Recent results on overlap fermions

Recent results on overlap fermions

Volker Weinberg

NIC / DESY-Zeuthen / FU-Berlin

[email protected]

QCDSF Collaboration

September 1, 2004

I3HP Topical Workshop, St. Andrews

Outline

• overlap fermions and their numerical implementation

• spectral density of overlap fermions on T=0 Luscher-Weisz gauge-field configurations and

Random Matrix Theory

• spectral density on T 6= 0 dynamical Wilson-Clover configurations

• hadron masses on large lattices, low mode averaging

• moments of nucleon structure functions

I3HP Topical Workshop, St. Andrews, 1.09.2004

Wilson Fermions

• Wilson term −r2

Pn Ψ(n)2Ψ(n) added to naive discretization of fermionic action to avoid

doublers

• chiral symmetry explicitly broken γ5D +Dγ5 6= 0

• improvement necessary to achieve O(a)

• additive renormalization of quark mass necessary

• no satisfactory definition of topological charge

• plagued by the appearance of exceptional configurations

• numerically not so expensive

• local


The Overlap Operator

• exact chiral symmetry on the lattice

• index theorem → allows investigation of the relation between topological properties of gauge

fields and the dynamics of chiral fermions

• no O(a) improvement necessary

• no exceptional configurations → simulation at small quark masses allows us to make contact

with (quenched) chiral perturbation theory

• numerically very expensive

• only local in generalized sense

• additional irrelevant parameter ρ


The massive overlap operator is defined by

D =ą1− amq

2ρ

ćDN +mq,

DN =ρ

a

ą1 +

X√X†X

ć, X = DW −

ρ

a,

• DW Wilson-Dirac operator

• 0 ≤ ρ ≤ 2 additional irrelevant parameter

• DN has n−+n+ exact zero modes, DNψn = 0, n− (n+) being the number of modes with

negative (positive) chirality

• Index of DN is thus given by ν = n− − n+

• ‘continuous’ modes λ, DNψλ = λψλ, having (ψ†λ, γ5ψλ) = 0, come in complex conjugate

pairs λ, λ∗

The Sign Function

sgn(X) =X√X†X

≡ γ5 sgn(Q), Q = γ5X,

To implement the sign function we use 2 different approaches:

• Zolotarev optimal rational approximation

• Minmax polynomial approximation


Zolotarev Optimal Rational Approximation

v. d. Eshof et al, hep-lat/0202025

sgn(x) ≈ xnX

i=1

σi

x2 + ωi

Coefficients σi, ωi known explicitly (Zolotarev formula)

D = ρ/a(1 + γ5Q

nX

i=1

σi

Q2 + ωi)

Computation of Dψ:

Solution of (Q2 + ωi)xi = Qψ

⇒ Dψ = ρ(ψ + γ5

Pi σixi)

with multimass CG (Beat Jegerlehner: hep-lat/9612014)


Minmax Polynomial Approximation

Luscher et. al. hep-lat/0212012

Construct an optimal polynomial approximation to sgn(x) in a range that excludes a small interval

around the origin:

Find polynomial P(y) of degree n that minimizes the error

δ = max0<ε≤y≤1

|h(y)|

h(y) = 1−√yP (y)

⇒ xP (x2) approximates sign(x) uniformly with a maximal deviation δ on√ε ≤ |x| ≤ 1

minmax polynomials: Polynomials that minimize the maximal relative error


Using Chebyshev Polynomials Tk:

P (y) =

nX

k=0

ckTk(z), z =2y − 1− ε

1− ε

h(y) has n+2 extremas of equal height u and alternating sign

Find ck and u by solving the system:

h(yl) = (−1)lu

for arbitrary values ε ≤ y0 < y1 . . . yn+1 ≤ 1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

outsign@xD

Speedup and Improvement of the Condition Number

Number of CG iterations determined by condition number of Q2

Speedup by:

• projecting out some (≈ 10) low eigenmodes and treat them exactly

• using an improved gauge action:

S[U ] =6

g2

hc0

X

plaquette

1

3Re Tr (1− Uplaquette)

+ c1

X

rectangle

1

3Re Tr (1− Urectangle) (1)

+ c2

X

parallelogram

1

3Re Tr (1− Uparallelogram)

i.

coefficients c1, c2 (c0 + 8c1 + 8c2 = 1) taken from tadpole improved perturbation theory.


