lee-yang zero distribution of high temperature qcd …...the arg(pol) is translated into a 4. this...
TRANSCRIPT
![Page 1: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/1.jpg)
1
Keitaro Nagata (KEK)
Reference KN, K.Kashiwa, A.Nakamura, S.M.Nishigaki [arXiv:1410.0783] also, PTEP2012, 01A103(2012), arXiv:1305.0760, etc.
Lee-Yang zero distribution of high temperature QCD and Roberge-Weiss phase transition
格子QCDと現象論模型による有限温度・有限密度の物理の解明 九州大学, 2015/02/19
wikipedia
![Page 2: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/2.jpg)
Taylor Canonical
RWphase transition
![Page 3: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/3.jpg)
Taylor Canonical
RWphase transition
Lee-Yang zero
![Page 4: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/4.jpg)
Taylor Canonical
RWphase transition
Lee-Yang zero
Linking them !
![Page 5: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/5.jpg)
Introduction
5
Beam energy scan experiments at RHIC
attempts to explore QCD phase diagram
information in finite density region and accessible in lattice QCD.
Finding of CEP has not been reported.
However, the obtained data are valuable
![Page 6: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/6.jpg)
Introduction
6
What we expect to understand from data (in principle)
baryon number distribution - canonical partition functions
cumulants - Taylor coefficients
Agaarwal et.al. PRL105, 022302(’10), arXiv:1004.4959.
![Page 7: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/7.jpg)
Determination of Zn even allows to study Lee-Yang zeros
Even the Roberge-Weiss phase transitions
Nagata, et. al.PTEP2012, 01A103(2012), arXiv:1204.1412
Nakamura, Nagata, arXiv:1305.0760
![Page 8: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/8.jpg)
Lee-Yang zeros from exprimental data
Nakamura, Nagata, arXiv:1305.0760Agaarwal et.al. PRL105, 022302(’10), arXiv:1004.4959.
![Page 9: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/9.jpg)
Lee-Yang zero calculation is difficultZeros of fugacity polynomial
the coefficients fluctuate statistically
truncation of the polynomial is inevitable.
!
Zeros of Z(µ) in Monte Carlo simulation with reweighting.
Partition function zero is the breakdown of the reweighting.
![Page 10: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/10.jpg)
Purpose
We want to present reliable result for Lee-Yang zeros.
focus on high temperature QCD
provide analytic calculations of Lee-Yang zeros
examine the effect of statistical fluctuation and convergence of Lee-Yang zeros in lattice QCD simulations
!
We also present a relation between RW phase transition and measurable quantities
![Page 11: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/11.jpg)
Lee-Yang zero theorem (Lee&Yang, PR87, 404 and 410(’52))
11
A grand canonical system
Expansion in terms of eigen vectors of N
Zn : coefficients
xi_i : roots
Product expression
![Page 12: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/12.jpg)
12
How zeros are related to phase transitions
Extend µ to complex and define the real part of the free energy as φ = Re[f]
Taking second order derivative w.r.t ξ
Poisson’s equation
Electrostatic analogue
![Page 13: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/13.jpg)
Distribution of Lee-Yang zeros
13
1. ξi also depends on T
2. None of ξi is real positive.
3. ξi and 1/ξi form a pair No LY zero
1. Zn depends on T
2. All Zn are real and positive.
3. Zn = Z-n
![Page 14: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/14.jpg)
Foundation, applications to Ising model, electrostatice analogue [Yang & Lee, Phys. Rev.87 404(1952), Lee & Yang, Phys. Rev.87 410(1952)] Extension to spin 3/2 [Asano, JPSJ25, 1220, ’68]
Extension to temperature [Fisher ] Scaling of zeros near CP [Itzkson, Pearson, Zuber NPB220, 415, ’83, Stephanov PRD73, 094508(’06)] Some other applications
non-equilibrium [Blythe, Evans, PRL89, 08061, (’02)] collective flow
Potts model and Campel-Blume model [Biskup et. al.] Random matrix model [Halasz, Jackson, Verbaaschot(’97)] LG model [Ejiri, Shinnno, Yoneyama, arXiv:1404.6004], Perhaps, there are more studies in the mathematical context of zeros of functions. etc…
14
References
![Page 15: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/15.jpg)
Some features of high temperature QCD
15
![Page 16: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/16.jpg)
Features of QCD at high temperatures
16
We use some features of the free energy of high T QCD to obtain Lee-Yang zeros. !
Chemical potential dependence of Free energy !
