「テンソル繰り込み群を用いたゼロ温度高密度領域におけるNJLモデルのカイラル相転移の解析」

筑波大学・計算科学研究センター
藏増 嘉伸
＠宇宙史研究センター構成員会議, 2020年11月30日

1内容

・ テンソル繰り込み群(TRG)­ 2次元Isingモデル
・ 特異値分解(SVD)・ モンテカルロ法との比較
・ TRG法の素粒子物理への応用
・ 低温・高密度下におけるNJLモデルの相転移・ まとめχ(L) =L2 ∂(1/2κ)2!
テンソル繰り込み群(TRG)
Z = DψDψ̄DU e−ψ̄D[U ]ψ−Sg [U ]Levin-Nave
PRL99(2007)120601

Dcut"
例としてNサイトを持つ2Dイジングモデルを考える
Ti,j,k,l "U(i,j),mσmVm,(k,l)

ハミルトニアンH =!

!i,j"Z =m=1si sj

si ± 1"H =si sj#i,j$!

si ± 1exp (−βH)"
分配関数 {Si }Z =exp (−βH)={si }

2!

Ti,m,n,l Ts,t,i,j
2 Tr,j,k,q Tk,l,o,p · · ·"i,j,k,l,···=1 =Tα,λ,ρ,δ Tσ,κ,α,β Tµ,β,γ,τ Tγ,δ,ν,χ · · ·Z ="α,β,γ,δ,···=1

DU det D({U!}) テンソルネットワーク表現
e−Sg ({U })Z = DU det D({U }) e−Sg ({U })モデルの詳細は初期テンソルのみに依存"−Sg ({U })!O" = DU O({U, D−1! }) det D({U }) e
計算アルゴリズムはモデルと独立#O$ = DU O({U, D−1}) det D({U }) e−Sg ({U })1
勿論，サイト数Nが大きくなれば添字の縮約の完全実行は不可能
g ({U })P =det D({U }) e−S
1Z⇒ どうやって分配関数を評価するか？
P =det D({U }) e−Sg ({U })Z
Z=!

i,j,k,...

−S(i,j,k,...)eZ=!

Tijkl TimnoTjpqr
Tksuv Tlwxy · · · .
"−S(i,j,k,...)
i,j,k,...
e=Tijkl TimnoTjpqr Tksuv Tlwxy · · · .

="TRGアルゴリズムの概略
1. サイト上のテンソルTに対する特異値分解2. 古い添字の縮約 (疎視化)
3. 手続きの反復

大きな特異値を持つ部分空間のみを残す

新しい添字を持つテンソル

サイト数は半減4m{j,k},m{l,i},m! √ "√ "Ti,j,k,l ⇒ T{k,l},{i,j}= U ΛV {k,l},{i,j} =U Λ {k,l},m V Λ {i,j},m2Dイジングモデルを使ったテスト!

t"#!

m#

=(S2){k,l},m (S4){i,j},mm
アルゴリズムの要諦は特異値分解を用いた低ランク近似
Ti,j,k,l #
$

D#
cut

m=1

U{k,l},mΛmV{i,j},m%22222∗∗4
誤差をコントロールするパラメーターはDS=dx|∂
φ|+(m−µ)|φ|+µ(φ∂φ−∂φφ)+λ|φ|
cut
cont
ρ
22&

転移点近傍での自由エネルギーの厳密解からの相対誤差,30〜50XIE, CHEN, QIN, ZHU,YANG,AND
XIANG
PHYSICAL REVIEW B 86, 045139 (201
格子サイズ=2, Dcut=24Z=$Z(original) =Dφ exp(−S)

$(a)

(n)
Xie et al.
T
PRB86(2012)045139

z(b)

(n)

T y'1x1
x'1
y'2
y1 ix2x'2(n)Ty2z'Onsagerの厳密解との比較
Dφ1Dφ2 exp(−S)
相対誤差：≤10−6z(n)

V(n)

(n)

(n)

