The QCD phase transition probed by fermionic boundary ......partly with E. Bilgici, C. Gattringer,...

The QCD phase transitionprobed by fermionic boundary conditions

Falk Bruckmann(Univ. Regensburg)

Nonperturbative Aspects of QCD at Finite Temperature and DensityTsukuba, Nov. 2010

partly with E. Bilgici, C. Gattringer, C. Hagen,Z. Fodor, K. Szabo, B. Zhang

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 0 / 16

Motivation

phase transition at finite temperature (x0 ∈ S1β=1/kBT ) phase diagram

Deconfinement: Polyakov loopprobes center symmetry for infinitely heavy quarksorder parameter: 0→ finite

Chiral restoration: quark condensate ρ(0)probes chiral symmetry for massless quarksorder parameter: finite→ 0

related? Dual quantities

idea and definitionorder parameter and mechanismslattice results plus random matrix model


Dual quantities: idea

lattice: gauge invariant quantities link products along closed loops

plaquettes(→ action)

how to distinguish these classes of loops?

⇒ phase factor eiϕ multiplying U0 at fixed x0-sliceFalk Bruckmann The QCD phase transition probed by fermionic boundary conditions 2 / 16



plaquettes Polyakov loop(→ action) U0(0, ~x)U0(a, ~x) . . .





plaquettes Polyakov loop “Polyakov loops(→ action) U0(0, ~x)U0(a, ~x) . . . with detours”





plaquettes Polyakov loop “Polyakov loops loops(→ action) U0(0, ~x)U0(a, ~x) . . . with detours” winding twice





plaquettes Polyakov loop “Polyakov loops loops(→ action) U0(0, ~x)U0(a, ~x) . . . with detours” winding twice


⇒ phase factor eiϕ multiplying U0 at fixed x0-slice & Fourier componentFalk Bruckmann The QCD phase transition probed by fermionic boundary conditions 2 / 16


phase factor eiϕ multiplying U0 at fixed x0-slice

≡ quarks with general boundary conditions Gattringer ’06

ψ(x0 + β) = eiϕψ(x0)

physical quarks are antiperiodic: ϕ = π

boundary angle ϕ: tool on the level of observables (valencequarks)

formally an imaginary chemical potential


Dual condensate: definition

various dual quantities possible cf. Synatschke, Wipf, Wozar, ’07

general quark propagator:1

γµDµϕ + m

general quark condensate:〈tr

1γµD

µϕ + m

〉≡ Σϕ [ lim

m→0Σπ : chiral condensate]

dual condensate: Bilgici, FB, Gattringer, Hagen ’08

Σ̃n ≡1

2π

∫ 2π0

dϕ e−inϕΣϕ

Fourier component picks out all contributions that wind n times





γµDµϕ + m


1γµD

µϕ + m

〉≡ Σϕ [ lim



Σ̃1 ≡1

2π

∫ 2π0

dϕ e−iϕΣϕ

Fourier component picks out all contributions that wind once





γµDµϕ + m


1γµD

µϕ + m

〉≡ Σϕ [ lim



Σ̃1 ≡1

2π

∫ 2π0

dϕ e−iϕΣϕ


≡ dressed Polyakov loop: chiral symmetry connected to confinement


Dual condensate: definitionvarious dual quantities possible cf. Synatschke, Wipf, Wozar, ’07


γµDµϕ + m


1γµD

µϕ + m

〉≡ Σϕ [ lim



Σ̃1 ≡1

2π

∫ 2π0

dϕ e−iϕΣϕ


≡ dressed Polyakov loop: chiral symmetry connected to confinement

dual quantities: same behavior under center symmetry (← special ϕ’s)⇒ order parameter


Dual condensate: order parameter I

SU(3) quenched: Bilgici, FB, Gattringer, Hagen 08

(bare) Σ̃1 with m = 100MeV

100 200 300 400 500 600T [MeV]

0.00

0.05

0.10

0.15

0.20

0.25

Σ1

[GeV

3 ]

83 x 4

103 x 4

103 x 6

123 x 4

123 x 6

143 x 4

143 x 6

143 x 8

(bare) Polyakov loop

100 200 300 400 500 600T [MeV]

0.00

0.05

0.10

0.15

0.20

0.25

<T

r P

>/3

V [

GeV

3 ]

83 x 4

103 x 4

103 x 6

123 x 4

123 x 6

143 x 6

less UV renormalisation in a ∼ 1Nt ← ‘detours’


Dual condensate: order parameter II

SU(3) with dynamical quarks: FB, Fodor, Gattringer, Szabo, Zhang prelim.

