有限密度格子ゲージ理論におけるセンター対称性による符号問題の回避法を用いた粒子密度確率分布関数

江尻 信司 新潟大理
熱場の量子論とその応用
2020年8月24-26日

Sign problems in the canonical approach• Canonical partition function: 𝑍𝐶(Fugacity expansion)𝑍𝐺𝐶 𝑇, 𝜇 = ෍ 𝑍𝐶 𝑇, 𝑁 exp 𝑁𝜇 Τ𝑇 ≡ ෍ 𝑊 𝑁𝑁𝑁• Effective potential as a function of the quark number N.
𝜇𝑉eff (𝑁ሻ = −ln𝑊(𝑁ሻ = −ln𝑍𝐶 (𝑇, 𝑁ሻ − 𝑁𝑇
• At the minimum,𝜕𝑉eff (𝑁൯𝜕ln𝑊(𝑁ሻ𝜕ln𝑍𝐶 (𝑇, 𝑁ሻ 𝜇=−=−− =0𝜕𝑁
𝜕𝑁𝜕𝑁𝑇
• First order phase transition: Two phases coexist.
𝑉eff (𝑁൯In the thermodynamic limit,𝜕ln𝑍𝐶 (𝑁ሻ−𝜕𝑁𝜇Τ𝑇𝜇𝜕ln𝑍𝐶 (𝑇, 𝑁ሻ
𝑁 =−𝑇𝜕𝑁

Maxwell construction𝑁Hopping parameter expansion, Fugacity expansion
𝑍 = ඲ෑ𝑑𝑈𝜇 (𝑥൯ det𝑀 𝑁f e−𝑆𝑔

• Grand partition function

𝑥,𝜇Quark matrix• Hopping parameter expansion [K~1/(quark mass)]

1   n ln det M 1nnln det M K    K
DKnK nn 0 n!
n 1 n!
 K 0• Dn： Sum of all n-step Wilson loops (connected)
winding number N=0N=1
N=2
Periodic B.C.
Periodic B.C.
No m-dependence

1/Tem /Tem /T

e2m /T

N... ePolyakov loop W• Classify the Wilson loops by the winding number.
• Fugacity expansion: expansion with the winding number N.
𝑍𝐺𝐶 𝑇, 𝜇 = ෍ 𝑍𝐶 𝑁, 𝑇 exp 𝑁𝜇Τ𝑇
𝑁N m /TCenter symmetry• Quenched QCD (no dynamical quarks, detM=1)2𝜋𝑖4𝜋𝑖• Center of SU(3) group: 𝑈center = 𝜔𝐼, 𝜔 = {1, 𝑒 3 , 𝑒 3 ቅ• On one time slice, 𝑈4 ⇒ 𝜔𝑈4• Polyakov loop W changes as 𝛺 ⇒ 𝜔 𝛺• 𝛺 = 0, when the symmetry is unbroken.
• winding number N loop

W in the complex planeZ(3) symmetric𝛺𝑁 ⇒ 𝜔𝑁 𝛺𝑁• Canonical partition
𝑍𝐶 𝑁, 𝑇 ⇒ 𝜔𝑁 𝑍𝐶 𝑁, 𝑇
𝑍𝐶 𝑁, 𝑇 = 0, when 𝑁 ≠ 3 × (integerሻProbability distribution at Tc• For U(1) gage theory, 𝑍𝐶 𝑁, 𝑇 ⇒ 𝑒 𝑖𝑁𝜃 𝑍𝐶 𝑁, 𝑇
𝑍𝐶 𝑁, 𝑇 = 0, for 𝑁 ≠ 00 ≤ 𝜃 < 2𝜋

We discuss U(1) gauge theory.

Center symmetry in U(1) gauge theory• Centers of group are all members U eiq.
• Under the center transformation,Z GC T , m    Z C  N , T  e iNq e Nm T ,N

• Except for N=0, the canonical partition function is zero.

1 iNq Nm T Z GC T , m  Z C N , T  e edq  Z C 0, T   0  0  
2  N

Charged particles cannot exist. No m-dependence.

Probability distribution of Polyakov loop W

Deconfinement
Broken phase

The distribution is always U(1) symmetry.
The expectation value of W is always zero.
To discuss the symmetry breaking,Explicit breaking term: required,e.g. dynamical quarks.