Motivation for Spectral Approach

Why computing small eigenvalues?

• Saturation of hadronic correlation function

• Low mode averaging

• Topological Susceptibility→ η′ Mass (Witten-Veneziano)

• phenomenological instanton models

• Banks-Casher-formula: < ΨΨ >= −πρ(0)

ρ(λ) : Spectral density of the Dirac operator:

ρ(λ) =1

V

*Xn

δ(λ− λn)+

chiral condensate Σ =< ΨΨ >: order parameter of chiral symmetry breaking

Finite Temperature QCD:

T < Tc: spontaneously broken chiral symmetry

T > Tc: chiral restored phase


10 Low Lying Modes on a Single Configuration

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Resource Requirements

VOL n Memory/Evct nev Total HDD-Space

44 3072 48 kB 3072 144 MB

43 × 8 6144 96 kB 6144 576 MB

163 × 32 1.572.864 24 MB 100 2.3 GB

243 × 48 7.962.624 121 MB 100 11.9 GB


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(4&5666678 +:9;+=<><9 + 8@? 9 ? <<�9 ?A8CBED D D<F< D D D D D D

G HHHHI -(J&

5666678 +LK,+�<M<9C+ 8 ? K ? <<N9 ? 8 BOD D D<P< D D D D D D

G HHHHI Q &

56666667R + + R + ?AD D D R + SR ? + R ? ?AD D D R ? S< D D D D D D DDD<T< D D D R S S

G HHHHHHI

U'V W�X�X DZY [ \ ]$\ ^`_`a2b$^`c -( Y [ \ ],\ ^`_2a`b$^`c Q'd,e$e$f [ Q�f V V f b2g f [ _#" �&' "( #% '&� "(�h # ! 3�&43%( ! #" �&' %/#Jij S & 9,S ),+ ij S ),+Ck 8 S ij Slk�9$S ij S m;+n o p qrs t#uij S & K�S )$+ ij S )$+Ck 8 S ij Slk�9$S ij S m;+n o p qrs t

#Jij S & S m;+vw x + R w S ij w#�!yiz S & 9,S ),+ iz S ),+Ck 8 S iz SZkuK2S iz S m$+n o p qr{ t

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Spectral Density of Dov

β = 8.45, a=0.095 fm, ρ = 1.4, V=163 × 32

0 0.1 0.2 0.3 0.4aλ

0

100

200

300

400a-1

ρ(λ)


Spectral Density and Random Matrix Theory

Theoretical expectations for the spectral density:

λ < Ec : ρ(λ) can be described by Random Matrix Theory

(Ec : Thouless energy, expectation: Ec ≈ πf2π

Σ√V

)

consider microscopic spectral density

ρS(x) = limV→∞

1

ΣVρ(

x

ΣV)

if sector of topological charge ±ν:

ρS(x) =x

2(J

2ν(x)− Jν+1(x)Jν−1(x))

(Verbaarschot, Zahed)


0

0.1

0.2

0.3

0.4

2 4 6 8 10

x

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2 4 6 8 10

x

0

0.05

0.1

0.15

0.2

0.25

0.3

2 4 6 8 10

x

ρs(0, x) ρs(1, x) ρs(3, x)

In sector with topological charge ±ν appears with weight wν:

ρ(λ) ≈ V ΣXν

wνρs(ν, V Σλ)

for λ small


Spectral Density and Random Matrix Theory

For λ < 0.06, ρ(x) seems to follow the RMT behaviour

0 0.02 0.04 0.06 0.08 0.1aλ

0

20

40

60

80

100

a-1ρ(

λ)

Fit to RMT prediction yields < ΨΨ >= (249(9)MeV )3 TI→< ΨΨ >MS= 261(9)MeV)3


Chiral Symmetry

Chiral Transformations Ψ→ e−iωγ5Ψ for vanishing Quark masses.

