Dirac Spectra in QCD - cgc.physics.miami.edu

Dirac Spectra in QCD

Jacobus Verbaarschot

[email protected]

Stony Brook University

Miami 2013, December 2013

Dirac Spectra – p. 1/40

Acknowledgments

Collaborators: Gernot Akemann (Bielefeld)Poul Damgaard (NBIA)Mario Kieburg (Bielefeld)Kim Splittorff (NBI)Savvas Zafeiropoulos (Stony Brook)


Relevant Papers

J. J. M. Verbaarschot, QCD, Chiral Random Matrix Theory and Integrability, hep-th/0502029.

J. J. M. Verbaarschot and T. Wettig, Random Matrix Theory and Chiral Symmetry in QCD,hep-ph/0003017

M. Stephanov, J. J. M. Verbaarschot and T. Wettig, Random Matrices, hep-ph/0509286.

M. Kieburg, K. Splittorff and J. J. M. Verbaarschot, QCD Dirac Spectra in Two Dimensions, arXiv

(2013).


Contents

I. Introduction

II. Chiral Symmetry and Dirac Spectra

III. Chiral Random Matrix Theory

IV. Dirac Spectra in Two Dimensions

V. Conformal Dirac Spectra

VI. Conclusions


I. Introduction

Dirac Spectra in QCD

Free Dirac Spectra

Banks-Casher Relation


QCD Partition Function and Dirac Eigenvalues

Euclidean QCD partition function

ZQCD = 〈det(D + m + µγ0)〉YM = 〈∏

k

(iλk + m)〉YM.

Dirac operator

D = γµ(dµ + iAµ).

Eigenvalues Dφk = λkφk.

Axial symmetry {γ5, D} = 0.

Eigenvalues are zero or occur in pairs ±λ .

Dirac eigenvalues are gauge invariant and the eigenvalue number isrenormalization group invariant. Giusti-Luscher-2008


The Free Dirac Spectrum

The eigenvalues of the free Dirac operator are given by

λnk= ±(

d∑

k=1

π2n2k

L2)1/2.

The total number of eigenvalues < λ is equal to

N(λ) ∼(

λL

π

)d

.

Then the eigenvalue density is

ρ(λ) =dN(λ)

dλ∼ V λd−1.


Free Dirac Spectrum

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Λ0

1

2

3

4

5ΡHΛL

d = 4

d = 2

The free Dirac spectrum in 2 and 4 dimensions for N = 1000 and 100,respectively.


Chiral Condensate

Chiral condensate:

Σ(m) ≡ −|〈q̄q〉| =1

V∂m log Z =

1

V

∑

k

1

m + λk.

m

l

−m

I

dsΣ(s) = il(Σ(m) − Σ(−m))

= 2πilρ(0)/V

eigenvalue density

Σ(m) =πρ(0)

V. Banks − Casher formula

Spacing of the eigenvalues: ∆λ = 1ρ(0) = π

ΣV .

Experimentally: 〈q̄q〉 = (−230MeV )3


Picture of the Dirac Spectrum

Zero Modes

λ

V Σπ

∼ V λ3

ρ( )λ

� Because of asymptotic freedom, the Dirac spectrum shouldapproximate the free one for λ ≫ ΛQCD .

� What is the origin of the small eigenvalues?


Instanton Picture of Dirac Spectrum

� Instantons have an exact zero mode.

� A band of low-lying modes is obtained from weakly interactinginstantons and anti-instantons.

� For a finite topological susceptibility, the number of these modesscale with the volume.


Disorder Picture of the Dirac Spectrum

� In the presence of gauge fields the eigenmodes of the free Diracoperator become coupled resulting in a repulsing of theeigenvalues.

� The eigenvalues move to the place where there are noeigenvalues, i.e. to zero.

� Since the free density is proportional to the volume, we expect thatthe interacting density is also proportional to the volume.


Microscopic Spectral Density

Since the smallest eigenvalue scale as 1/V we can can define themicroscopic spectral density Shuryak-JV-1993, JV-1994

ρS(z) = limV →∞

1

ΣVρ

( z

ΣV

)

.

