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The QCD phase transition probed by fermionic boundary conditions

Falk Bruckmann (Univ. Regensburg)

Bogoliubov readings Dubna, September 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 / 19

QCD

need to understand confinement and chiral symmetry breaking

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 1 / 19

QCD and its phase diagram

need to understand confinement and chiral symmetry breaking but also deconfinement and chiral symmetry restoration at finite temperature and/or density ⇒ new phases of matter

250 500 750 1000 1250 1500 1750 2000 Baryon chemical potential MeV 25 50 75 100 125 150 175 200 Temperature MeV Quarkgluon plasma Early Universe Hadron phase 2SC NQ CFL

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 1 / 19

QCD and its phase diagram

need to understand confinement and chiral symmetry breaking but also deconfinement and chiral symmetry restoration at finite temperature and/or density ⇒ new phases of matter

250 500 750 1000 1250 1500 1750 2000 Baryon chemical potential MeV 25 50 75 100 125 150 175 200 Temperature MeV Quarkgluon plasma Early Universe Hadron phase 2SC NQ CFL

here finite temperature: x0 ∈ S1

β . . . Eucl. and compact, β ≡ 1/T

both effects related? Dual quantities generic? Random Matrix Theory

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 1 / 19

Theory challenge: Deconfinement

Polyakov loop: P( x) = P exp

• i

β

0 dx0A0(x0,

x)

• ∈ SU(3)

trP

x in complex plane [one point per configuration]

• 0.2 -0.1 0

0.1 0.2 0.3

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3

• 0.2 -0.1 0

0.1 0.2 0.3

• 0.2 -0.1 0

0.1 0.2 0.3

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 T < Tc T > Tc T ≈ Tc

• rder parameter like magnetization, but inverse behavior

free energy of infinitely heavy quarks tr P ∼ e−βFquark = e−∞ = 0 T < Tc e−# = 0 T > Tc breaks center symmetry

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 2 / 19

Theory challenge: Chiral symmetry restoration

spectral density ρ(λ) of the Dirac operator:

0.02 0.04 0.06 0.12 0.18 0.24 0.3 0.02 0.04 0.02 0.04 0.06 0.12 0.18 0.24 0.3 T < Tc T > Tc T ≈ Tc

• rder parameter of chiral symmetry: ρ(0) ∼ ¯

ψψ

Banks-Casher

i.e. for massless quarks [mass breaks chiral symmetry explicitly] Confinement and chiral symmetry related? Dual quantities

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 3 / 19

Dual quantities: idea and definition

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-slice

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19

Dual quantities: idea and definition

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-slice

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19

Dual quantities: idea and definition

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-slice

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19

Dual quantities: idea and definition

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-slice

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19

Dual quantities: idea and definition

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

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19

Dual quantities: idea and definition

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

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19

Dual quantities: idea and definition

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 component

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 4 / 19

≃ quarks with general boundary conditions

Gattringer ’06

ψ(x0 + β) = eiϕψ(x0) physical quarks are antiperiodic: ϕ = π general quark propagator:

• cf. Synatschke, Wipf, Wozar, ’07

1 γµDµ

ϕ + m

(physical) chiral condensate: ρ(0) = ¯ ψψ = lim

m→0 lim V→∞

1 V

• tr

1 γµDµ

ϕ=π + m

• ≡ Σϕ=π

dual condensate:

Bilgici, FB, Gattringer, Hagen ’08

˜ Σ1 ≡ 1 2π 2π dϕ e−iϕ 1 V

• tr

1 γµDµ

ϕ + m

• 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 5 / 19

Dual quantities and center symmetry

center: commutes with all group elements for SU(3): {1, e2πi/3 ≡ z, e−2πi/3 ≡ z∗} · 13 center transformation: non-periodic gauge transformation, e.g. U0 → zU0 in some time slice invariance: action invariant, Polyakov loop: tr P → z tr P center symmetric = confined phase: tr P = 0 at low T center broken = deconfined phase: tr P ≈ {1, z, z∗} = 0 at high T [transform into each other] dual quantities like dual condensate ˜ Σ1: same behaviour under center: ˜ Σ1 → z ˜ Σ1

Synatschke, Wipf, Langfeld ’08

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 6 / 19

Dual condensate: order parameter I

SU(3) quenched:

Bilgici, FB, Gattringer, Hagen ’08

(bare) ˜ Σ1 with m = 100MeV

100 200 300 400 500 600

T [MeV]