S=



#
n

(4 + m2)|φn|2 + λ|φn|4 −M

2 !
#ρ=1U x'x U

"y (n)Vn eµδρ,2 φ∗nφn+ρ̂ + e−µδρ,2 φ∗n+ρ̂φy'(n)z'M

特異値分解(Singular Value Decomposition)
任意のm×n実行列Aは A=UΣVT と分解できる
U: m×mの直交行列
V: n×nの直交行列
Σ=diag(σ1, σ2, σ3, σ4, …, σn)(σ1≥σ2≥ σ3≥σ4≥…≥σn≥0)

σ1, σ2, σ3, σ4,・・・,σnはAの特異値で非負Uの各列u1, u2, …, unとVの各列v1, v2, …, vnを用いたランク1の行列和に分解,
A=σ1u1v1T+σ2u2v2T+…+σnunvnT
σiuiviT=××
σiuiviT6

行列の近似行列Aの近似A=σ1u1v1T+σ2u2v2T+…+σkukvkT+…+σnunvnT

Ak=σ1u1v1T+σ2u2v2T+…+σkukvkT(行列k個の和で近似)

近似誤差は||A-Ak||Fで定義||A-Ak||F=(σk+12+σk+22+…+σn2)1/2
ただし, ||A||F=(Tr(ATA))1/2=(ΣiΣj aij2)1/2
画像圧縮などに利用
7

特異値分解(SVD)を用いた画像圧縮
200x320ピクセルの画像データ ⇒ 200x320実行列A行列を特異値分解
A=σ1u1v1T+σ2u2v2T+…+σnunvnT (n=200)

サンプル画像(200x320ピクセル)

特異値の分布(大→小)J. Demmel: Applied Numerical Linear Algebra, SIAM 19978

復元画像の品質
Ak=σ1u1v1T+σ2u2v2T+…+σkukvkT (k≪200)

J. Demmel: Applied Numerical Linear Algebra, SIAM 19979

モンテカルロ法との相違点
モンテカルロ法
確率的手法

テンソル繰り込み群決定論的手法
コペルニクス的転換

・ モンテカルロ法における符号問題および複素作用問題がない
Z=!

Dφ exp(−SRe[φ] + iSIm[φ])$ ! π dϕx,µ $・ LDのシステムサイズに対する計算コスト∝D×log(L)Z=
T (ϕx,1, ϕx+1̂,2, ϕx+2̂,1, ϕx,2)−π 2πx,µx
・ グラスマン数を直接扱うことが可能
・ 分配関数Zそのものを計算可能

T (ϕx,1, ϕx+1̂,2, ϕx+2̂,1, ϕx,2) = exp β cos px + iθ qx2πdϕx,µ 
Z=exp (−S)
x,µ −π 2π
素粒子物理：軽いクォークのダイナミクス，有限密度QCDの相構造解析'(' √ (Strong CP問題などの研究に応用可能)' √ (tTi,j,k,l ⇒ T{j,k},{l,i} = U ΛV {j,k},{l,i} =U Λ {j,k},m V Λ {l,i},m$ ! πm
物質科学：強相関量子系，金属絶縁体転移，高温超伝導などの)=(S1){j,k},m (S3){l,i},m
研究に応用可能 (ハバードモデル)
mT⇒T'= U ΛVt(=√ (U Λ)'√ (V Λ'TRG法の素粒子物理への応用(1)2次元モデルCP(1)モデル：Kawauchi-Takeda, PRD93(2016)114503
実φ4理論：
Shimizu, Mod.Phys.Lett.A27(2012)1250035,
Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP1905(2019)184
有限密度における複素φ4理論：
Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, , JHEP2002(2020)161
θ項(トポロジカル項)を持つU(1)ゲージ理論：
YK-Yoshimura, JHEP2004(2020)089
Schwingerモデル(2次元QED), θ項(トポロジカル項)を持つSchwingerモデル：
Shimizu-YK, PRD90(2014)014508, PRD90(2014)074503,
PRD97(2018)034502
有限密度におけるGross-Neveuモデル：
Takeda-Yoshimura, PTEP2015(2015)043B01
N=1 Wess-Zuminoモデル(超対称性理論)：
Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP1803(2018)141
符号問題解決の検証，スカラー場・フェルミオン場・ゲージ場の計算手法開発
11