This is a double scaling limit of the spectral density which is universal.

The surprising result is that this quantity can be obtained analyticallyfor a strongly interacting field theory such as QCD.


II. Chiral Symmetry and Dirac Spectra

Resolvent

Chiral Lagrangian

Thouless Energy

Scales in the Dirac Spectrum


Resolvent

A useful way to study spectra is by means of the resolvent

G(z) =1

V

∑

k

1

λk + z,

with z complex. The spectral density is given by

ρ(λ) =1

πReG(iλ + ǫ)

Note that the resolvent lives outside of the theory. It cannot beobtained from derivatives of the partition function. To find the resolventwe have to introduce additional valence quarks

G(z) = limn→0

1

n

d

dz〈det(D + m)detn(D + z)〉


Chiral Lagrangian

For M ≪ ΛQCD the QCD partition function is equivalent to a chiralLagrangian with the same transformation properties.

If the pion Compton wavelength is much larger than the size of the box,the static modes factorize from the partition function so that the massdependence of the QCD partition function is given by

Z(M) =

∫

U∈SU(Nf )

dUe1

2ΣTr(M†U+MU†).

The mass of the pseudo Goldstone bosons corresponding to z , theargument of the resolvent, is given by

m2π =

2zΣ

F 2π

.

For the partition function of the resolvent, the quark mass z can bechosen as small as we like.


Gell-Mann–Oakes–Renner Relation

� Therefore we can always find a range of z for which thecorresponding pion Compton wavelength is much larger than thesize of the box and the z -dependence of the resolvent is given bya unitary matrix integral. This integral is completely determined bythe pattern of chiral symmetry breaking.

� The energy scale for which the Compton wave length is equal tothe size of the box is known as the Thouless energy

zTh =F 2

π

2Σ√

V.

We conclude that because of the spontaneous breaking of chiralsymmetry, eigenvalue fluctuations for z < zTh are universal anddepend only on the global symmetries. The same correlations arefound in any theory with same pattern of chiral symmetry breaking anda mass gap. The simplest such theory is chiral random matrix theory.


Scales in the Dirac Spectrum

PT

V1 Fπ

V1/2 FFFπ0 λ

Microscopic Domain Chiral Domain Macroscopic Domain

chRMT

χ

ΣΣ

χ PT

� Eigenvalue spacing: ∆λ = πΣV .

� Thouless energy in units of the eigenvalues spacing

Eth

∆λ=

~D

ΣL2−d.

No separation of scales takes place for d = 2 . The reason is theabsence of spontaneous symmetry breaking and Goldstone bosons.


The Resolvent in Lattice QCD

Analytical result for the resolventG(z)V Σ = x(I0(x)K0(x) + I1(x)K1(x)), z = xV Σ

The resolvent of quenched QCD. The points represent lat-

tice data obtained by the Columbia group, and the theo-

retical prediction is given by the solid curve.

JV-1995


III. Chiral Random Matrix Theory

Chiral Random Matrix Theories

Other Symmetry Classes

Anti-Unitary Symmetries

Spontaneous Symmetry Breaking in DifferentDimensions


Chiral Random Matrix Theory

This is a theory with the global symmetries of QCD, but matrixelements. of the Dirac operator replaced by random numbers(JV-1994, Shuryak-JV-1992). In the sector of topological charge ν

the random matrix Dirac operator is given by

D =

m iW

iW † m

, P (W ) ∼ e−NTrW †W

where W a n × (n + ν) matrix so that D has exactly ν zeromodes.

The chRMT partition function is given by

ZνchRMT =

∫

dWdetNf (D + m)P (W ).

The theory simplifies because it has a large invariance group.


Flavor Topology Duality

chRMT prediction:

In the chiral limit, the averageposition of the small Diraceigenvalues only depends onthe combination Nf + Q.

JV-2000

P (λ) ∼ λ2Nf λ2Q+1

fermiondeterminant

JacobianDkl → λk

CERN COURIER, June 2007Lattice simulations: Fukaya-et al-2007

Based on work by: Giusti-Lüscher-Weisz-Wittig-2003


Other Symmetry Classes

In addition to an ensemble of complex matrices (Dyson index βD = 2

), we can also have an ensemble of real matrices ( βD = 1 ) or anensemble of quarternion real matrices ( βD = 4 ). This corresponds toQCD with two colors with quarks in the fundamental representation,and QCD with two or more colors and quarks in the adjointrepresentation, respectively.

