Dirac Operator Eigenvalues from Field Theory
e-Science Institute Edinburgh
8. March 2005
Gernot Akemann
Brunel University West London
collaborators:
P.H. Damgaard Phys. Lett. B 583 (2004) 199
Y. Fyodorov work in progress
Plan:
• motivation
/ eigenvalues λi
• what is known about D
• generating functionals:
– ρ densities and pk individual eigenvalues from FT
• explicit examples: –regime of chiral Perturbation Theory
– ρ, p1, p2, all?
• open problems
2
y
Motivation
/ eigenvalues are a sensitive tool to test
•D
(−→ Fig. 2 hep-th/9902117 [Edwards et al.])
– topology
– chiral symmetry breaking pattern
in a setting with exact or approximate Lattice chiral symmetry
• “easier” than density (but related)
(−→ Fig. 1 hep-th/9803007 [Nishigaki, Damgaard, Wettig])
– can be normalised
– ∃ integrated version −→ binning independent
• What can be said about them from Field Theory?
3
Setup: what is known about λi’s
R
ZQCD ≡ [dA]
Q Nf
/
f =1 det[iD(A)
− mf ] exp [−S[A]]
/ k = iλk ψk , k = 1, . . . , N finite #
Lattice regularisation: iDψ
−→ order λ1 < λ2 < . . . < λN
properties:
• ∃ ν λk = 0:
zeromodes f. given config (index Th.)
/ γ5} = 0 (continuum!)
• ρ(λ) = ρ(−λ): symmetry from {D,
• λ1 ≤ C/L ≤ free fields [Vafa,Witten 84]
assume χSB ⇒
• ρ(λ ≈ 0) =
[Banks,Casher 80]
• λ1 ≤ C̃/L4
1 4
πL Σ
+ |λ|Ĉ(Nf − 2)(Nf + b)/bNf
b = 1, 2, 4 for SU (2), QCD, adj.
+
[Stern, Smilga 93; Toublan, Verbaarschot 99]
ρs(ζ) ≡
1
ΣL4
ρ λ=
ζ
ΣL4
microscopic density
[Shuryak,Verbaarschot 93]
4
Density Correlations from Field Theory
Define:
density ρ(λ) ≡ h
PN
density – density ρ2(λ, η) ≡ h
k=1 δ(λ
− λk )iA
PN
k,n=1 δ(λ
− λk )δ(λ − λn)iA etc.
⇒ ASSUMES existence of joint probability distribution function
PN (λ1, . . . , λN ; A) symmetric
where ρk (λ1, . . . , λk ) ≡
R
dλk+1 . . . dλN PN
generating functional:
Z(Nf + 1|1) ∼
*
/
det[iD(A)−λ]
+
/
det[iD(A)−µ]
A
∂λ Z(Nf + 1|1)|λ=µ ∼
*
for the density
+
Tr 1
/
iD(A)−λ
A
≡ Σ(λ) for λ 6= λk
⇒ =m(Σ(iλ)) = ρ(λ)
in general: Z(Nf + k|k) ⇒ k-density
example: compute in –regime of chPT
/
(Alternative: Replicas det[iD(A)
− λ]n in the limit n → 0)
[Damgaard, Splittorff, Verbaarschot]
5
Individual eigenvalues from FT: how?
• all pk (λ) ⇒ ρ(λ) =
PN
k=1 pk (λ)
(−→ Fig. 1 hep-th/0006111 [Damgaard,Nishigaki ])
•⇐ ?
need all k-densities !
Define:
R
• Ek (s) ∼ 0s dλ1 . . . dλk
gap probability
R∞
s dλk+1 . . . dλN PN
R
R
• pk (s) ∼ 0s dλ1 . . . dλk-1 s∞ dλk+1 . . . dλN PN (. . . , λk = s, . . .)
k-th eigenvalue
• ∂sEk (s) ∼ (pk (s) − pk+1(s))
R
R
R n
trick: ( s∞)n = ( 0∞ − 0s) as a binomial (a − b)n
⇒
Ek (s) =
PN −k
l=0
(−)l R S
l! 0
dλ1 . . . dλk+l ρk+l (λ1, . . . , λk+l )
• pk (s) from all k-densities: have generating funct.
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Examples
∂
• p1(s) = − ∂s
E0(s) = ρ1 (s) −
first eigenvalue
• p2(s) =
Rs
0
Rs
0
dλ ρ2(λ, s) + . . .
dλ ρ2(λ, s) + . . .
second eigenvalue
• ⇒ we understand the relation p1(s) ←→ ρ(s)
• true ∀ gauge theories:
– any # of colours and reps.
– anywhere in the spectrum
• Convergence – need to know all k-densities ?
−→ pictures from –chPT
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Example: first eigenvalue for ν = 0, 1, 2 in QCD
ν=0
0.4
0.3
0.2
0.1
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
-0.1
ν=1
0.4
0.3
0.2
0.1
-0.1
ν=2
0.4
0.3
0.2
0.1
-0.1
• exact result from chiral Random Matrix Theory vs.
ρ1 and ρ2 term from –chPT: good convergence
8
Second eigenvalue for ν = 0 in QCD
ν=0
0.4
0.3
0.2
0.1
2
4
6
8
10
-0.1
• exact result from chiral Random Matrix Theory vs.
ρ2 term from –chPT
⇒ need more and more k-density correlations
9
Other theories
ν=0
0.5
SU (2)
0.4
0.3
0.2
0.1
2
4
6
8
10
2
4
6
8
10
ν=0
0.5
adjoint
0.4
0.3
0.2
0.1
• exact result vs.
ρ1 term, both from chiral Random Matrix Theory
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Chiral Perturbation Theory & -regime
R
R
ZχP T ≡ [dU (x)] exp[− TrLef f (U, ∂U )]
Lef f (U, ∂U ) =
Fπ2
†
4 ∂U (x)∂U (x)
− 21 ΣMf U (x) + U (x)† + . . .
parametrise
U ∈ SU (Nf ) for QCD