TRG法の素粒子物理への応用(2)3次元モデル
自由Wilsonフェルミオン：
Sakai-Takeda-Yoshimura, PTEP2017(2017)063B07,
Yoshimura-YK-Nakamura-Takeda-Sakai, PRD97(2018)054511
有限温度におけるZ2 ゲージ理論：
YK-Yoshimura, JHEP1908(2019)023
4次元モデル
Isingモデル：
Akiyama-YK-Yamashita-Yoshimura, PRD100(2019)054510
有限密度における複素φ4理論：
Akiyama-Kadoh-YK-Yamashita-Yoshimura, JHEP2002(2020)161
有限密度におけるNambu−Jona-Lasinio(NJL)モデル：
Akiyama-YK-Yamashita-Yoshimura, arXiv:2009.11583⇒ 研究の重心は2次元モデル・理論から4次元モデル・理論へ移行中
12easilythermodynamic
obtained withproperties
the TRG
of method.
dense QCD.
Thispaper
paper weis organized
as follows.
In Sec.structure
2 we explainof the lattice (NJL)
In thisinvestigate
the phaseof thetheformulation
Nambu–Jona-Lasinio
with
the Kogut–Susskind
and the algorithmic
the lattice
GrassmannmodelNJL[16,model17] 低温・高密度下におけるNJLモデルの相転移(1)atfinite
temperature Tfermion
and chemical
potentialdetailsµ onofthe
developing
ATRG (GATRG). Numerical results for the chiral condensate and the equation of state
the Grassmannversion of the ATRG algorithm. The Lagrangianof the arXiv:2009.11583
NJL model in theAkiyama+,
are presented in Sec. 3. Section 4 is devoted to summary and outlook.
continuum is defined as follows:
連続時空におけるNJLモデル
22.1

Formulation¯ and numerical algorithm¯L = (x) ⌫ @⌫ (x)NJL model on the latticeg0 ( (x) (x))2 + ( ¯(x)i 5 (x))2 ,(1.1)

which has the U(1) chiral symmetry with (x) ! ei↵ 5 (x) and ¯(x) ! ¯(x)ei↵ 5 . This
We use the Kogut–Susskind fermion to formulate the NJL model on the lattice. Following
is an Refs.
e↵ective
theoryof QCD
which
describes
thepotential
dynamical
格子上における有限密度NJLモデル(Kogut-Susskindフェルミオン)[21, 22],
we definethe modelat finite
chemicalµ as chiral symmetry breaking:once the strength of the coupling constant g0 exceeds a certain critical value the system
4hiXX¯13µaµagenerates a non-trivialwith6 ) 0.e The⌫,4 chiral phase structure of theS = a vacuum⌘⌫ (n)e h⌫,4 ¯(x)(n) (x)i(n + ⌫ˆ=¯(n + ⌫ˆ) (n)2⌫=1 is discussed by some analytical methods, e.g., the mean-field
NJL model on the T -µn2⇤
plane
4
Xand the functional
XXapproximation (MFA) [18]renormalizationgroup (FRG) [19]. Figure 1+ ma4¯(n) (n) g0 a4¯(n) (n) ¯(n + ⌫ˆ) (n + ⌫ˆ),(2.1)
shows a schematic view ofn2⇤the expected phasen2⇤ ⌫=1structure, whose characteristic feature is
the first-order chiral phase transitionin the dense region at very low temperature [20].
where n = (n1 , n2 , n3𝜇：化学ポテンシャル(密度をコントロール), n4 )(2 Z4 ) specifies a position in lattice ⇤, whose spacing is a. (n)This phasetransition is our primary target to investigate, employing the chiral condensate
and ¯(n) are Grassmann-valued fields without the Dirac structure. Since they describe
𝑚：フェルミオンの質量h ¯(x)the(x)i
as an order fermions,parameter.
thearechiralsymmetry plays
a crucialrole in this
Kogut–Susskind(n) Sinceand ¯(n)single-component
Grassmannvariables.
𝑔! ：4フェルミ相互作用の結合定数study, we use the Kogut–Susskind
fermion to formulate the NJL model on the lattice.
𝑎：格子間隔The analysis of the phasestructure with the TRG method would help us understand thethermodynamic properties of dense QCD.
This paper is organized as follows. In Sec. 2 we explain the formulation of the lattice低温・高密度下におけるNJLモデルの相転移(2)Akiyama+, arXiv:2009.11583

NJLモデルはQCDのプロトタイプ
QCDで期待されている相図

NJLで期待されている相図𝑇

MC計算

Critical end point2nd𝜓𝜓1st0𝜓𝜓0𝜇低温・高密度での一次相転移を検証することが重要

re 1. Schematic view of expected phase diagram of the NJL model on the T -µ plane. Solidroken curves represent the first- and second-order phase transitions, respectively. Closed circle
es the critical end point (CEP) where the first-order phase transition line terminates.