U(3)V × U(3)A = SU(3)L × SU(3)R × U(1)V × U(1)A

• Explicit breaking of chiral Symmetry:

SU(3)L×SU(3)Rmu=md=ms 6=0−→ SU(3)Flavour

ms 6=mu−→ SU(2)Isospinmu 6=md−→ U(1)charge

• spontaneous breaking of chiral symmetry by the dynamical creation of a nonvanishing chiral

condensateŋΨΨ

ő 6= 0

Related to the density of the Eigenvalues of the Dirac-Operator near zero by the

Banks-Casher Relation:

ŋΨΨ

ő= −πρ(0)

• chiral symmetry restoration above critical temperature TCwith the chiral condensate being the order parameter of the chiral phase transition


Finite Temperature Simulation Parameters

Gauge action: Clover improved Wilson Fermions with Nf = 2 flavours of dynamical quarks

Lattice size: 163 × 8

β 5.2 (5.25)

κ 0.1330, . . . , 0.1360 (9 different values)

a ≈ 0.12 fm

Smearing 5 iterations of APE-smearing, α=0.45

Fermions: valence overlap

Eigenvalues: 50


Transition Temperature from the Polyakovop Loop (DIK Collaboration)

Increasing κ at a fixed value of β increases the temperatur T ∝ r0a . κt identified as the point,

where the Polyakov loop susceptibility: χ = N3s (< L2 > − < L >2) with Polyakov loop

L(s) = 13 Tr

QNts4=1U(s, 4)

assumes its maximum.

0.133 0.134 0.135 0.136 κ0

0.1

0.2

0.3

χβ=5.2β=5.25

0.133 0.134 0.135 0.136 κ0

20

40

60

80

100

χMonopole10✕Abelian100✕Photon

β = 5.2 β = 5.25

κt 0.1344(1) 0.1341(1)

Tc 210(3) MeV 219(3) MeV


Density on Dynamical Finite Temperature Configurations

0

50

100

150

200

250

300

350

400

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

rho(

lam

bda)

lambda

0.13430.13480.1360


Getting Hadron Masses from Correlation Functions

C(x) = 〈0|T{Of(x)O†i(0)}|0〉=

DOf(x)O†i(0)

E

C(t, p) =Xx

eipx〈0|T{Of(x, t)O†i(0, 0)}|0〉 t→∞−→

p=0

〈0|Of |H〉〈H|O†i |0〉2mH

e−mHt

on the lattice: C(t, 0) = Ae−mHt + Ae−m

H(T−t)

Mesonic Correlation Functions:

OFΓ (t, p) = a3

Xx

x4=t

X

a,bα,βi

e−ipx

FabΨiaα(x)ΓαβΨ

ibβ(x)


C(∆t, p) = a6

Xx

Xy

e−ipx+ipy

X

α,βα′,β′i,j

FabF′a′b′ΓαβΓ

′α′β′

��

= a6

Xx

Xy

e−ipx+ipy

X

α,βα′,β′i,j

FabF′a′b′ΓαβΓ

′α′β′

� �� ! " # $ %

�� ! $ # " %

& �� ! $ # $ %& �� ! " # " %

'

= a6

Xx

Xy

e−ipx+ipy

(−(Tr FF′)Tr CDΓG(x, y)Γ

′G(y, x)) +

(Tr F )(Tr F′)(Tr CDΓG(x, x))(Tr CDΓ

′G(y, y)))

= a6

Xx

Xy

e−ipx+ipy

(−(Tr FF′)Tr CDγ5ΓG(x, y)Γ

′γ5G(x, y)

†) +

(Tr F )(Tr F′)(Tr CDΓG(x, x))(Tr CDΓ

′G(y, y)))

= −��

Recent Results on Hadron Masses

The pion is computed from the correlation functions of

P = ψγ5ψ

A4 = ψγ4γ5ψ

We used Jacobi smeared sources (κs = 0.21, Ns = 50) in order to improve the ground state

overlap.

