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0.5setgray1 FRIDAY:

The Banks Casher RelationKim Splittorff

What The QCD Phase Diagram & Dirac Spectra

Activity The Banks Casher relation (µ = 0)

What The Banks Casher relation (µ 6= 0)

The Sign Problem

Why Understand unquenched lattice QCD

FRIDAY:The Banks Casher Relation – p.1/37

QCD Phase diagram & Dirac Spectra

T

T

Chiral symmetry broken

Pions Protons Neutrons ...

Chirally symmetricQuarks Gluons

c

QCD

µ

1st oder

FRIDAY:The Banks Casher Relation – p.2/37

The sign problema problem for lattice QCD

ZNf=2 =

∫

dA det2(iD + µγ0 +m) e−SYM

Anti Hermitian Hermitian

det2(iDηγη + µγ0 +m) = |det(iDηγη + µγ0 +m)|2e2iθ

The measure is not real and positive

FRIDAY:The Banks Casher Relation – p.3/37

Yesterday: Why don’t we just quench the Determinant ?T

µm /2 m /3π N

Bose Einstein phase transition inQuenched QCD

Today: What is the effect of the Determinant ?How difficult is it to include it ?

OFRIDAY:The Banks Casher Relation – p.4/37

Yesterday: Why don’t we just quench the Determinant ?T

µm /2 m /3π N

Bose Einstein phase transition inQuenched QCD

Today: What is the effect of the Determinant ?How difficult is it to include it ?

FRIDAY:The Banks Casher Relation – p.4/37

Activity: Banks Casher relation at µ = 0

FRIDAY:The Banks Casher Relation – p.5/37

What is the effect of the Determinant ?

FRIDAY:The Banks Casher Relation – p.6/37

Quark mass & the eigenvalue distribution

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Re(λ)

Im(λ)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Re(λ)

Im(λ)

X

quark mass

inside

X

quark mass

outside

µ < mπ/2 µ > mπ/2

(iDηγη + µγ0)ψk = zkψk det(iDηγη + µγ0 +m) =∏

k

λk +m

Bloch Wettig Lattice 2006 Gibbs PRINT-86-0389

Davies Klepfish PLB 256 (1991) 68 Lombardo Kogut Sinclair PRD 54 (1996) 2303

FRIDAY:The Banks Casher Relation – p.7/37

Electrostatic analogy suggests

Im(z)

Re(z)X

X

X

X

X

X

X

X

X

X

m

< >=0

< >(m)

ψ ψ

ψ ψ_

_

FRIDAY:The Banks Casher Relation – p.8/37

The big pictureT

µm /2 m /3π N

inside eigenvaluesQuark mass

NOT a phase transition in ZNf

Chiral condensate is independent of µ at T = 0 and µ < mN/3

OFRIDAY:The Banks Casher Relation – p.9/37

The big pictureT

µm /2 m /3π N

inside eigenvaluesQuark mass

NOT a phase transition in ZNf

Chiral condensate is independent of µ at T = 0 and µ < mN/3

FRIDAY:The Banks Casher Relation – p.9/37

The Silver Blaze Problem

Sir Arthur Conan Doyle The Memoirs of Sherlock Holmes: Silver Blaze

Thomas D . Cohen PRL (2003) 222001

FRIDAY:The Banks Casher Relation – p.10/37

µ = 0 Banks Casher

X

X

X

X

X

X

X

X

Re(z)

Im(z)

< >

< >(m)

ψ ψ

ψ ψ_

_

m

〈ψψ〉 =π

Vρ(0)

Banks Casher NPB 169 (1980) 103

FRIDAY:The Banks Casher Relation – p.11/37

µ 6= 0 The silver blaze problemIm(z)

Re(z)X

X

X

X

X

X

X

X

X

X

< >

< >(m)

ψ ψ

ψ ψ_

_

m

Eigenvalues move into the complex planethe discontinuity of the chiral condensate remains