0.06
−<χχ>

低温・高密度下におけるNJLモデルの相転移(3)0.04

Akiyama+, arXiv:2009.11583

0.02

3.2Chiral phase transition
0.00

We investigate the chiral phase transition employing the chiral condensate h ¯(n) (n)i, as-0.02
カイラル相転移のオーダーパラメータan order parameter, which
is defined by
0.0
1.0
2.0
3.0
4.0
5.0
µ1 @
ln Z,
m!0 V !1 V @mh ¯(n) (n)i = lim lim(3.2)

Figure 4. Chiral condensate at m = 0.01 and 0.02 on 10244 lattice as a function of µ with D = 55.

in the cold region. We calculate h ¯(n) (n)i with the numerical derivative of thermodynamic
potential andthe chiral extrapolation with the corresponding体積依存性@m=0results at finite mass in the
質量依存性@L=1024
0.10
thermodynamic
limit6 . In this study, the partial0.10derivative in Eq. (3.2) is numericallym = 0.01
L = 128
evaluatedvia
m = 0.02
L = 1024
0.08
0.08@ 𝑉 = 1024ln! Z(m +ln Z ⇡@mm)
0.06
m
−
<χχ>

−χ>
<χ

0.06

ln Z(m),(3.3)

0.04
0.04 of the chiral condensate at m = 0.01
withm = 0.01. In Fig. 4 we plot the µ dependence
and0.02 on the L4 = 10244 lattice. The signals0.02show slight fluctuations as a function
0.02
of µ around the transition point. Away from the transition point, we have found little0.00
response
in h ¯(n) (n)i to changes in mass. Figure0.00
5 presents the results in the chiral limit
obtainedby1.0 the chiralextrapolation
with5.0those at-0.020.0
m = 0.01
and 0.02
on 3.0two volumes
of5.0
-0.02
0.0
2.0
3.0
4.0
1.0
2.0
4.0
µµ
L4 = 1284 and 10244 . It is hard to find the di↵erence between the L = 128
and 1024 results.
This allows us to consider the L = 1024 result to be essentially in the thermodynamic
𝜇 0.02≈3.0付近での不連続性(トビ)⇒ 一次相転移gure 4. Chirallimit.
condensate
m = 0.01 andon
10244 lattice fromasa functionof µ with
D =to55.zero
Weatobserve
thediscontinuity
a5.finite
value
for thechiralat of µ with D = 55 on
FigureChiralcondensate
extrapolated
in thechiralcondensate
limit as a function
44128 and 1024 lattices.
µc = 3.0625 ± 0.0625, which is a clear
indication of the first-order phase transition.

and 1024 . It is hard to find the di↵erence between the L = 128 and 1024 results.
ln Z
ws us to consider the L = 1024 result to be essentially in the thermodynamic
3.0
P =,Ve observe the discontinuity from a finite value to zero for the chiral condensate at
2.5
25 ± 0.0625, which is a clear indication of the first-ordertransition.
where thephasevast homogeneous
system is assumed. In Fig. 6 we plot the µ dependenc
pressure at m = 0.01. We find a kink behavior at µc = 3.0625 ± 0.0625, where thuation of statecondensate shows the2.00.0
discontinuity.
Notem = 0.02
resultAkiyama+,
1.0
2.0 that the
3.0 arXiv:2009.11583
4.0
5.0 shows little dµof state is a relation between the pressure and theparticle
density. Herefromthe m number= 0.01 one.
nts both results as a function of µ, respectively.
In thelimit,
the is obtained by the numerical derivative of pre
Thethermodynamic
particle number
density状態方程式：圧力と粒子数密度
P is directly obtained from the thermodynamic terms
potential:of the Figure
chemicalpotential:
6. Pressure at m = 0.01 as a function of µ on 1284 and 10244 lattices.