The sinks were either points or Jacobi smeared. In general the point sinks give a better signal.

V 123 × 24 163 × 32 243 × 48

β 8.1 8.45 8.45

a 0.125 fm 0.095 fm 0.095 fm


Low Mode Averaging

DeGrand, Schaefer: hep-lat/0401011, Giusti et al.: hep-lat/0402002

• for small quark masses hadron propagators from a single source show irregularities due to the

low eigenmode part of the propagator

→ much larger statistical fluctuations

• To smooth the signal, the contribution of the low-lying eigenmodes is averaged over all

positions of the quark source

• C(t) = CLL(t) + CHL(t) + CLH + CHH(t)

• CHL not computed directly, since it requires an inversion of the Dirac operator for every

eigenvalue

• Replace spectral 1-to-all Propagator with an all-to-all propagator in the traditionally computed

1-to-all propagator


1e+11

1e+12

1e+13

0 5 10 15 20 25 30 35

C(t

)

t

Inversion 1-allEV 1-all

EV all-allimproved


0 5 10 15 20 25

t

1e+11

1e+12

ma=0.0112ma=0.0196ma=0.028

PS correlator on the 243 × 48 lattice, black symbols: low mode averaged propagator.


0 5 10 15 20

t

1e+12

1e+14

1e+16

ma=0.0112ma=0.0196

nucleon correlator on the 243 × 48 lattice, black symbols: low mode averaged propagator


Pseudoscalar Masses from A4 Correlators

Pseudoscalar masses from cosh fits to the A4 correlators in the plateau range of the effectivemass.

lattice amq amPS mPS MeV

163 × 32 0.0280 0.211( 3) 437( 7)

0.0560 0.288( 2) 598( 5)

0.0980 0.383( 2) 795( 5)

0.1400 0.466( 2) 968( 4)

243 × 48 0.0112 0.144( 3) 299( 6)

0.0196 0.182( 3) 377( 6)

0.0280 0.212( 2) 441( 4)

0.0560 0.293( 2) 607( 4)

0.0980 0.387( 2) 803( 4)

0.1400 0.469( 2) 973( 4)


Comparison with Chiral Perturbation Theory

Quenched chira l perturbation theory predicts in infinite volume (Bernard, 1992)

m2π/mq = A

ş1− δ

şln

şAmq/Λ

2χ

ť+ 1

ťť+O

şm

2q

ť

0 0,05 0,1 0,15

a mq

1,5

1,6

1,7

1,8

1,9

a m

π2 /mq

163x32

243x48

Fits to m2π/mq


0 0,02 0,04 0,06 0,08 0,1 0,12 0,14

a mq

0

0,05

0,1

0,15

0,2

0,25

(a m

π)2

Fits to m2π for both lattice sizes.

Using Λχ = 4πfπ(fπ = 93 MeV), we obtain δ = 0.27(13) for the small and δ = 0.21(6)

for the large lattice.


Nucleon Correlator

The nucleon mass is computed from non-relativistic nucleon propagators. The nucleon mass is

extracted according to

〈NN〉 = A exp (−mNt) + B expą−m∗ (T − t)ć

0 10 20 30 40 50

t

0,1

1

10

100

1000

10000

1e+05

1e+06

ma=0.0112ma=0.0196ma=0.028ma=0.056ma=0.098ma=0.14

nucn2_24_pt_sm

fit range: 5 .. 24, -5 .. -1046 configs


Recent Nucleon Masses

lattice amq amnucl mnucl/MeV

163 × 32 0.0280 0.607( 9) 1260(19)

