lattice qcd at low temperature and ﬁnite density - early ... · silver blaze 6 • the effect of...

Lattice QCD workshop at KEK, 2014/01/20 -/22

Lattice QCD at low temperature and finite density��- early onset/Silver Blaze problem

Keitaro Nagata KEK, Theory Center

1

ü  KN, arXiv:1204.6480 ü  KN, Motoki, Nakagawa, Nakamura, Saito, PTEP01A103(’12). ü  KN, Nakamura, PRD82, 094027(‘10)


Outline

•  Introduction •  Framework : reduction formula, reduced matrix •  Zero temperature limit of the fermion determinant •  mπ/2 ? or MN/3 ?

2


Introduction

3


Introduction

4

εF

•  Hadronic phase at low-T -  d.o.f is nucleons and pions

ε>εF

•  In order to add one particle, energy larger than fermi energy is needed.

T=0


Early onset problem

5

•  Hadronic phase at low-T : d.o.f is nucleons and pions •  µ for the onset of quark number at T=0 –  expectation : µ=MN/3 –  lattice calculations : µ=mπ/2

Glasgow method -  propagator matrix -  reweigh5ng

[Barbour et. al. PRD56, 7063 (1997)] 6^4, staggered, mq=0.1


Silver Blaze

6

•  The effect of µ does not appear, although the action explicitly depends on µ. -  well-known property of fermion at T=0. -  But it is unclear how it is realized in field theory).

•  Cohen(‘04) considered isospin chemical potential case and showed. -  fermion determinant of QCD is independent of µ at T=0 for µ

less then half the pion mass. •  (baryon Silver Blaze (µ<MN/3) has not been solved)


Motivation

7

•  In previous studies by Glasgow method, it was unclear what is the origin of the early onset problem.

•  Isospin Silver Blaze by Cohen provides an explanation of the early onset observed in Glasgow method.

•  Motivation 1.  lattice study is important to confirm it non-perturbatively. 2.  It is still unclear why it is half the pion mass, not one-third of

the nucleon mass.

•  we mostly concentrate on -  µ<mπ/2 -  property of fermion determinant at configuration level


Motivation

•  Question –  Why does the quark number start at µ=mπ/2 at T=0 ?

•  We discuss the early onset problem in taking the zero temperature limit of the fermion determinant

•  relation between fermion determinant and pion mass

8


Idea and outline of talk

9

µ-independence of fermion determinant

meson mass (meson propagator)

zero T (large Nt) limit

µ-independent lightest meson (pion)

?

? propagation of quarks


Framework

10


Fermion matrix

11

•  Fermion path integral and fermion determinant

-  Δ : matrix with the rank 4 Nc Nx Ny Nz Nt -  it consists of fermion loops -  chemical poten5al comes from temporal loops


Reduction formula

12

[Gibbs ('86). Hasenfratz, Toussaint('92). Adams(’03, ‘04), Borici('04). KN&AN(‘10), Alexandru &Wenger(’10)]

← det Δ can be calculcated analy5cally.

•  A formula to perform t-‐part of the fermion determinant. -  A fermion matrix in t-‐t matrix rep.


Reduction formula

13

•  det Δ is reduced to

⇠ = e�µ/T

det� = ⇠�Nred/2C0 det(⇠ +Q)

-  C0 and Q are func5ons of gauge fields. -  chemical poten5al and gauge fields are separated -  Q is a matrix with rank Nred=12 Ns

3


Reduction formula

14

•  det Δ is reduced to

⇠ = e�µ/T

det� = ⇠�Nred/2C0 det(⇠ +Q)

•  Using eigenvalues of Q, det Δ is given by

⇠ = e�µ/T

•  eigenvalues of Q control the µ-‐dependence of det Δ


Reduced matrix (Wilson fermions)

15

•  It can be applied to e.g. -  winding number expansions (Q, QQ, QQQ...) -  hadron masses (QQ^+, QQQ ...) [Gibbs(‘86), Fodor, Szabo,

Toth(‘07)]

spa5al temporal r+-‐=(1+-‐gamma4)/2


Spectral properties of reduced matrix 1

16

-800-600-400-200

0 200 400 600 800

-600-400-200 0 200 400 600 800

Im[h

]

Re[h]

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

Im[h

]

Re[h]

10-310-210-1100101102103

0 1 2 3 4 5 6n/1000

|hn|

T/Tc=1.08 •  Eigenvalues λ of Q - pair (γ5-‐hermi5cy)

�n $ 1/�⇤n

Eigenvalue distribu5on L: large scale R: small scale

independent


10-6

10-4

10-2

100

102

104

106

0 1 2 3 4 5 6n/1000

|h| , |h|1/Nt

Nt=4Nt=8Nt=4 (scaled)Nt=8 (scaled)


17

•  Q follows Nt-‐scaling -  Q is temporal quark line,

analogous to the Polyakov loop

Nt=4 original

Nt=8 original

Nt=4, 8 scaled


10-6

10-4

10-2

100

102

104

106

0 1 2 3 4 5 6n/1000

|h| , |h|1/Nt

Nt=4Nt=8Nt=4 (scaled)Nt=8 (scaled)


18

•  The gap gives the minimum value of epsilon

•  Lajce setup β=1.86(same value of a) 8^3xNt, mps/mV=0.8, RG-‐improved gauge and clover-‐Wilson fermion with Nf=2


zero temperature limit

19


Fermion determinant •  Let us rewrite det Δ, using the proper5es of λ

20



21

�n $ 1/�⇤n



22

�n $ 1/�⇤n

analy5c in µ and T -‐>possible to take T=0 limit


Fermion determinant

23

•  Consider zero temperature limit


Fermion determinant

•  Inequality

24

•  Consider zero temperature limit

-  If µ < εmin , then all the µ-‐dependence vanishes.


Is εmin mπ/2 or MN/3?

25


Hadron masses in reduction formula

26

•  Correla5on func5on for charged meson

•  Small eigenvalues contribute to hadron correlators. •  low temperature limit (large Nt) -  m goes to the lightest mass. -  Q is dominated by the eigenvalue near the gap.

Phase vanishes


Hadron masses in reduction formula

27

•  Correla5on func5on for baryon

Phase survives


Numerical result

•  μ-‐dependence appears at μa = 0.5(= about mπ/2)

28

R =

✓det�(µ)

det�(0)

◆2

= |R|ei✓

0.0001 0.001

0.01 0.1

1 10

100 1000

0 0.2 0.4 0.6 0.8 1

<ln

|R|>

µa

T/Tpc=0.465T/Tpc=0.5T/Tpc=0.54T/Tpc=0.76T/Tpc=1T/Tpc=1.35 low T


Numerical result

29

10-810-710-610-510-410-310-210-1100101

0.4 0.5 0.6 0.7 0.8 0.9T/Tc

<|Q|2> (exact)<|Q|2> (LME)<Q3> (exact)<Q3> (LME)

Low Mode approxima5on(LME)


Summary •  Using the reduc5on formula, we consider the property of the fermion determinant at T=0.

–  fermion determinant is µ-‐independent at T=0 –  threshold for µ-independence is given by mπ/2

•  a lot of things to do – numerical confirma5on – why does the quark number keep to be zero up to 1/3 MN

30

lattice qcd at low temperature and ﬁnite density - early ... · silver blaze 6 • the effect of...

Documents