低温・高密度下におけるNJLモデルの相転移(4)P =ln Z,
V(3.4)=
hn(µ)i@P (µ)
P (µ +⇡@µµ)µP (µ).

P<n>e vast homogeneoussystem is assumed. In Fig. 6The
we plot
the µ dependence
the in Fig. 7. We observe an abrupt jump from4.5µ dependenceof1.2hni isofshownat m = 0.01. We find aL =kinkbehavior at µc = 3.0625
the chiral128
128
to hni ±=0.0625,
1 at µcwhere= 2.9375±LL ==0.0625.
This is another indication of the first-orde1024
L = 1024
1.0
te shows the discontinuity.
Note that the m = 0.02
result shows
littleshiftdi↵erence
4.0transition.
The smallof µc compared to the chiral condensate case is attrib
0.8
m = 0.01 one.
𝑚 = 0.01
𝑚=0.01 in Eq. (3.5).
the definition of the numericalderivative
3.5 density is obtained by the numerical derivative of pressure inparticle number
6
It is also possible to 0.6
evaluate the chiral condensate with the impurity tensor method [15, 2the chemical potential:
3.0hn(µ)i =@P (µ)
P (µ +⇡@µµ)µEq. (2.9) consists of eight0.4types of tensor, however,we need to coarse-grain several kinds of tenso

P (µ)

.

0.2(3.5)

2.50.0pendence of hni is shown in Fig. 7. We observe an abrupt jump fromhni = 01 at µc = 2.9375± 0.0625. This is another indication of the first-orderphase-0.2
2.0
0.0
1.0
0.0
1.0
2.0
3.0
4.0
5.0
µn. The small shift of µc compared to thechiral condensate case is attributed totion of the numerical derivative in Eq. (3.5).

– 10 –
2.0

µ3.0

4.0

5.0

so possible to evaluate the chiral condensate with the impurity4 tensor method[15, 26]. Since
447. 4Particle
Figure 6. Pressure at m = 0.01 as a function of µ on 128 Figureand 1024
lattices.number density at m = 0.01 as a function of µ on 128 and 1024 lattice
onsists of eight types of tensor, however,we need to coarse-grain several kinds of tensor network𝜇 ≈ 3.0付近での不連続性(トビ) ⇒ 一次相転移
TRG法では状態方程式も容易に得られる
4 Summary and outlook

1.2
L = 128
L = 1024– 10 –We have investigated the restoration of the chiral symmetry of the NJL model indense region at very low temperature employing the Kogut–Susskind fermion actionまとめ・ モンテカルロ法における符号問題および複素作用問題がない
Z=!

Dφ exp(−SRe[φ] + iSIm[φ])$ ! π dϕx,µ $・ LDのシステムサイズに対する計算コスト∝D×log(L)Z=T (ϕx,1, ϕx+1̂,2, ϕx+2̂,1, ϕx,2)−πx,µx2π
・ グラスマン数を直接扱うことが可能
・ 分配関数Zそのものを計算可能

T (ϕx,1, ϕx+1̂,2, ϕx+2̂,1, ϕx,2) = exp β cos px + iθ qx
2πdϕx,µ Z =  −πexp (−S)
x,µ2π
素粒子物理：軽いクォークのダイナミクス，有限密度QCDの相構造解析'(' √ (Strong CP問題などの研究に応用可能)' √ (tTi,j,k,l ⇒ T{j,k},{l,i} = U ΛV {j,k},{l,i} =U Λ {j,k},m V Λ {l,i},m$ ! πm
物質科学：強相関量子系，金属絶縁体転移，高温超伝導などの)=(S1){j,k},m (S3){l,i},m
研究に応用可能 (ハバードモデル)
m'Ti,j,k,l ⇒ T{k,l},{i,j} = U ΛVt(=' √ (√ (U Λ {k,l},m V Λ {i,j},m)'m現段階：)=(S2){k,l},m (S4){i,j},mm
4次元モデル・理論の計算が可能になった
有限密度NJLモデルの解析に成功)(new)
To,n,m,p=(S4){l,k},o (S3){k,j},n (S2){j,i},m (S1){i,l},p⇒i,j,k,l
有限密度QCD・ハバードモデルの相構造解析{k,l},{i,j}