0.0560 0.678( 6) 1406(13)

0.0980 0.785( 4) 1628( 9)

0.1400 0.887( 4) 1840( 8)

243 × 48 0.0112 0.493(17) 1022(36)

0.0196 0.528( 9) 1095(20)

0.0280 0.556( 8) 1153(17)

0.0560 0.654( 5) 1357(11)

0.0980 0.771( 4) 1601( 9)

0.1400 0.876( 4) 1818( 9)


Structure Functions

Operator product expansion:

2

Z 1

0

dxxn−1

F1(x,Q2) =

X

f

E(f)F1,n

v(f)n (µ) +O(

1

Q2)

Z 1

0

dxxn−2

F2(x,Q2) =

X

f

E(f)F2,n

v(f)n (µ) + O(

1

Q2)

with

< p, s|O{µ1...µn}f − tr|p, s >= 2v

(f)n [p

µ1...pµn − tr]

Oµ1...µnq = qγ

µ1↔Dµ2...↔Dµnq


Structure Functions on the Lattice

Nucleon matrix elements of quark bilinear operators O

〈N |O|N〉, 〈N |N〉 = 2mN

derived from ratios

R ≡ 〈N(t)O(τ)N(0)〉〈N(t)N(0)〉 ' 1

2mN

〈N |O|N〉,

Operator Moment Mn(f) =R 1

0dxxnf(x) R

Ov2a = O{14} M1(F1) ∝< x > ip1v2a

Ov2b=O44 − 13

P3i=1Oii M1(F1) ∝< x > −E2

p+1/3p2

Epv2b

Ov2a requires non-zero momentum nucleon states

→ much larger statistical errors


v2b for Quenched Clover Fermions

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07(a/r0)

2

0.1

0.2

0.3

0.40.0 2.0 4.0 6.0 8.0 10.0

0.1

0.2

0.3

0.4(r0mps)

2

mπ √2mK

hep-lat/0311017(QCDSF)

vMS2b (2GeV) = 0.25(1) vs. phenom. ≈ 0.18


v2b for Unquenched Clover Fermions

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07(a/r0)

2

0.1

0.2

0.3

0.40.0 2.0 4.0 6.0 8.0 10.0

0.1

0.2

0.3

0.4(r0mps)

2

mπ √2mK

hep-lat/0311017 (QCDSF)

vMS2b (2GeV) = 0.27(2)


v2b for Quenched Overlap Fermions

0 10 20 30 40t

-0.2

-0.15

-0.1

-0.05

0

0.05

mq=0.028

0 10 20 30 40t

-0.2

-0.15

-0.1

-0.05

0

0.05

mq=0.056

0 10 20 30 40t

-0.2

-0.15

-0.1

-0.05

0

0.05

mq=0.098

0 10 20 30 40t

-0.2

-0.15

-0.1

-0.05

0

0.05

mq=0.14


v2b for Quenched Overlap Fermions

Renormalization: perturbative, tadpole-improved (MS, 2GeV)

Zv2b = 1.4566

No dependance on mass or volume visible

Lower result than previous Wilson calculations


Comparison v2a − v2b

V = (1.5fm)3

0 0.05 0.1 0.15m

q

0.1

0.15

0.2

0.25

0.3

0.35

0.4

v2b

v2a

Results compatible within errors


Summary

• overlap fermions have many advantages over Wilson and staggered fermions

• they offer the opportunity to investigate the topological properties of gauge fields through

exact zero modes

• numerical results for low eigenmodes agree with Random Matrix Theory

• through computations on large 243 × 48 lattices and small quark masses amq = 0.0196,

mPS = 299(6) MeV we hope to touch the region, where (quenched) chiral perturbation

theory is valid

• perturbatively TI-renormalized first moment of the nuclear structure function smaller than

previous quenched and unquenched Wilson+Clover results


Documents

Recent results on overlap fermions