0 −1  T
U
for SU (2)
1 0


0
1

T

U
U
for SU (NC ) adj.
1 0
Mf = diag(mu, md, . . .)

U
• -regime : zeromodes U0 dominate U = U0 exp[ξ(x)]
Leutwyler 87] for ν fixed:
Z ≡
R
U (Nf ) dU0 det[U0 ]
ν
exp[− 12 ΣL4Tr
Mf (U0 +
U0†)
[Gasser,
] × free fields
• valid for 1/Λ V 1/4 1/mπ unphysical finite Volume limit
⇒ analytic solution for group integral [Brower et al.
11
81]
Generating partition function + bosons: all k
Z (Nf + k|k) ∼


where M = 
R
dU0 exp[− 21 ΣL4 STr M (U0 + U0−1 ) ]
Mf 0
0 Mb



and U0 ∈ Gl(Nf + k|k) for QCD
• Mf is an (Nf + k) × (Nf + k) matrix containing the fermion
masses
• Mb is an k × k matrix containing the bosons
• the group of the coset integral over the supersymmetric extension
of SU (Nf ) is fixed by convergence
• know for k = 1 and Nf massless as well as for k = 2 and Nf = 0
⇒ ρ1(x) = x2 (JNf +ν (x)2 − JNf +ν−1 (x)JNf +ν+1 (x))
[Damgaard, Osborn, Toublan, Verbaarschot 98]
• all k?: express in terms of larger group integral reducing to it in
a saddle point approximation (work in progress)
result (known): Z(Nf + k|k) ∼ det[{Iν (xf ), Kν (xb)}]/∆(xf )∆(xb)
⇒ generates all k-densities
= Random Matrix Theory !
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[Fyodorov, G.A. 03]
Open problems
∗ group integrals for other χSB classes: SU (2) and adjoint
(b = 1, 4)
– only perturbative map between ρ1(λ) from RMT and -chPT
– difficult coset integral Gl /OSp
– Can improved staggered fermions for (b = 1, 4) see the continuum symmetry?
∗ QCD with chemical potential:
– ∃ results for Z and for ρ quenched from -chPT
−→ pK for complex eigenvalues ??
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