RW periodicity and phase transition [Roberge-Weiss (’86)]
![Page 17: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/17.jpg)
1. Free energy at high temperature Free energy of QCD is quartic w.r.t µ at high T.
1. massless free quark
f(µ) is the quartic function of µ. c2 dominates at small µ
(c2= Nf/2, c4 = Nf/(4π2) => c2/c4 ~ 10)
![Page 18: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/18.jpg)
Nagata, Nakamura JHEP(2012)
2. Lattice simulations suggested that this equation is valid at temperature slightly above Tc (10-20%).
Consistent results have been obtained for several lattice setup
![Page 19: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/19.jpg)
2. QCD at imaginary µ
SU(Nc) gauge theory has a periodicity
This periodicity is realized smoothly at low T, and discontinuously at high T.
pcT
TI /
T
),3/( RWT
A
C
D 3/2
-0.2
-0.1
0
0.1
0.2
-0.2 -0.1 0 0.1 0.2 0.3Re[Pol]
`=1.95
Im[Pol]
µI=0.00µI=0.16µI=//12µI=0.28
I II III
Nagata, Nakamura PRD (2012)
![Page 20: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/20.jpg)
Calculation of Lee-Yang zeros
20
![Page 21: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/21.jpg)
Lee-Yang零点を2通りの方法で計算します。
Cancellation of the free energy
Zeros of the fugacity polynomial
![Page 22: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/22.jpg)
1. Cancellation of free-energy
RW phases are distinguished by the argument of Pol.
The arg(Pol) is translated into A4.
This modifies the free energy as
Free energy is different in different RW phases
![Page 23: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/23.jpg)
1. Cancellation of free-energy
Cancellation of free energies [Biskup et al(’01)]
Approximate solution for c2>>c4
Possible to solve it with c4
fI and fIII, and fII and fIII.
Cancellation of two types of free energy allows Z=0
![Page 24: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/24.jpg)
2. Zeros of the fugacity polynomial
We need Zn. : Fourier integral with the free energy as input (analytic)
(We use the fugacity expansion in lattice simulation. )
We show that the fugacity polynomial of high T QCD is well approximated by a well-known polynomial.
![Page 25: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/25.jpg)
Zn from the Fourier transformation
We use the quartic form of f(µ) as input.
f(µ) is obtained for real µ. On the other hand, the Fourier integral requires complex µ.
First, we derive Zn using the Fourier transformation.
![Page 26: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/26.jpg)
We decompose the integral into three domains.
Shift of θ leads to
(This relation holds for any temperature, regardless of the RW phase transition)
Next, we use the RW periodicity
![Page 27: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/27.jpg)
At T/Tc > 1.1~1.2, c2/c4 ~ 10. We use the saddle point approximation.
We assume the Gaussian Zn is valid for large n. SPA is valid for small Re[µ]
Now, we use the quartic expression of f(µ) and SPA
0 0.5
1 1.5
2 2.5
3 3.5
4
0 0.5 1
- (f (µ
/T) -
f(0)
)/T4
µ/T
ExactSPA
![Page 28: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/28.jpg)
If Zn is Gaussian, then Z(µ) is a Jacobi-theta function.
Method1
Theta function
Zeros of theta function
![Page 29: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/29.jpg)
Lee-Yang zero distribution of theta function
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[j
]
Re[j]
Jacobi Theta"jtzerofinite.dat"
Zeros approaches to the RW transition point as 1/V. Spacing of zeros is a prediction
k
![Page 30: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/30.jpg)
Calculation of Lee-Yang zero (lattice QCD)
30
gauge configurations are generated at µ = 0 and use reweighing volume : 8^3x4, 10^3x4 ( (2~3 fm)^3 mass : mps/mv~0.8 (mπ~800 MeV) action : clover-improved Wilson fermion +renormalization improved gauge # of statistics : 400 (20 trajectoriy-intervals, 3000 therm.)
![Page 31: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/31.jpg)
1e-50
1e-40
1e-30
1e-20
1e-10
1
-40 -30 -20 -10 0 10 20 30 40
Zn
B
nB
Ns=10Ns=8
a) Ns=10 n0=37 b) Ns=8 n0=32 c) Ns=8 n0=19
a
bc
Lattice simulation vs analytic result
We reanalyze previous lattice data. Estimation of statistical error of Lee-Yang zeros
bootstrap analytis with 1000 BS samples. Estimation of convergence and finite size scaling
![Page 32: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/32.jpg)
Error analysis and Lattice vs Analytic
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-1 -0.9 -0.8 -0.7 -0.6 -0.5
∠ ξ Ns=8, n0=32 Ns=8, n0=19 Ns=10, n0=37
A scaling predicted by theta function
0 1 2
0 1Relevant zero
Zeros near the RW phase transition point are statistically stable.
