Non-Hermitian RMT applied to · ONon-Hermitian RMT applied to QCD – p.6/35. The QCD partition...

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0.5setgray1 Non-Hermitian RMT applied to

QCDKim Splittorff

with: Jac Verbaarschot

Martin Zirnbauer

James Osborne

Gernot Akemann

Niels Bohr Institute

CRM, Montreal, August 30, 2008

Setup RMT for QCD

Feature Chemical potential and Non-Hermiticity

Focus Chiral symmetry breaking

Key-words Orthogonal polynomials in the complex planeSign problemNon-positive densitiesToda lattice equationsCauchy transformsBosonic theories

Non-Hermitian RMT applied to QCD – p.1/35

The Big Picture

T

µm /3N

B Sχ

m /2π


RMT for QCD


The QCD partition function

Path integral

ZQCD =

Z

dA det(iDηγη)Nf e

−SYM(A)

The Dirac operator iDηγη will be the random matrix.


Properties of the QCD Dirac operator

Anti-Hermitian

(iDηγη)† = −iDηγη

Axial-Symmetry

{iDηγη, γ5} = 0

No additional symmetries

(iDηγη) =

0

@

0 iW

iW † 0

1

A

W has complex matrix elements.

ONon-Hermitian RMT applied to QCD – p.5/35

Properties of the QCD Dirac operator

Anti-Hermitian

(iDηγη)† = −iDηγη

Axial-Symmetry

{iDηγη, γ5} = 0

No additional symmetries

(iDηγη) =

0

@

0 iW

iW † 0

1

A

W has complex matrix elements.



ZQCD =

Z


−SYM(A)

The chGUE partition function

Z ≡

Z

dW det(D)Nf e−NTrW †W

where

D =

0

@

0 iW

iW † 0

1

A

same flavor symmetriesShuryak, Verbaarschot, NPA 560, 306 (1993), Verbaarschot, PRL 72, 2531 (1994)

Selvin, Nagao, PRL 70 (1993) 635, Zirnbauer, J. Math. Phys. 37 (1996) 4986


The quark mass m and chemical potential µ

ZQCD =

∫

dA detNf (iDηγη + µγ0 +m) e−SYM

Anti Hermitian Hermitian

The sign problem: The measure is not real and positive

A problem for Monte Carlo


The quark mass m and chemical potential µ

ZQCD =

∫

dA detNf (iDηγη + µγ0 +m) e−SYM

Anti Hermitian Hermitian

The sign problem: The measure is not real and positive

A problem for Monte CarloNon-Hermitian RMT applied to QCD – p.7/35

chRMT by Stephanov

Z ≡

Z

dW det(D(m;µ))Nf e− N

2TrW †W

where

D(m;µ) =

0

@

m iW + µ

iW † + µ m

1

A

Same explicit symmetry breaking of flavor symmetriesSame hermiticity properties as in QCD

No eigenvalue representation known

Stephanov, PRL 76, 4472 (1996)


chRMT by Stephanov

Z ≡

Z

dW det(D(m;µ))Nf e− N

2TrW †W

where

D(m;µ) =

0

@

m iW + µ

iW † + µ m

1

A


No eigenvalue representation known

Stephanov, PRL 76, 4472 (1996)


chRMT by Osborn

ZNf

N (m;µ) ≡

∫

dWdΨdetNf (D(µ) +m )e−NTrW †W e−NTrΨ†Ψ

where the Dirac operator is given by

D(µ) =

0

@

0 iW + µΨ

iW † + µΨ† 0

1

A


Eigenvalue representation known

Osborn PRL 93 (2004) 222001


chRMT by Osborn

Eigenvalue representation

ZNf

N(m;µ) =

Z NY

k=1

d2zk˛

˛∆N ({z2l })˛

˛

2|zk|

2ν+2

×Kν

„

N(1 + µ2)

2µ2|zk|

2

«

e−

N(1−µ2)

4µ2 (z2k+zk∗2)

(m2 − z2k)Nf

The eigenvalue density from the orthogonal polynomials

ρNf =1

N= 2w(z, z∗;µ)

N−1X

k=0

pk(z∗)(pk(z) − pN (z)pk(m)/pN (m))

rk

where

pk(z;µ) =

„

1 − µ2

N

«k

k!Lνk

„

−Nz2

1 − µ2

«

G. Akemann Int.J.Mod.Phys. A22 (2007) 1077

Not real and positive

Osborn PRL 93 (2004) 222001


chRMT by Osborn

Eigenvalue representation

ZNf

N(m;µ) =

Z NY

k=1

d2zk˛

˛∆N ({z2l })˛

˛

2|zk|

2ν+2

×Kν

„

N(1 + µ2)

2µ2|zk|

2

«

e−

N(1−µ2)

4µ2 (z2k+zk∗2)

(m2 − z2k)Nf

The eigenvalue density from the orthogonal polynomials

ρNf =1

N= 2w(z, z∗;µ)

N−1X

k=0


rk

where

pk(z;µ) =

„

1 − µ2

N

«k

k!Lνk

„

−Nz2

1 − µ2

«

G. Akemann Int.J.Mod.Phys. A22 (2007) 1077

Not real and positiveOsborn PRL 93 (2004) 222001


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)


Sign problem ⇒ ρ complex valued

What is this good for ?


