A surprise with many-flavor staggered fermions in the ... · WolfgangUnger,ETHZürich...

A surprise with many-flavor staggered fermions in thestrong coupling limit

Wolfgang Unger, ETH Zürichwith Philippe de Forcrand, ETH Zürich/CERN, Seyong Kim, Sejong University

Lattice 2012, Cairns28.06.2012

Wolfgang Unger, ETH Zürich () Many flavor staggered fermions 28.06.2012 1 / 10

Outline

OutlineOutlineOutlineOutlineOutline

��1 Large Nf Strong Coupling QCD: A Case Against Mean Field��2 Chiral Restoration for Large Nf

��3 Tests of Conformality��4 Discussion


Large Nf Strong Coupling QCD: A Case Against Mean Field

Mean Field: chiral symmetry is always broken in the strong-coupling limit ofstaggered fermions at T = 0 for all values of Nf and Nc

chiral condensate well known to be independent of Nf and Nc,i.e. in d spatial dimensions:[Kluberg-Stern et al., 1983]

⟨ψ̄ψ⟩

(T = 0) =((1+d2)1/2−1)/2)

1/2

d

we also found, following [Damgaard et al., 1985]:chiral restoration temperature is Tc = d

4 + d8

NcNf

+O( 1N2

f)

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.1 0.2 0.3 0.4 0.5

Tc

1/Nf

Nc=1

Nc=2

Nc=3

mean field expected to work well for large number of d.o.f. per site,e.g. exact results in the Gross-Neveu model for Nf →∞


Chiral Restoration for large Nf

On the other hand: loop expansion of the determinant shows that dynami-cal fermions induce a plaquette coupling ∝ Nf/m4

q, as studied numerically by[A. Hasenfratz, T. DeGrand PRD49 (1994)]⇒ suggests chiral symmetry restoration for sufficiently large Nf ?





Answer from Monte Carlo: Surprise! strong first order Nf-driven bulk transitionfor strong-coupling limit of staggered fermions found

0 20

40 60

80 100

120 0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0

0.5

1

1.5

2

β=0.0, 44

⟨ ψψ- ⟩

Chiral CondensateMC data

Nf

amq

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

chiral symmetryrestored!←−−−−

Ncf ' 52 continuum flavors for mq = 0, Nc

f increases with mq(heavy fermions → less ordering)






0 20

40 60

80 100

120 0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0

0.5

1

1.5

2

β=0.0, 64

⟨ ψψ- ⟩


Nf

amq

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Ncf ' 52 continuum flavors for mq = 0, Nc

f increases with mq(heavy fermions → less ordering), almost no finite size effects






0 5

10 15

20 25

30 0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

β=5.0, 44

⟨ ψψ- ⟩


Nf

amq

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


β = 5 similar but Ncf smaller






0 5

10 15

20 25

30 0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

β=5.0, 64

⟨ ψψ- ⟩


Nf

amq

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


β = 5 similar but Ncf smaller, stronger finite size effects






0 5

10 15

20 25

30 0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

β=5.0, 64

⟨ ψψ- ⟩


Nf

amq

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Explanation for failure of mean field: terms of O( NfNc, Nf

d2 ) are neglected



The Chirally Restored Phase for large βThe Chirally Restored Phase for large βThe Chirally Restored Phase for large βThe Chirally Restored Phase for large βThe Chirally Restored Phase for large β

smooth variation with β → Nf-driven transition extends to weak couplingNc

f ' O(10) at weaker couplingconnection with Nf-driven transition to conformal window?

0 10

20 30

40 50

60 70

80 0

1 2

3 4

5

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

amq=0.025, 44

⟨ ψψ- ⟩


Nfβ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Tests of Conformality

Characterizing the chirally restored phase: hadron massesCharacterizing the chirally restored phase: hadron massesCharacterizing the chirally restored phase: hadron massesCharacterizing the chirally restored phase: hadron massesCharacterizing the chirally restored phase: hadron masses

Hadron spectrum obtained from simulations with Nf = 56 and Nf = 96at zero quark mass

hadron masses measured for mq = 0 are non-zerobut masses decrease as the lattice size L is increasedparity partners degenerate (c.f. chiral symmetry restoration)mass ratios ∼ independent of L:

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

mrh

o

mpion

L=4

L=6

L=8

L=10

preliminary

L=4

L=6

L=8

L=10

L=12

Nf=56Nf=96

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 0.05 0.1 0.15 0.2 0.25

mrh

o/m

pio

n

1/L

Rho/Pion mass ratio vs 1/L, Nf=56 and 96, mq=0

preliminary

Nf=56Nf=96



Characterizing the chirally restored phase: Dirac SpectrumCharacterizing the chirally restored phase: Dirac SpectrumCharacterizing the chirally restored phase: Dirac SpectrumCharacterizing the chirally restored phase: Dirac SpectrumCharacterizing the chirally restored phase: Dirac Spectrum

