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0.5setgray0
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
Non-Hermitian RMT applied to QCD – p.2/35
RMT for QCD
Non-Hermitian RMT applied to QCD – p.3/35
The QCD partition function
Path integral
ZQCD =
Z
dA det(iDηγη)Nf e
−SYM(A)
The Dirac operator iDηγη will be the random matrix.
Non-Hermitian RMT applied to QCD – p.4/35
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.
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.
Non-Hermitian RMT applied to QCD – p.5/35
The QCD partition function
ZQCD =
Z
dA det(iDηγη)Nf e
−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
ONon-Hermitian RMT applied to QCD – p.6/35
The QCD partition function
ZQCD =
Z
dA det(iDηγη)Nf e
−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
Non-Hermitian RMT applied to QCD – p.6/35
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
ONon-Hermitian RMT applied to QCD – p.7/35
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)
ONon-Hermitian RMT applied to QCD – p.8/35
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)
ONon-Hermitian RMT applied to QCD – p.8/35
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)
Non-Hermitian RMT applied to QCD – p.8/35
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
Same explicit symmetry breaking of flavor symmetriesSame hermiticity properties as in QCD
Eigenvalue representation known
Osborn PRL 93 (2004) 222001
ONon-Hermitian RMT applied to QCD – p.9/35
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
Same explicit symmetry breaking of flavor symmetriesSame hermiticity properties as in QCD
Eigenvalue representation known
Osborn PRL 93 (2004) 222001
ONon-Hermitian RMT applied to QCD – p.9/35
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
Same explicit symmetry breaking of flavor symmetriesSame hermiticity properties as in QCD
Eigenvalue representation known
Osborn PRL 93 (2004) 222001
Non-Hermitian RMT applied to QCD – p.9/35
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
ONon-Hermitian RMT applied to QCD – p.10/35
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
ONon-Hermitian RMT applied to QCD – p.10/35
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 positiveOsborn PRL 93 (2004) 222001
Non-Hermitian RMT applied to QCD – p.10/35
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)
ONon-Hermitian RMT applied to QCD – p.11/35
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)
Non-Hermitian RMT applied to QCD – p.11/35
Sign problem ⇒ ρ complex valued
What is this good for ?
ONon-Hermitian RMT applied to QCD – p.12/35
Sign problem ⇒ ρ complex valued
What is this good for ?
Non-Hermitian RMT applied to QCD – p.12/35
The Silver Blaze Problem
Sir Arthur Conan Doyle The Memoirs of Sherlock Holmes: Silver Blaze
Thomas D . Cohen PRL (2003) 222001
Non-Hermitian RMT applied to QCD – p.13/35
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
Non-Hermitian RMT applied to QCD – p.14/35
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
Non-Hermitian RMT applied to QCD – p.15/35
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
Non-Hermitian RMT applied to QCD – p.17/35
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
Non-Hermitian RMT applied to QCD – p.19/35
The unquenched eigenvalue density
Structure: ρNf=1 = ρQ + ρU
Non-Hermitian RMT applied to QCD – p.20/35
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)
Non-Hermitian RMT applied to QCD – p.21/35
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
Non-Hermitian RMT applied to QCD – p.22/35
Observation: The complex oscillations of the spectralcorrelation functions take part on the microscopic scale:period ∼ 1/V amplitude ∼ exp(V )
Non-Hermitian RMT applied to QCD – p.23/35
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
ONon-Hermitian RMT applied to QCD – p.24/35
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
Non-Hermitian RMT applied to QCD – p.24/35
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
ONon-Hermitian RMT applied to QCD – p.25/35
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
ONon-Hermitian RMT applied to QCD – p.25/35
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
Non-Hermitian RMT applied to QCD – p.25/35
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
Osborn Splittorff Verbaarschot arXiv:0805.1303
Non-Hermitian RMT applied to QCD – p.26/35
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 finite N from the orthogonal polynomials
ρNf =1N = 2w(z, z∗;µ)
N−1∑
k=0
pk(z∗)(pk(z) − pN (z)pk(m)/pN (m))
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
ONon-Hermitian RMT applied to QCD – p.27/35
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 finite N from the orthogonal polynomials
ρNf =1N = 2w(z, z∗;µ)
N−1∑
k=0
pk(z∗)(pk(z) − pN (z)pk(m)/pN (m))
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
ONon-Hermitian RMT applied to QCD – p.27/35
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 finite N from the orthogonal polynomials
ρNf =1N = 2w(z, z∗;µ)
N−1∑
k=0
pk(z∗)(pk(z) − pN (z)pk(m)/pN (m))
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
Non-Hermitian RMT applied to QCD – p.27/35
OSV relation at finite N
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
Osborn Splittorff Verbaarschot arXiv:0805.1303
Non-Hermitian RMT applied to QCD – p.28/35
Alternative way to compute the spectral density
Non-Hermitian RMT applied to QCD – p.29/35
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
Non-Hermitian RMT applied to QCD – p.30/35
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
Akemann Osborn Splittorff Verbaarschot NPB 712 (2005) 287
ONon-Hermitian RMT applied to QCD – p.31/35
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
Akemann Osborn Splittorff Verbaarschot NPB 712 (2005) 287
Non-Hermitian RMT applied to QCD – p.31/35
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
ONon-Hermitian RMT applied to QCD – p.32/35
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
Non-Hermitian RMT applied to QCD – p.32/35
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
ONon-Hermitian RMT applied to QCD – p.33/35
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
Non-Hermitian RMT applied to QCD – p.33/35
Additional slides
Non-Hermitian RMT applied to QCD – p.34/35
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)
ONon-Hermitian RMT applied to QCD – p.35/35
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)
ONon-Hermitian RMT applied to QCD – p.35/35
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 QCD – p.35/35
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