象
磁化と角運動量間の交差応答：磁場が誘起する自発的な回転• トポロジカル物質との類似性
QCD

Nc 2
T r⌘0 · ⌦6f

部分積分

Huang, Nishimura and Yamamoto, JHEP (2018)

Topological supercondoctor/superﬂuid

2
kB✓T rT · ⌦2
12~v
Nomura, Ryu, Furusaki and Nagaosa, PRL (2012)

LSポテンシャルによる中性子3P2
超流動における強磁場効果
安井繁宏, Chandrasekhar Chatterjee, 新田宗土  (慶應義塾大学)

マルチメッセンジャー天文学(可視光,  電磁波,  ニュートリノ,  重力波)

中性子3P2超流動
Wikipedia
熱場の量子論とその応用@理研, 28-30 Aug. 2018

中性子星の強磁場効果
Ginzburg-Landau方程式による解析〜磁場の高次項〜
中性子3P2超流動の温度-磁場(T-B)相図-1≤r

Magneter              今回の結果  

強磁場のとき中性子33PP22超流動  
の相転移温度が増加する！  →  D4  BN1015  G

非常に強い磁場が存在する！

B〜1015  G

D2  BN

Wikipedia

UN