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

Taylor Canonical

RWphase transition

Taylor Canonical

RWphase transition

Lee-Yang zero

Taylor Canonical

RWphase transition

Lee-Yang zero

Linking them !

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

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.

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

Lee-Yang zeros from exprimental data

Nakamura, Nagata, arXiv:1305.0760Agaarwal et.al. PRL105, 022302(’10), arXiv:1004.4959.

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.

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

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

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

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

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

Some features of high temperature QCD

15

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)]

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)

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

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)

Calculation of Lee-Yang zeros

20

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

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

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

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.

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.

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

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

If Zn is Gaussian, then Z(µ) is a Jacobi-theta function.

Method1

Theta function

Zeros of theta function

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

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.)

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

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.

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)

-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

-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

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

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

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

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

Lattice QCD simulations

40

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

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

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

!

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

!

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

Buckup

46

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

Buckup : Numerical calculation of Lee-Yang zero : CBK

48

How to achieve LY zeros ?

Calculation of Zn : truncation is inevitable

49

How to achieve LY zeros ?

Calculation of Zn : truncation is inevitable

50

Cauchy integral + recursive division + multi-precision arithmetic

Calculation of Zn Reduction formula

51

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.

Lattice Data

53

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

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

-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

-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.

-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.

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

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.

Lee-Yang zeros - 20 years

61

c2-dominance at high T and small µ We use a saddle point approximation to solve this integral.

incomplete Gamma function is suppressed as V^k /exp(V)