Barbour et al. NPB 275 (1986) 296Gibbs PLB 182 (1986) 369

Cohen PRL 91 (2003) 222001FRIDAY:The Banks Casher Relation – p.12/37

We need Microscopic-regime of QCDThe eigenvalue density z〈ψψ〉 1√

V

SBχS The basic assumption

Chiral limit m〈ψψ〉 1√V

Small chemical potential µ2F 2π 1√

V

Notice µ ∼ mπ

Gasser, Leutwyler, PLB 184 (1987) 83, PLB 188 (1987) 477Neuberger, PRL 60 (1988) 889

Leutwyler, Smilga, PRD 46 (1992) 5607Shuryak, Verbaarschot, NPA 560 (1993) 306

Stephanov PRL 76 (1996) 4472Akemann PRL 89 (2002) 072002, J.Phys. A36 (2003) 3363

Splittorff, Verbaarschot, NPB 683 (2004) 467Osborn PRL 93 (2004) 222001

Akemann Osborn Splittorff Verbaarschot NPB 712 (2005) 287

FRIDAY:The Banks Casher Relation – p.13/37

The unquenched eigenvalue densitym〈ψψ〉V = 100 increasing 2µ2F 2

πV

-1000100

-100

-50

0

50

100

-0.001

0

0.001

0.002

0.003

0.004

-1000100

-100

-50

0

50

100

-1000100

-100

-50

0

50

100

-0.001

0

0.001

0.002

0.003

0.004

-1000100

-100

-50

0

50

100

y〈ψψ〉V

Re[ρNf =1(x,y,m;µ)]

〈ψψ〉2V 2

2µ < mπ 2µ > mπ

x〈ψψ〉Vx〈ψψ〉V

For 2µ > mπ the density is complex and oscillatesOsborn PRL 93 (2004) 222001

Akemann Osborn Splittorff Verbaarschot NPB 712 (2005) 287FRIDAY:The Banks Casher Relation – p.14/37

Definition of the eigenvalue density

Eigenvalue equation

(iDηγη + µγ0)ψj = zjψj

Eigenvalue density

ρNf (z, z∗,m;µ) ≡

*

X

j

δ2(z − zj)

+

QCD

〈O〉QCD ≡

R

dA O det(iDηγη + µγ0 +m)Nf e−SYM(A)

R

dA det(iDηγη + µγ0 +mf )Nf e−SYM(A)

OFRIDAY:The Banks Casher Relation – p.15/37

Definition of the eigenvalue density

Eigenvalue equation

(iDηγη + µγ0)ψj = zjψj

Eigenvalue density

ρNf (z, z∗,m;µ) ≡

*

X

j

δ2(z − zj)

+

QCD

〈O〉QCD ≡

R

dA O det(iDηγη + µγ0 +m)Nf e−SYM(A)

R

dA det(iDηγη + µγ0 +mf )Nf e−SYM(A)

FRIDAY:The Banks Casher Relation – p.15/37

The unquenched eigenvalue densitym〈ψψ〉V = 60 and 2µ2F 2

πV = 124

50 100 150-100

-50

0

50

100

-0.002-0.001

00.001

0.002-100

-50

0

50

100

Oscillations: Period ∼ 1/V

Amplitude ∼ e+V

Osborn PRL 93 (2004) 222001

Akemann Osborn Splittorff Verbaarschot NPB 712 (2005) 287

FRIDAY:The Banks Casher Relation – p.16/37

The chiral condensate from the eigenvalue density

〈ψψ〉(m) =1

V∂m logZ(m;µ)

=1

V

∫

dxdy ρ(x, y)1

x+ iy +m

The oscillations of the density areresponsible for chiral symmetry breaking

Osborn Splittorff Verbaarschot PRL 94 (2005) 202001

Ravagli Verbaarschot arXiv:0704.1111

FRIDAY:The Banks Casher Relation – p.17/37

The unquenched eigenvalue density

Structure: ρNf=1 = ρQ + ρU

Osborn, Splittorff, Verbaarschot hep-lat/0510118

FRIDAY:The Banks Casher Relation – p.18/37

The unquenched chiral condensate

−160 −120 −80 −40 0 40 80 120 160

−1.2

−0.9

−0.6

−0.3

0

0.3

0.6

0.9

1.2

Σ(m

ΣV)

−160 −120 −80 −40 0 40 80 120 160

−1.2

−0.9

−0.6

−0.3

0

0.3

0.6

0.9

1.2

Σ Q(m

ΣV,µ

FV

1/2 )