Lattice
Nt=4
Ns=12
Ns=16
Ns=24Confinement
Symmetric phaseSpontaneous Symmetry Breaking
Adding dynamical quark with a small K• Center symmetry: explicitly broken.
• Double limit: V   and K  0.
Polyakov loop ReΩ
Confinement
Symmetric phaseb=0.8

ReΩ ~𝐾𝑁𝑡

K~1/(quark mass)
V: volumeU(1) gauge theoryReΩ ~𝑉𝐾𝑁𝑡
Deconfinement
Broken phase
b=1.2

Lattice
Nt=4
Ns=24
Ns=16
Ns=12In V  , 𝐾 = 0 limit,
Symmetric phase: ReΩ =0, Broken phase: ReΩ ~𝑉𝐾𝑁𝑡 (finite)Integrate over the complex phase q• Introducing the distribution function W(|W|)U(1) symmetry
 A function of only |W|, independent of qln det 𝑀 = 6 × 2𝑁𝑡 𝑁𝑠3 𝐾𝑁𝑡 Ω + Ω∗ + ⋯1ReΩ = න 𝐷𝑈 ReΩ𝑒 𝜀𝑉ReΩ 𝑒 −𝑆𝑔 = න Ω cos𝜃𝑒 𝜀𝑉 Ω cos𝜃 𝑊 Ω 𝑑𝜃𝑑 Ω𝑍= 𝜀𝑉𝜋 න Ω 2 𝑊 Ω 𝑑 Ω + ⋯[ 𝜀~◍𝐾𝑁𝑡 ]𝑛𝜋1𝜀𝑉2𝑛 𝑊 Ω 𝑑 Ω + ⋯ReΩ𝑛 = න 𝐷𝑈 ReΩ𝑛 𝑒 𝜀𝑉ReΩ 𝑒 −𝑆𝑔 =නΩ𝑍𝑛! 2𝑛−1

No complex phase, No sign problem

Canonical partition function by Saddle pointapproximation (S.E., Phys. Rev. D78, 074507 (2008))

mI
Saddle point• Inverse Laplace transformation•m01 - N  m0 T i mI T Z C T , N  dmTe
ZGC T , m0  i mI 
I
2𝑁f 𝜕 lndet𝑀
𝑧Saddle point: 0𝐷′ 𝑧0 ==𝜌
𝑉 𝜕 𝜇Τ𝑇𝜇=𝑧𝑇mR

IntegralArbitrary m00• Canonical partition function in a saddle point approximation
𝑍𝐶 𝑇, 𝜌11=exp 𝑉 𝐷 (𝑧0 ሻ − 𝜌𝑧0 𝑒 −𝑖𝛼Τ2𝑍quench 𝑇𝑉 𝐷′′ 𝑧02𝜋≡12𝜋𝑒 𝐹+𝑖𝜑 quenchquench

𝐷𝜇𝑇

𝑁f=ln det𝑀𝑉• Derivative of ln Zc
Similar to the reweighting method(sign problem & overlap problem)

𝑁f 𝜕 2 ln det𝑀
𝜇𝑖𝛼𝐷′′=≡𝐷′′𝑒𝑇𝑉 𝜕 𝜇Τ𝑇 2𝑧0 𝑒 𝐹+𝑖𝜑 quench−1 𝜕 ln𝑍𝐶 (𝑇, 𝜌ሻ≈𝑉𝜕𝜌
𝑒 𝐹+𝑖𝜑 quench

saddle point

reweighting factor

Heavy quark region (K: small) U(1) theory
Approximation: ln det 𝑀 ≈ 6 × 2𝑁𝑡 𝑁𝑠3 𝐾𝑁𝑡 𝑒 𝜇/𝑇 Ω + 𝑒 −𝜇/𝑇 Ω∗𝑧0 𝑒 𝐹+𝑖𝜑 quench 𝜀 𝑁 ∫ 𝑥0 𝑒 −𝑉eff 𝑑|Ω|−1 𝜕 ln𝑍𝐶 (𝑇, 𝜌ሻ≈≈ 𝑁𝑉𝜕𝜌𝑒 𝐹+𝑖𝜑 quench𝜀 ∫ 𝑒 −𝑉eff 𝑑|Ω|𝜑 = −𝑁𝜃𝑁 = 𝜌𝑉(Ω =Ω 𝑒 𝑖𝜃 )U(1) symmetric𝑒 𝐹+𝑖𝜑 = න 𝑒 𝐹 cos (𝑁𝜃ሻ 𝑒 𝜀𝑉 Ω cos𝜃 𝑊 Ω 𝑑𝜃 𝑑 Ω