The reality of the matrix elements is determined by the anti-unitarysymmetries of the Dirac operator.

[AK, D] = 0.

with A unitary and K the complex conjugation operator.


Anti-Unitary Symmetry for QCD withFundamental Quarks

For three or more colors, QCD in the fundamental representation doesnot have any anti-unitary symmetries and βD = 2 . QCD with twocolors is exceptional. The reason is the pseudo-reality of SU(2) .

[KCγ5τ2, D] = 0

with C = γ2γ4 and K the complex conjugation operator. Because(KCγ5τ2)

2 = 1, we can construct a basis such that the Dirac matrix isreal for any Aµ denoted by βD = 1


Anti-Unitary Symmetry for QCD with AdjointQuarks

For QCD with gauge fields in the adjoint representation the anti-unitarysymmetry of the Dirac operator is given by

[Cγ5K, D] = 0.

Because

(Cγ5K)2 = −1,

the eigenvalues of D are doubly degenerate (Kramers degeneracy).This corresponds to the case βD = 4 , so that it is possible to organizethe matrix elements of the Dirac operator into real quaternions.


Anti-Unitary Symmetry and Chiral Symmetry

For QCD in four dimensions we have that

[γ5, KCγ5τ2] = 0, [γ5, Cγ5K] = 0,

so that the chiral block structure is not affected is when we change thebasis to make the Dirac operator real or quaternion real.

In two dimensions,

γ5 → σ3 and γ2γ4 → iσ2

so that the anti-unitary symmetry and the axial symmetry do notcommute, and it is not possible to to make the Dirac operator real orquaternion real while maintaining the chiral block structure.

In three dimensions, there is no axial symmetry, and the Dirac operatoris a Hermitian real, complex or quaternion real matrix.


Spontaneous Symmetry Breaking andDimensionality

D Theory βD Symmetry Breaking Pattern RMT

2 Nc = 2 fund. 1 USp(2Nf ) × USp(2Nf ) → USp(2Nf ) (CI)

2 Nc ≥ 3 fund. 2 U(Nf ) × U(Nf ) → U(Nf ) chGUE (AIII)

2 Nc ≥ 2 adj. 4 O(2Nf ) × O(2Nf ) → O(2Nf ) (DIII)

3 Nc = 2 fund. 1 USp(4Nf ) → USp(2Nf ) × USp(2Nf ) GOE

3 Nc ≥ 3 fund. 2 U(2Nf ) → U(Nf ) × U(Nf ) GUE

3 Nc ≥ 2 fund. 4 O(2Nf ) → O(Nf ) × O(Nf ) GSE

4 Nc = 2 fund. 1 U(2Nf )/Sp(2Nf ) chGOE

4 Nc ≥ 3 fund. 2 U(Nf ) × U(Nf ) → U(Nf ) chGUE (AIII)

4 Nc ≥ 2 adj.. 4 U(2Nf )/O(2Nf ) chGSEChiral symmetry breaking in two, three and four dimensions for different values

of the Dyson index βD. Also indicated is the random matrix theory with the

same breaking pattern and the corresponding symmetric space.

Kieburg-Zafeiropoulos-JV-2013Dirac Spectra – p. 27/40

Dirac Spectra in Two Dimensions

Dirac Spectra of Lattice QCD in Two Dimensions

Dirac Spectra of the Schwinger Model

The Mermin-Wagner-Coleman Theorem

Possible Solutions


Two and Four Dimensional Dirac Spectra forβD = 1

Microscopic spectral density of

quenched staggered Dirac opera-

tor in 4 dimensions for QCD with

two colors and quarks in the adjoint

representation. Edwards-Heller-

Narayanan-1999

Microscopic spectral density of the

quenched QCD Dirac operator in 2

dimensions for QCD with three col-

ors and adjoint quarks ( β = ∞ ).