Nf = 2 + 1 improved staggered fermions Aoki et al. 06⇒ crossover with Tc = 155(2)(3) − 170(4)(3)MeV

(bare) Σ̃1 with m = 60MeV

75 100 125 150 175 200 225 2500

0.01

0.02

0.03

0.04

0.05

0.06

(bare) Polyakov loop

75 100 125 150 175 200 225 250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

similar behaviour (center symmetry not an exact symmetry anymore)mass dependence of Σ̃1?


Dual condensate: mechanism I

Σ̃1 =1

2π

∫ 2π0

dϕ e−iϕ ·〈

tr1

γµDµϕ + m

〉Fourier integrand 〈...〉 as a function of ϕ: Bilgici, FB, Gattringer, Hagen 08

0 π/2 π 3π/2 2πϕ

0.20

0.25

0.30

0.35

0.40

0.45I(

ϕ)

T < Tc, am = 0.10

T < Tc, am = 0.05

T > Tc, am = 0.10

T > Tc, am = 0.05

[for real Polyakov loops, others shift plot by 2π/3]

⇒ depends on ϕ only at high temperatures⇒ Σ̃1 6= 0 Xin particular: chiral condensate survives at high T for periodic bc.sdummy several lattice works, class. dyons [FB @ Mishima]


Dual condensate: mechanism IItr means sum over all eigenmodes:

Σ̃1 ≡∫ 2π

0

dϕ2π

e−iϕ〈

tr1

γµDµϕ + m

〉=

∫ 2π0

dϕ2π

e−iϕ〈∑

k

1iλkϕ + m

〉truncate the ev sum: IR dominance, strongest for small m

0.000

0.005

0.010

0.015

0.020

0.025

T < TcT > Tc

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

T < TcT > Tc

0 2000 4000|λ| [MeV]

0.0

0.5

1.0

1.5

T < TcT > Tc

0 2000 4000|λ| [MeV]

0.0

0.5

1.0

1.5

T < TcT > Tc

Individual contributions Individual contributions

Accumulated contributions Accumulated contributions

m = 100 MeV m = 1 GeV

m = 100 MeV m = 1 GeV

expected: λ in denominator, lowest modes most sensitive to bc.sFalk Bruckmann The QCD phase transition probed by fermionic boundary conditions 8 / 16

Dual condensate: mechanism III

Σ̃1 ≡∫ 2π

0

dϕ2π

e−iϕ〈

tr1

γµDµϕ + m

〉;

∞∑k=0

(−1)k tr(γµDµϕ)k

mk;

(U/am

)k

⇒ large mass suppresses long loops (many U ’s⇒many 1/m’s)

m→∞: conventional Polyakov loop reproduced since shortest,

but UV dominated FB, Gattringer, Hagen 06

mass gives width of dressed Polyakov loops indep. of lattice spacing


Dual condensate: other applications

in gauge group G2 (no center) Danzer, Gattringer, Maas 08

in SU(2) with adjoint quarks (Tchiral ' 4Tdeconf)dummy Bilgici, Gattringer, Ilgenfritz, Maas 09

through Schwinger-Dyson equations Fischer, Mueller 09

at imag. µ through functional RG Braun, Haas, Marhauser, Pawlowski 09

in Polyakov loop extended NJL modeldummy Kashiwa, Kouno, Yahiro 09, Mukherjee, Chen, Huang 09incl. magnetic field Gatto, Ruggieri 10