The Silver Blaze Problem

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

Thomas D . Cohen PRL (2003) 222001


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


Electrostatic analogy suggests

Im(z)

Re(z)X

X

X

X

X

X

X

X

X

X

m

< >=0

< >(m)

ψ ψ

ψ ψ_

_

Electrostatic analogy:Eigenvalues = charges, quark mass = test charge

Barbour et al. NPB 275 (1986) 296


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) 222001Non-Hermitian RMT applied to QCD – p.16/35

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


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) 287Non-Hermitian RMT applied to QCD – p.18/35

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


The unquenched eigenvalue density

Structure: ρNf=1 = ρQ + ρU


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: 〈ψψ〉Nf=1(m) = 〈ψψ〉Q(m) + 〈ψψ〉U (m)


Banks-Casher µ = 0

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

OSV µ 6= 0

The oscillations of the eigenvalue density areresponsible for chiral symmetry breaking


Observation: The complex oscillations of the spectralcorrelation functions take part on the microscopic scale:period ∼ 1/V amplitude ∼ exp(V )


Exact microscopic result

Exact microscopic partition function and condensate

ZNf=1(m) = Iν(m) 〈ψψ〉Nf =1(m) =I ′ν(m)

Iν(m)

〈ψψ〉Nf(m) =

∫

dxdyρNf

(x, y)

x+ iy −m

Exact microscopic eigenvalue density

ρ(ν)Nf =1(z, z∗) =

|z|2

2πµ2Kν

„

|z|2

4µ2

«

e− z2+z∗ 2

8µ2

×

„Z 1

0dt t e−2µ2t2Iν(zt)Iν(z

∗t) −Iν(z)

Iν(m)

Z 1

0dt t e−2µ2t2Iν(mt)Iν(z

∗t)

«

Osborn Splittorff Verbaarschot arXiv:0805.1303


Exact microscopic result

Exact microscopic partition function and condensate

ZNf=1(m) = Iν(m) 〈ψψ〉Nf =1(m) =I ′ν(m)

Iν(m)

〈ψψ〉Nf(m) =

∫

dxdyρNf

(x, y)

x+ iy −m

Exact microscopic eigenvalue density

ρ(ν)Nf =1(z, z∗) =

|z|2

2πµ2Kν

„

|z|2

4µ2

«

e− z2+z∗ 2

8µ2

×

„Z 1

0dt t e−2µ2t2Iν(zt)Iν(z

∗t) −Iν(z)

Iν(m)

Z 1

0dt t e−2µ2t2Iν(mt)Iν(z

∗t)

«



Chiral Random Matrix Theory

The partition function - eigenvalue representation

ZNf

N(m;µ) =

Z NY

k=1

d2zk˛

˛∆N ({z2l })˛

˛

2|zk|

2ν+2

×Kν

„

N(1 + µ2)

2µ2|zk|

2

«

e−

N(1−µ2)

4µ2 (z2k+zk∗2)

(m2 − z2k)Nf

The complex orthogonal polynomial method

ZNf =1

N(m;µ) = mνpN (m;µ)

pk(z;µ) =

„

1 − µ2

N

«k

k!Lνk

„

−Nz2

1 − µ2

«

Osborn PRL 93 (2004) 222001


OSV relation at finite N

ZNf =1N (m) = mνpN (m) 〈ψψ〉

Nf =1N (m) =

dpN (m)/dm

pN (m)+ν

m

The eigenvalue density at N = 20




ZNf=1N (m) = mνpN (m) 〈ψψ〉

Nf =1N (m) =

dpN (m)/dm

pN (m)+ν

m

The eigenvalue density at finite N from the orthogonal polynomials

ρNf =1N = 2w(z, z∗;µ)

N−1∑

k=0


rk

The integral

〈ψψ〉Nf =1N (m) =

∫

dxdyρ

Nf=1N (x, y)

x+ iy +m

Osborn Splittorff Verbaarschot arXiv:0805.1303, to appear in PRD



The integral

〈ψψ〉Nf =1N (m) =

1

V

∫

dxdyρ

Nf =1N (x, y)

x+ iy +m=dpN (m)/dm

pN (m)+ν

m

can be done using the orthogonality of the polynomials

∫

C

d2z w(z, z∗;µ) pk(z;µ) pl(z;µ)∗ = δkl rνk



Alternative way to compute the spectral density


The replica method

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

Girko Theory of Random Determinants


The Toda Lattice Equation

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

[ZNf ,n]2

The Replica Limit (n→ 0) of the Toda lattice equation n→ 0

ρNf(z, z∗,m;µ) = 4zz∗

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

[ZNf(m;µ)]2

Verbaarschot, 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/9903338Kanzieper, PRL 89, 250201 (2002)

Splittorff, Verbaarschot, PRL 90, 041601 (2003)Splittorff, Verbaarschot, Nucl.Phys. B683 (2004) 467



Bosonic quarks = average inverse determinants

ZNf=−1 =

⟨

1

det(D + µγ0 +m)

⟩

From Cauchy transform of orthogonal polynomials

ZNf =−1 = −1

rN−1m−ν

Z

d2zw(z, z∗;µ)pN−1(z)∗1

z2 −m2

From σ-model: Because of convergence requirements

ZNf =−1 =

*

det(D + µγ0 +m)∗

det

0

@

ε D + µγ0 +m

(D + µγ0 +m)∗ ε

1

A

+

Akemann, Pottier, J.Phys. A37 (2004) 453Bergère arXiv:hep-th/0404126

Feinberg Zee NPB 504 (1997) 578Splittorff, Verbaarschot Nucl.Phys. B757 (2006) 259

Splittorff, Verbaarschot, Zirnbauer arXiv 0802.2660


Conclusions

Eigenvalue density of Non Hermitian chRMT is complex valued

Chiral symmetry breaking linked to oscillations at the microscopic scale

Shows the numerical difficulties in dealing with the sign problem

At finite N cancellations are due to orthogonality of the polynomials

Replica limit of the Toda Lattice equation as alternative to OP

Averages of inverse determinants of non hermitian operators


Additional slides


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)


Non-Hermitian RMT applied to · ONon-Hermitian RMT applied to QCD – p.6/35. The QCD partition...

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