Kieburg-JV-Zafeiropoulos-2013




quenched staggered Dirac operator

in 4 dimensions for QCD with three

colors. Wettig-etal-1999


quenched QCD Dirac operator in 2 di-

mensions for QCD with three colors

and fundamental quarks ( β = ∞ ).





quenched staggered Dirac opera-

tor in 4 dimensions for QCD with

two colors and fundamental quarks.

Wettig-JV-etal-1999


quenched QCD Dirac operator in 2

dimensions for QCD with two colors

and fundamental quarks ( β = ∞ ).



Schwinger Model

Cumulative eigenvalue density of the two flavor Schwinger model

Bietenholz-Hip-Scheredin-Volkholz-2011Eigenvalues are rescaled by λ → λV 5/8 because the eigenvaluedensity of the two flavor Schwinger model ρ(λ) ∼ V λ3/5.

See also Damgaard-Heller-Narayanan-Svetitsky-2005


Dirac Spectra in Two and Four Dimensions

� Universal spectral correlations arise as a consequence ofspontaneous symmetry breaking.

� Dirac spectra in two dimensions and four dimensions show thesame degree of agreement with random matrix predictions.

� Yet according to the Mermin-Wagner-Coleman theorem acontinuous symmetry cannot be broken spontaneously in twodimensions.


Possible Solutions

The generating function for the resolvent is given by

G(z) =d

dz

∣

∣

∣

∣

z′=z

⟨

detNf (D + m)det(D + z)

det(D + z′)

⟩

.

� This partition function has both fermionic and bosonic“ghost”-quarks.

� The flavor group is a supergroup and the boson-boson part to benoncompact. Otherwise the integrals in the partition diverge.

� A trivial ground state cannot exist for a noncompact Goldstonemanifold because the integration over the noncompact group willbe divergent.


Mermin-Wagner-Coleman Theorem

� We conclude that the Mermin-Wagner-Coleman theorem does notapply to noncompact continuous symmetries and the flavorsymmetry remains spontaneously broken in two dimensions.

Zirnbauer,Niedermaier-Seiler

� Since the compact part of the symmetry group is not brokenspontaneously, one would expect different universal eigenvaluecorrelations. At this moment it is not yet clear what is going on.


Localization and Correlations

� In two dimensions all states are localized but the localizationlength may be long.

� Eigenvalues of localized state are uncorrelated, but that is not whatwe are seeing for two dimensional Dirac spectral.

� If the localization length, ξ , is sufficiently long, we may be in adomain

a ≪ L ≪ ξ.

This can be tested by studying larger lattices.


Conformal Dirac Spectra


Random Matrix Behavior



In the conformal limit

ρ(λ) ∼ V λα

Then the mode number

N ∼ Ldλα+1

so that eigenvalues scale as

λk ∼ 1

Ld/(1+α).

Since λ has the anamolous dimension, γ of the quark mass we have

1 + γ =d

1 + α⇒ α =

d − 1 − γ

1 + γ.

The anomalous mass dimension can be determined by the volumescaling of the eigenvalues.


Dirac Spectra for Large Nf

QCD with three colors and fundamental fermions is not conformal for Nf = 4

and Nf = 8 . Fodor-Holland-Kuti-Nógrádi-Schroeder-2009

Given the results for the Schwinger model, we could have randommatrix behavior in the conformal limit.

For the Nf = 2 Schwinger model we have that α = 1/3 and γ = 1/2 .


VI. Conclusions

� We have seen that the behavior of the low-lying Dirac eigenvaluescan be understood analytically despite the fact that QCD is astrongly interacting theory.


VI. Conclusions


� Dirac spectra are classified according to anti-unitary symmetries,and below the Thouless energy the eigenvalue fluctuations aregiven by the corresponding random matrix theory.


VI. Conclusions



� Eigenvalue correlations is two dimensions are similar to eigenvaluecorrelations in four dimensions.


VI. Conclusions



� Eigenvalue correlations is two dimensions are similar to eigenvaluecorrelations in four dimensions.

� One possible explanation is that this is due to the fact thatnoncompact symmetries are broken in any dimension.


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