0.00 0.05 0.10 0.15 0.20 0.25

Σ1 [GeV

3] 8

3 x 4

10

3 x 4

10

3 x 6

12

3 x 4

12

3 x 6

14

3 x 4

14

3 x 6

14

3 x 8

(bare) Polyakov loop

100 200 300 400 500 600

T [MeV]

0.00 0.05 0.10 0.15 0.20 0.25

<Tr P>/3V [GeV

3] 8

3 x 4

10

3 x 4

10

3 x 6

12

3 x 4

12

3 x 6

14

3 x 6

less renormalisation ← detours = dressing

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 7 / 19

Dual condensate: order parameter II

SU(3) with dynamical fermions:

FB, Fodor, Gattringer, Szabo, Zhang preliminary

Nf = 2 + 1 staggered fermions at phys. masses

Aoki et al. ’06

⇒ crossover with T ¯

ψψ c

= 155(2)(3)MeV and T P

c = 170(4)(3)MeV

(bare) ˜ Σ1 with m = 60MeV

75 100 125 150 175 200 225 250 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)

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 8 / 19

Dual condensate: mechanism I

˜ Σ1 = 1 2π 2π dϕ e−iϕ · 1 V

• tr

1 γµDµ

ϕ + m

• Fourier integrand ... as a function of ϕ:

Bilgici, FB, Gattringer, Hagen ’08

π/2 π 3π/2 2π

ϕ

0.20 0.25 0.30 0.35 0.40 0.45

I(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 ⇒ ˜ Σ1 = 0 in particular: chiral condensate survives at high T for periodic bc.s dummy

several lattice works

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 9 / 19

Dual condensate: mechanism II

tr means sum over all eigenmodes: ˜ Σ1 ≡ 2π dϕ 2π e−iϕ 1 V

• tr

1 γµDµ

ϕ + m

• =

2π dφ 2π e−iϕ 1 V

k

1 iλk

ϕ + m

• truncate the ev sum: IR dominance

Bilgici, FB, Gattringer, Hagen ’08

0.000 0.005 0.010 0.015 0.020 0.025

T < Tc T > Tc

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025

T < Tc T > Tc

2000 4000 |λ| [MeV] 0.0 0.5 1.0 1.5

T < Tc T > Tc

2000 4000 |λ| [MeV] 0.0 0.5 1.0 1.5

T < Tc T > 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.s

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 10 / 19

Summary so far

the dual condensate ˜ Σ1 is an order parameter under center symmetry ˜ Σ1 = 0 at low T ← similar to the Polyakov loop ˜ Σ1 > 0 at low T limit of large mass: detours suppressed ⇒ conventional (straight) Polyakov loop limit of small mass: Fourier component of chiral condensate wrt. fermionic boundary conditions mechanism: lowest modes respond to boundary conditions at high T [boundary angle ≃ imag. chemical potential, but only at the level of

• bservables, not for dynamical quarks]

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 11 / 19

Summary so far

the dual condensate ˜ Σ1 is an order parameter under center symmetry ˜ Σ1 = 0 at low T ← similar to the Polyakov loop ˜ Σ1 > 0 at low T limit of large mass: detours suppressed ⇒ conventional (straight) Polyakov loop limit of small mass: Fourier component of chiral condensate wrt. fermionic boundary conditions mechanism: lowest modes respond to boundary conditions at high T [boundary angle ≃ imag. chemical potential, but only at the level of

• bservables, not for dynamical quarks]

relax . . . change subject!

Bogoliubov

How generic are these features? Random Matrix Theory

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 11 / 19

Random matrix theory in a nutshell

≡ replace dynamics of a given physical system by random matrices (“0-dim. field theory”) with the correct symmetry

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 12 / 19

Random matrix theory in a nutshell

≡ replace dynamics of a given physical system by random matrices (“0-dim. field theory”) with the correct symmetry showcase: distribution of (neighbouring) level spacings s = ∆λ P(s) =

• dX exp(−trXX †) prob.(s)X

where X is N × N and real complex quaternionic ≡ Gaussian Orthogonal Unitary Symplectic

• Ensemble ← different anti-unitary symm.s

Dyson index βD = 1 2 4

• ∼ number of real d.o.f.