Dirac eigenvalue spectrum, measured at zero quark mass, β = 0:

integrated eigenvalue density:∫ λ0 ρ(λ̄)d λ̄ = rank(λ)

rank(Dirac matrix) ∈ [0, 1]

derivative gives ρ(λ)







Compare Nf = 0 (quenched configurations) and Nf = 56 (chirally symmetric phase)similar for large eigenvalues (UV)the Nf = 56 curve shows a gap for small eigenvalues (IR), consistent with chiral symmetryrestoration: ρ(0) = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Lambda x a

Integrated eigenvalue density, 104, mq=0

Nf=0 Nf=56

0

0.02

0.04

0.06

0.08

0.1

0 0.1 0.2 0.3 0.4

Nf=0 Nf=56







Compare different volumes for Nf = 56:large eigenvalues (UV) are L-independent,the IR spectral gap shrinks as L increases

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

Lambda x a

Integrated eigenvalue density, mq=0, Nf=56

44

64

84

104







Compare different volumes for Nf = 56:IR spectrum invariant after rescaling by L: spectral gap ∝ 1/LIR physics only depends on L, while the UV physics depends on ano other scale in the system ⇒ we have an IR-conformal theory!

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25

Lambda x L

Integrated eigenvalue density, rescaling the infrared, mq=0, Nf=56

44

64

84

104

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.4 0.6 0.8 1 1.2

44

64

84

104



Conjecture: β = 0 IR-conformal phase is analytically connected with the weak-coupling, continuum IR-conformal phase

Study of continuum limit is much more difficult:for a given lattice size L4, the scales are ordered as a� 1/ΛQCD � Lat strong-coupling the hierarchy is a ' 1/ΛQCD � Lrange of conformal invariance (LΛQCD) maximized at β = 0 for given lattice sizeL/a

weak coupling: strong coupling:

a 1/ΛQCD L=aN a≈1 /ΛQCD L=aN

strong-coupling limit is the laboratory of choice to study a4d IR-conformal gauge theory


Discussion

SummarySummarySummarySummarySummary

Shown: for β = 0, a strong first order bulk transition exists which is Nf-driven to achirally symmetric phase

in the chiral limit: Ncf = 52(4) continuum flavors

finding in contrast to meanfield predictionchirally restored phase extends to weak coupling

Also shown: for β = 0, “large-Nf QCD” is IR-conformal:strong-coupling limit is the laboratory of choice to study a 4d IR-conformal gaugetheorysimulations at large Nf and zero quark mass can be performed without too muchcomputer effort


Discussion

SummarySummarySummarySummarySummary

Shown: for β = 0, a strong first order bulk transition exists which is Nf-driven to achirally symmetric phase

in the chiral limit: Ncf = 52(4) continuum flavors

finding in contrast to meanfield predictionchirally restored phase extends to weak coupling

Also shown: for β = 0, “large-Nf QCD” is IR-conformal:strong-coupling limit is the laboratory of choice to study a 4d IR-conformal gaugetheorysimulations at large Nf and zero quark mass can be performed without too muchcomputer effort

no evidence for additional T = 0 phase transition as β is increased⇓

Conjecture: strong coupling chirally symmetric, IR-conformal phase isanalytically connected with the continuum IR-conformal phase


Discussion

DiscussionDiscussionDiscussionDiscussionDiscussion

Outlookquark mass breaks conformality; chiral condensateis proportional to mq ⇒ γ∗ = 1check for IRFP ?glueballs, static potential, . . .

-0.2 0

0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

0 0.02 0.04 0.06 0.08 0.1 0.12

⟨ψψ-

⟩

a mq

Mass deformation: condensate vs mq

Nf=56, 44

linear fit

Single parameter of this IR-conformal theory is Nf :increasing Nf increases the magnitude of the spectral gap,decreasing Nf brings us back to a chirally broken phase, via a first-order transitionas in Sannino’s "Jumping dynamics" [hep-ph/1205.4246] → no walking dynamics

Our findings are consistent with the literature:Kogut, Sinclair, Nucl. Phys. B295 (1985) bulk transition for Nf = 12Damgaard, et al., Phys. Lett. B400 (1997): bulk transition for Nf = 16Jin, Mawhinney, PoS Lattice2011: bulk transition for Nf = 12 with improved actionA. Hasenfratz, hep-lat/1111.2317 (2012): bulk transition for Nf = 8, 12


Discussion

Backup Slide: Time Histories for Chiral CondensateBackup Slide: Time Histories for Chiral CondensateBackup Slide: Time Histories for Chiral CondensateBackup Slide: Time Histories for Chiral CondensateBackup Slide: Time Histories for Chiral Condensate

0

0.5

1

1.5

2

2.5

3

0 200 400 600 800 1000

p-b

ar-

p

Nf = 48, cold startNf = 48, hot start


Discussion


0

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1

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2.5

3

0 200 400 600 800 1000

p-b

ar-

p



A surprise with many-flavor staggered fermions in the ... · WolfgangUnger,ETHZürich...

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