the qcd phase transition probed by fermionic boundary ......partly with e. bilgici, c. gattringer,...
TRANSCRIPT
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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
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 1 / 16
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 1 / 16
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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
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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
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Dual quantities: idea
lattice: gauge invariant quantities link products along closed loops
plaquettes Polyakov loop(→ action) U0(0, ~x)U0(a, ~x) . . .
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
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Dual quantities: idea
lattice: gauge invariant quantities link products along closed loops
plaquettes Polyakov loop “Polyakov loops(→ action) U0(0, ~x)U0(a, ~x) . . . with detours”
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
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Dual quantities: idea
lattice: gauge invariant quantities link products along closed loops
plaquettes Polyakov loop “Polyakov loops loops(→ action) U0(0, ~x)U0(a, ~x) . . . with detours” winding twice
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
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Dual quantities: idea
lattice: gauge invariant quantities link products along closed loops
plaquettes Polyakov loop “Polyakov loops loops(→ action) U0(0, ~x)U0(a, ~x) . . . with detours” winding twice
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
-
Dual quantities: idea
lattice: gauge invariant quantities link products along closed loops
plaquettes Polyakov loop “Polyakov loops loops(→ action) U0(0, ~x)U0(a, ~x) . . . with detours” winding twice
how to distinguish these classes of loops?
⇒ phase factor eiϕ multiplying U0 at fixed x0-slice & Fourier componentFalk Bruckmann The QCD phase transition probed by fermionic boundary conditions 2 / 16
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Dual quantities: idea
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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 3 / 16
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 16
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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
Σ̃1 ≡1
2π
∫ 2π0
dϕ e−iϕΣϕ
Fourier component picks out all contributions that wind once
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 16
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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
Σ̃1 ≡1
2π
∫ 2π0
dϕ e−iϕΣϕ
Fourier component picks out all contributions that wind once
≡ dressed Polyakov loop: chiral symmetry connected to confinement
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 16
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Dual condensate: definitionvarious 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
Σ̃1 ≡1
2π
∫ 2π0
dϕ e−iϕΣϕ
Fourier component picks out all contributions that wind once
≡ dressed Polyakov loop: chiral symmetry connected to confinement
dual quantities: same behavior under center symmetry (← special ϕ’s)⇒ order parameter
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 16
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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’
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 5 / 16
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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?
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 6 / 16
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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]
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 7 / 16
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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
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 9 / 16
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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]
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 10 / 16
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 11 / 16
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 11 / 16
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 12 / 16
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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”
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 13 / 16
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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 =∞
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 14 / 16
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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 =∞
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 14 / 16
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 15 / 16
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 15 / 16
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 16 / 16
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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
Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 16 / 16
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dummy c© NICA at JINR Dubna
dummyFalk Bruckmann The QCD phase transition probed by fermionic boundary conditions 16 / 16