![Page 33: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/33.jpg)
Error analysis and Lattice vs Analytic
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-1 -0.9 -0.8 -0.7 -0.6 -0.5
∠ ξ Ns=8, n0=32 Ns=8, n0=19 Ns=10, n0=37
A scaling predicted by theta function
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 1 2 3 4 5
|Re[j]
|VT3
l
c) Ns=8, n0=19exp(-3(2l+1)/(4c2))
0 1 2
0 1
Underestimation is caused by the SPA (effect of c4)
![Page 34: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/34.jpg)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-1 -0.9 -0.8 -0.7 -0.6 -0.5
∠ ξ Ns=8, n0=32 Ns=8, n0=19 Ns=10, n0=37
ConvergenceA scaling predicted by theta function
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 1 2 3 4 5
|Re[j]
|VT3
l
c) Ns=8, n0=19b) Ns=8, n0=32exp(-3(2l+1)/(4c2))
0 1 2
0 1
多項式の最大次数を増やしても, 零点は不変=>収束OK
![Page 35: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/35.jpg)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-1 -0.9 -0.8 -0.7 -0.6 -0.5
∠ ξ Ns=8, n0=32 Ns=8, n0=19 Ns=10, n0=37
Volume scalingA scaling predicted by theta function
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 1 2 3 4 5
|Re[j]
|VT3
l
c) Ns=8, n0=19b) Ns=8, n0=32a) Ns=10, n0=37exp(-3(2l+1)/(4c2))
0 1 2
0 1
体積を増やしても, 零点は不変=>体積スケーリングを満す
![Page 36: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/36.jpg)
SummaryWe have studied Lee-Yang zeros of high temperature QCD.
two types of analytic calculation
reanalysis of lattice data
At high temperature, the canonical partition function is well approximated by the Gaussian.
If the Gaussian with RW periodicity, then the RW phase transition occurs.
(Note that the converse is not always true)
36
![Page 37: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/37.jpg)
From the view point of Lee-Yang zeros
a)Ising b) high T QCD c) high T QCD
Gaussian type of canonical partition functions are an exceptional case of Lee-Yang zero circle theorem.
e.g. free fermions at small density
![Page 38: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/38.jpg)
Roberge-Weiss periodicity
Grand partition function satisfies [Roberge & Weiss (1986)]
Presence of quark breaks center (Z(Nc)) invariance explicitly. ! However, this breaking can be absorbed into the shift of imaginary part of quark chemical potential
![Page 39: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/39.jpg)
Roberge-Weiss periodicity
RW periodicity is realized low T : smoothly high T : discontinuously
-0.2
-0.1
0
0.1
0.2
-0.2 -0.1 0 0.1 0.2 0.3Re[Pol]
`=1.95
Im[Pol]
µI=0.00µI=0.16µI=//12µI=0.28
-0.2
-0.1
0
0.1
0.2
-0.2 -0.1 0 0.1 0.2 0.3Re[Pol]
`=1.80
Im[Pol]
µI=0.00µI=0.16µI=//12µI=0.28
低温
高温At high T
![Page 40: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/40.jpg)
Lattice QCD simulations
40
![Page 41: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/41.jpg)
Lee-Yang零点と符号問題1 : 一般的に
Lee-Yang 零点 : Z(µ)=0
ボルツマン因子の相殺が必要.
熱力学パラメータを複素数へ拡張することで発生.
Lee-Yang零点の持つ一般的な性質.
QCDだけでなく, Ising模型などでも発生
41
![Page 42: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/42.jpg)
Lee-Yang零点と符号問題2 : QCDでは
格子QCD : 実化学ポテンシャルにおいて、フェルミオン行列式が複素数となる.
実µに対して, Z(µ)が複素数になりうる
符号問題のために、物理軸上において非物理的零点が発生することがある.
符号問題は体積とともに激しくなり、非物理的零点が実軸に近づく.
(符号問題とLee-Yang零点の判別は難しい)
42
![Page 43: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/43.jpg)
Calculation of Lee-Yang zero in LQCD
43
格子QCDにおけるLee-Yang zero点の計算 Z(µ) = 0 をreweightingを用いて求める. Z(µ)をフガシティξの多項式として表して根を求める. 自由エネルギーの漸近形を求める etc
!
![Page 44: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/44.jpg)
Calculation of Lee-Yang zero in LQCD
44
格子QCDにおけるLee-Yang zero点の計算 Z(µ) = 0 をreweightingを用いて求める. Z(µ)をフガシティξの多項式として表して根を求める. 自由エネルギーの漸近形を求める etc
!
![Page 45: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/45.jpg)
Questions
truncation is valid ?