◦ not yet: correlator of dressed Polyakov loops[eigenmodes not just eigenvalues]dummy [Synatschke, Wipf, Langfeld 08, Bilgici, Gattringer 08]


Summary so far

the dual condensate Σ̃1 is an order parameter under center symmetry

Σ̃1 = 0 at low T ← similar to the Polyakov loopΣ̃1 > 0 at high Tlimit of large mass: conventional (straight) Polyakov looplimit of small mass: Fourier component of chiral condensate wrt.fermionic boundary conditions

and connects confinement and chiral symmetry

mechanism: lowest modes respond to boundary conditions at high T

How generic are these features? Random Matrix Model


Random matrix theory for QCD at finite T

replace dynamics by random matrices (with correct symmetry)chiral ensembles mimicking the Dirac operatortemperature through Matsubara frequencies (quarks antiperiodic)

Random matrix model: Jackson, Verbaarschot 96, Stephanov 96dummy Wettig, Schaefer, Weidenmueller 96

Z =∫

dXN×N exp(−NC2trXX †) det(

m iX + iπT · 1NiX † + iπT · 1N m

)

Gaussian weight

lowest Matsubara frequency as non-random trace part

schematic (crit. exponents like mean field)

model parameter: C


numerical simulations: ρ(λ) from 500 30× 30 matricesT = 0

−6 −4 −2 0 2 4 60

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

T = 0.45/C

−6 −4 −2 0 2 4 60

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

T = 1/C

−6 −4 −2 0 2 4 60

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

semicircle of width ∼ 1/C, shift ±T ⇒ ρ(0) vanishes for high T

saddle point method (large matrix size N):

0.0 0.2 0.4 0.6 0.8 1.0 1.2TT_c

0.2

0.4

0.6

0.8

1.0

ρ(0) = C√

1− (πTC)2

vanishes above Tc ≡ 1πC

chiral phase transition, 2nd order

⇒ chiral condensate ρ(0) ∼ Σ and its absence at high T “generic”


Random matrix theory for dual condensate

general bc.s ϕ⇒modified Matsubara frequencies FB in preparation

ωϕ ≡ minn |(ϕ+ 2πn)T | ={ϕT ϕ ∈ [0, π](2π − ϕ)T ϕ ∈ [π,2π]

ωπ = πT as before, for other boundary conditions less shifted ...

saddle point similar to before:

ρ(0)ϕ = Σϕ = C√

1− (T/Tc,ϕ)2

with

Tc,ϕ ≡

{1ϕC ϕ ∈ [0, π]

1(2π−ϕ)C ϕ ∈ [π,2π]

... hence survives up to higher critical temperature, Tc,0 =∞


chiral condensate under general bc.s:

0

1

2

3

4

TT_c0

2

4

6j

0.0

0.5

1.0

Σϕ(T ) in units of C

dual chiral condensate:

0.0 0.5 1.0 1.5 2.0 2.5 3.0TT_c

0.5

1.0

1.5

2.0

Σ̃1(T ) plus its T -derivative

changes most at the chiral phase transition massive


Summary

the behavior of the dual condensate/dressed Polyakov loop can beunderstood from random matrices:

removed gauge dynamics

boundary angle ϕ incorporated through Matsubara frequencies(like temperature)

condensate under angle ϕ agrees qualitatively lattice results,functional methods and QCD models: “generic” behaviorRM model the simplest

dual condensate: deconfinement transition near the chiral one

outlook:canonical determinant dual to det(γµD

µϕ) FB, Gattringer in progress


condensate with valence mass under general bc.s:

0

1

2

3

4

TT_c0

2

4

6j

0.0

0.5

1.0

Σϕ(T ) in units of C

dual chiral condensate:

0 1 2 3 4TT_c

0.5

1.0

1.5

2.0

Σ̃1(T ) massive (massless)

mass weakens transition


dummy c© NICA at JINR Dubna

dummyFalk Bruckmann The QCD phase transition probed by fermionic boundary conditions 16 / 16

The QCD phase transition probed by fermionic boundary ......partly with E. Bilgici, C. Gattringer,...

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