Phase of W

𝑁f = 2, 𝐾 = 0.02

𝑧0 = 𝑥0 + 𝑖𝑦0

243 × 6 latticeExternal source
Solving thesign problem

𝜇
Quark number 𝑁 vs.
𝑇• In the thermodynamic limit,−1 𝜕 ln𝑍𝐶 (𝑇, 𝜌ሻ 𝜇=𝑉𝜕𝜌𝑇Critical 𝛽 at 𝜇=0𝛽 = 1.0096
𝑁f = 2, 𝐾 = 0.02
243 × 6 lattice
𝑁 = 𝜌𝑉Application to SU(3) gauge theory• SU(3) gauge theory in the low temperature phase:
The Polyakov loop is U(1) symmetric (not Z(3)) in thelarge volume limit, if the Z(3) center symmetry isunbroken.
• In the high temperature phase of SU(3) gauge theory:
The probability distribution of complex phase is wellapproximated by a gaussian function.
Value of |Ω|Distribution ofthe phase ΦSummary• Quark number distribution function 𝑊(𝑁ሻZGC T , m   ZC T , N  expNm T   W N NN• Considering the Center symmetry,– Canonical partition function of QCD is zero for 𝑁 ≠ 3𝑛 𝑛: integer .
– For U(1) gauge theory, 𝑍𝐶 = 0 for 𝑁 ≠ 0.

• It is important to break the center symmetry adding an externalfield term.
• Using the U(1) center symmetry, complex phase can be removed.

• For some cases, the sign problem is solved– U(1) gauge theory. Quarks are heavy.
– Deep confinement phase– Deconfinement phase (the sign problem is not serious.)フラッシュトーク
• 有限密度格子ゲージ理論におけるセンター対称性による符号問題の回避法を用いた粒子密度確率分布関数

有限密度格子ゲージ理論におけるセンター対称性による符号問題の回避法を用いた粒子密度確率分布関数江尻 信司 (新潟大理) 熱場の量子論とその応用 (2020/8/24-26)• 粒子数確率分布関数 𝑊(𝑁ሻ ： 熱浴中で生成される粒子数𝑁
𝑍𝐺𝐶 𝑇, 𝜇 = ෍ 𝑍𝐶 𝑇, 𝑁 exp 𝑁𝜇Τ𝑇 ≡ ෍ 𝑊 𝑁𝑁

大分配関数

カノニカル分配関数

𝑁

フガシティー 確率分布関数• 閉じ込め相転移で自発的に破れる対称性：センター対称性– QCDのカノニカル分配関数𝑍𝐶 𝑇, 𝑁 は 𝑁が3の倍数以外、ゼロ– U(1)格子ゲージ理論の場合、𝑁 = 0以外、厳密に 𝑍𝐶 = 0→ 究極の符号問題• U(1)格子ゲージ理論に注目。
• 微小な外場を加えてセンター対称性を破ることが重要。
• ついでに、U(1) センター対称性を利用して、 符号問題を取り除く。

U(1)格子ゲージ理論で、カノニカル分布関数の微分を計算
生成確率が最大の粒子数が𝑁となる μ/T−1 𝜕 ln𝑍𝐶 (𝑇, 𝜌𝑉ሻ 𝜇=𝜇=0 で閉じ込め相𝑉𝜕ρ𝑇
𝜇=0 で非閉じ込め相

𝑁 = 𝜌𝑉
Critical 𝛽 at 𝜇=0𝛽 = 1.0096

粒子数

𝑁f = 2, 𝐾 = 0.02
243 × 6 lattice

ポスター番号：35 江尻信司

• 普通に計算したら 𝑍𝐶 は厳密にゼロ。
• 微小な外場を入れて、0の極限をとる。
• 鞍点近似を使う。（体積が大きい場合の近似）• 粒子の質量が重い場合を考える。
（ホッピングパラメータ展開による近似）• U(1)ゲージ理論の特殊性を利用して符号問題を取り除く。
• QCD（つまりSU(3)）への適用、粒子の質量が軽い場合も考察。
• 有限密度格子QCDの符号問題の新しい回避方法を提案。