−160 −120 −80 −40 0 40 80 120 160mΣV

−1.2

−0.9

−0.6

−0.3

0

0.3

0.6

0.9

1.2

Σ U(m

ΣV,µ

FV

1/2 )

Structure: ΣNf=1(m) = ΣQ(m) + ΣU (m)

Splittorff hep-lat/0610072

FRIDAY:The Banks Casher Relation – p.19/37

Banks-Casher µ = 0

Accumulation of eigenvalues on the y-axis isresponsible for chiral symmetry breaking

New mechanism µ 6= 0

The oscillations of the eigenvalue density areresponsible for chiral symmetry breaking

FRIDAY:The Banks Casher Relation – p.20/37

How to calculate the eigenvalue density

FRIDAY:The Banks Casher Relation – p.21/37

The replica way of writing the eigenvalue density

ρNf (z, z∗,m;µ) = limn→0

1

πn∂z∗∂z logZNf ,n(m, z, z∗;µ)

generating functionals for the eigenvalue density

ZNf ,n(m, z, z∗;µ) =∫

dA det(iDηγη + µγ0 +m)Nf | det(iDηγη + µγ0 + z)|2n e−SYM(A)

Stephanov PRL 76 (1996) 4472

FRIDAY:The Banks Casher Relation – p.22/37

Central observation

The eigenvalue z and its complex conjugate z∗ appears as the mass of

two conjugate fermions.

very small eigenvalues ↔ very light quarks

⇒ Compton wavelength of the pions boxsize

⇒ Zero mode of the pions dominates

ZNf ,n =

∫

U(Nf+2n)

dUe−V4F 2πµ

2Tr[U,B][U−1,B] + 12m〈ψψ〉V Tr(U+U−1)

OFRIDAY:The Banks Casher Relation – p.23/37

Central observation

The eigenvalue z and its complex conjugate z∗ appears as the mass of

two conjugate fermions.

very small eigenvalues ↔ very light quarks

⇒ Compton wavelength of the pions boxsize

⇒ Zero mode of the pions dominates

ZNf ,n =

∫

U(Nf+2n)

dUe−V4F 2πµ

2Tr[U,B][U−1,B] + 12m〈ψψ〉V Tr(U+U−1)

OFRIDAY:The Banks Casher Relation – p.23/37

Central observation

The eigenvalue z and its complex conjugate z∗ appears as the mass of

two conjugate fermions.

very small eigenvalues ↔ very light quarks

⇒ Compton wavelength of the pions boxsize

⇒ Zero mode of the pions dominates

ZNf ,n =

∫

U(Nf+2n)

dUe−V4F 2πµ

2Tr[U,B][U−1,B] + 12m〈ψψ〉V Tr(U+U−1)

FRIDAY:The Banks Casher Relation – p.23/37

The replica way of writing the eigenvalue density

ρNf (z, z∗,m;µ) = limn→0

1

πn∂z∗∂z logZNf ,n(m, z, z∗;µ)

generating functionals for the eigenvalue density

ZNf ,n(m, z, z∗;µ) =∫

dA det(iDηγη + µγ0 +m)Nf | det(iDηγη + µγ0 + z)|2n e−SYM(A)

Stephanov PRL 76 (1996) 4472

FRIDAY:The Banks Casher Relation – p.24/37

The Replica Limit of the Toda Lattice Equation

∂z∂z∗ logZNf ,n = 4zz∗nZNf ,n+1ZNf ,n−1

[ZNf ,n]2

Take n→ 0 in this equation

ρNf(z, z∗,m;µ) = 4zz∗

ZNf ,n=1(m, z, z∗;µ)ZNf ,n=−1(m|z, z∗;µ)

[ZNf(m;µ)]2

ProblemsVerbaarschot, Zirnbauer, J. Phys. A 18, 1093 (1985)

Kamenev Mézard J.Phys.A 32 4373 (1999); PRB 60 3944 (1999)Yurkevich, Lerner, PRB 60, 3955 (1999)

M.R. Zirnbauer, cond-mat/9903338Solution

Kanzieper, PRL 89, 250201 (2002)