45
Zeros show expected volume scaling ?
(e.g. 1/V for 1st )
Lee-Yang 零点の妥当性に関するいくつかの疑問
![Page 46: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/46.jpg)
Buckup
46
![Page 47: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/47.jpg)
Canonical approach
Canonical and grand canonical partition functions follow
47
Once we have {Zn}, then Z(µ) is obtained at any µ.
This provides a tool to extrapolate data at µ=0 to nonzero µ in LQCD [e.g. Babour et.al.(’91) ].
= probability for an n-particle state
![Page 48: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/48.jpg)
Buckup : Numerical calculation of Lee-Yang zero : CBK
48
![Page 49: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/49.jpg)
How to achieve LY zeros ?
Calculation of Zn : truncation is inevitable
49
![Page 50: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/50.jpg)
How to achieve LY zeros ?
Calculation of Zn : truncation is inevitable
50
Cauchy integral + recursive division + multi-precision arithmetic
![Page 51: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/51.jpg)
Calculation of Zn Reduction formula
51
![Page 52: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/52.jpg)
Zn =
⌧C
Nf
0 dn(det�(0))Nf
�
[Gibbs ('86). Hasenfratz, Toussaint('92). Adams(’03, ‘04), Borici('04). KN&AN(‘10), Alexandru &Wenger(’10)]eigenvalues of a transfer matrix
we use gauge configurations at µ=0 (reweighting)
Calculation of Zn in lattice QCD simulation
52
KN, S. Motoki, Y. Nakagawa, A. Nakamura, T. Saito [PTEP01A103(2012)], references therein.
![Page 53: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/53.jpg)
Lattice Data
53
![Page 54: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/54.jpg)
Lee-Yang zeros - convergence
54
1e-50
1e-40
1e-30
1e-20
1e-10
1
-40 -30 -20 -10 0 10 20 30 40
Zn
B
nB
Ns=10Ns=8
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
� j Ns=8, n0=32 Ns=8, n0=19
![Page 55: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/55.jpg)
Lee-Yang zeros - convergence
55
1e-50
1e-40
1e-30
1e-20
1e-10
1
-40 -30 -20 -10 0 10 20 30 40
Zn
B
nB
Ns=10Ns=8
Zeros relevant for the RW phase transition are stable.
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
� j Ns=8, n0=32 Ns=8, n0=19
![Page 56: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/56.jpg)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
� j Ns=8, n0=32 Ns=8, n0=19
Lee-Yang zeros - convergence
56
1e-50
1e-40
1e-30
1e-20
1e-10
1
-40 -30 -20 -10 0 10 20 30 40
Zn
B
nB
Ns=10Ns=8
![Page 57: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/57.jpg)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
� j Ns=8, n0=32 Ns=8, n0=19
Lee-Yang zeros - convergence
57
1e-50
1e-40
1e-30
1e-20
1e-10
1
-40 -30 -20 -10 0 10 20 30 40
Zn
B
nB
Ns=10Ns=8
Zeros relevant for the RW phase transition converge.
![Page 58: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/58.jpg)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
� j Ns=8, n0=32 Ns=8, n0=19
Lee-Yang zeros - convergence
58
1e-50
1e-40
1e-30
1e-20
1e-10
1
-40 -30 -20 -10 0 10 20 30 40
Zn
B
nB
Ns=10Ns=8
Zeros relevant for the RW phase transition converge.
Zeros near the origin do not converge.
![Page 59: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/59.jpg)
Lee-Yang zeros - Volume dependence (T/Tc=1.2)
59
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
� j Ns=8, n0=19 Ns=10, n0=37
1e-50
1e-40
1e-30
1e-20
1e-10
1
-40 -30 -20 -10 0 10 20 30 40
Zn
B
nB
Ns=10Ns=8
Zeros of different volume are located on the common trajectory.
No RW-like SPA …x
RW-like SPA … o
![Page 60: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/60.jpg)
Convergence on the negative real axis
c.f. Leibnitz’s test for an alternating series
60
Infinite sum of higher order terms is bounded on the negative real axis.
![Page 61: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/61.jpg)
Lee-Yang zeros - 20 years
61
![Page 62: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/62.jpg)
c2-dominance at high T and small µ We use a saddle point approximation to solve this integral.
![Page 63: Lee-Yang zero distribution of high temperature QCD …...The arg(Pol) is translated into A 4. This modifies the free energy as Free energy is different in different RW phases 1. Cancellation](https://reader033.vdocuments.net/reader033/viewer/2022042316/5f0584647e708231d4135ab0/html5/thumbnails/63.jpg)
incomplete Gamma function is suppressed as V^k /exp(V)