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 12 / 19

P(s) for large matrices, βD =1, 2, 4:

0.5 1.0 1.5 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0 1.2

⇒ typical eigenvalue repulsion depending on ensemble well described by 2×2 matrices:

Wigner

P(s) ∼ sβDe−#s2 ⇋ independent eigenvalues: P(s) ∼ e−s

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 13 / 19

Random matrix theory for QCD

random entries of the Dirac operator: ev.s(X) → ev.s m iX iX † m

• mimics γ’s in chiral representation: ‘chiral ensembles’, same P(s)

0.2 0.4 0.6 0.8 1 1 2 3 P(s) s

lattice vs. chGUE prediction (βD = 2)

Pullirsch, Rabitsch, Wettig, Markum ’98

universal ‘bulk’ property, exact in ǫ-regime . . .

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 14 / 19

Random matrix theory for QCD at finite T

quarks are antiperiodic in x0 ∈ [0, β] ⇒ Dirac eigenvalues shifted by Matsubara frequencies πT + 2πnT (exact in free case: waves with certain frequencies) Random matrix model:

Jackson, Verbaarschot ’96

Z =

• dXN×N exp(−NC2trXX †) det
• m

iX + iπT · ✶N iX † + iπT · ✶N m

• 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 15 / 19

numerical simulations: ρ(λ) from 500 30 × 30 matrices

T = 0

−6 −4 −2 2 4 6 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 2 4 6 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 2 4 6 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:

0.0 0.2 0.4 0.6 0.8 1.0 1.2 TTc 0.2 0.4 0.6 0.8 1.0

ρ(0) = C

• 1 − (πTC)2

vanishes above Tc ≡

1 πC

chiral phase transition, 2nd order ⇒ chiral condensate ρ(0) ∼ ¯ ψψ and its absence at high T “generic”

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 16 / 19

Random matrix theory for dual condensate

general bc.s ϕ ⇒ modified Matsubara frequencies

FB in preparation

ωϕ ≡ min

n |(ϕ + 2πn)T| =

ϕT ϕ ∈ [0, π] (2π − ϕ)T ϕ ∈ [π, 2π] ωπ = πT as before, for other boundary conditions less shifted ...

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 17 / 19

Random matrix theory for dual condensate

general bc.s ϕ ⇒ modified Matsubara frequencies

FB in preparation

ωϕ ≡ min

n |(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 ≡

• 1

ϕC

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 17 / 19

chiral condensate with general bc.s:

1 2 3 4 TTc 2 4 6

• 0.0

0.5 1.0

Σϕ(T) in units of C

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 18 / 19

chiral condensate with general bc.s:

1 2 3 4 TTc 2 4 6

• 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 3.0 TTc 0.5 1.0 1.5 2.0

˜ Σ1(T) plus its T-derivative

changes at the chiral phase transition

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 18 / 19

Summary

the chiral condensate and the chiral phase transition at high T can easily be obtained in Random matrix theory: are ‘generic’ the boundary condition can be incorporated in RMT by virtue of Matsubara frequencies the chiral condensate as a function of the boundary angle agrees qualitatively with results from lattice, functional methods and QCD modelsdummy

Fischer, Müller ’09, Braun et al. ’09, Kashiwa, Kouno, Yahiro ’09, . . .

the dual condensate ˜ Σ1 shows a phase transition at the chiral Tc [but no exact center symmetry . . . ] deconfinement transition ‘generic’ and near the chiral transition

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 19 / 19

Relevant excitations!?

calorons ≡ class. solns. of Yang-Mills (instantons) at finite temperature dummy

Harrington, Shepard ’78; Kraan, van Baal; Lee, Lu ’98

topological (action) density for total charge Q = 1 in SU(3)

substructure: Nc constituents = magn. monopoles/dyons masses governed by asymptotic Polyakov loop P∞ = lim

| x|→∞ P(

x) . . . holonomy conjecture: holonomy tr P∞ ⇌ order parameter tr P ⇒ dyon masses sensitive to the phase of QCD

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 19 / 19

dyon masses sensitive to the phase of QCD, in SU(2) with 2 dyons: confined phase equal mass constituent dyons deconfined phase heavy + light constituent dyon fermionic zero modes: ψϕ≃0 at light dyon, ψϕ≃π at heavy dyon make up condensates in a caloron gas model mechanism above Tc: heavy dyons suppressed

FB ’09

⇒ ¯ ψψϕ≃π suppressed, ¯ ψψϕ≃0 stays

Bornyakov et al. ’09

⇒ top. susceptibility suppressed

Falk Bruckmann The QCD phase transition probed by fermionic boundary conditions 19 / 19