Splittorff, Verbaarschot, PRL 90, 041601 (2003)FRIDAY:The Banks Casher Relation – p.25/37

Today: What is the effect of the Determinant ? (DONE)How difficult is it to include it ? (NOW)

FRIDAY:The Banks Casher Relation – p.26/37

The Big Picture

T

µm /3N

B Sχ

m /2π

Mon

te C

arlo

No Monte Carlo

Lattice Pioneers

FRIDAY:The Banks Casher Relation – p.27/37

The lattice Pioneers

4.84.824.844.864.884.9

4.924.944.964.98

55.025.045.06

0 0.5 1 1.5 2

1.0

0.95

0.90

0.85

0.80

0.75

0.70

0 0.1 0.2 0.3 0.4 0.5

β

T/T

c

µ/T

a µ

confined

QGP<sign> ~ 0.85(1)

<sign> ~ 0.45(5)

<sign> ~ 0.1(1)

D’Elia, Lombardo 163

Azcoiti et al., 83

Fodor, Katz, 63

Our reweighting, 63

This work, 63

de Forcrand Philipsen JHEP PoS LAT2005 (2005) 016

FRIDAY:The Banks Casher Relation – p.28/37

The average phase factor

〈sign〉 = 〈e2iθ〉 ≡

⟨

det(D + µγ0 +m)

det(D + µγ0 +m)∗

⟩

is a ratio of two partition functions

〈e2iθ〉1+1∗ =Z1+1

Z1+1∗

FRIDAY:The Banks Casher Relation – p.29/37

The average phase factor in the microscopic limit

〈e2iθ〉1+1∗ =Z1+1

Z1+1∗

=I0(m)2 − I1(m)2

2e2µ2∫ 1

0dtte−2µ2t2I0(mt)2

0 0.5 1 1.52µ/mπ

0

0.2

0.4

0.6

0.8

1

<ex

p(2i

θ)>

1+1*

mΣV = 1 mΣV >> 1

Splittorff Verbaarschot PRL 98 (2007) 031601FRIDAY:The Banks Casher Relation – p.30/37

The average phase factor

〈sign〉 = 〈e2iθ〉 ≡

⟨

det(D + µγ0 +m)

det(D + µγ0 +m)∗

⟩

is a ratio of two partition functions

〈e2iθ〉1+1∗ =Z1+1

Z1+1∗= e−V∆Ω

Phase transition at µ = mπ/2

FRIDAY:The Banks Casher Relation – p.31/37

The big pictureT

µm /2 m /3π N

inside eigenvaluesQuark mass

Phase transition in Z1+1∗

FRIDAY:The Banks Casher Relation – p.32/37

Lattice measurements of the endpoint

0 0.5 1 1.5 22µ/mπ

0

0.2

0.4

0.6

0.8

1

1.2

T/T

0

Fodor Katz JHEP 0203 (2002) 014, JHEP 0404 (2004) 050

Splittorff PoS LAT2006:023

FRIDAY:The Banks Casher Relation – p.33/37

The current questionT

µm /2 m /3π N

inside eigenvaluesQuark mass

Sign problem ?

FRIDAY:The Banks Casher Relation – p.34/37

Summary

FRIDAY:The Banks Casher Relation – p.35/37

Equivalence chRMT and QCD through low energy effective theory

Mean field chRMT & QCD Phase Diagram

Microscopic spectral correlation functions

The sign problem

OFRIDAY:The Banks Casher Relation – p.36/37

Equivalence chRMT and QCD through low energy effective theory

Mean field chRMT & QCD Phase Diagram

Microscopic spectral correlation functions

The sign problem

OFRIDAY:The Banks Casher Relation – p.36/37

Equivalence chRMT and QCD through low energy effective theory

Mean field chRMT & QCD Phase Diagram

Microscopic spectral correlation functions

The sign problem

OFRIDAY:The Banks Casher Relation – p.36/37

Equivalence chRMT and QCD through low energy effective theory

Mean field chRMT & QCD Phase Diagram

Microscopic spectral correlation functions

The sign problem

FRIDAY:The Banks Casher Relation – p.36/37

Thanks !!

FRIDAY:The Banks Casher Relation – p.37/37