Inverse magnetic catalysis and the Polyakovloop

Tamas G. KovacsInstitute of Nuclear Research, Debrecen

with

Falk Bruckmann and Gergely EndrodiUniversitat Regensburg

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Is there always magnetic catalysis?

Common wisdom:chiral condensate always increases with the magnetic field

First lattice study → confirms D’Elia et al. PRD (2010)

Larger than physical quark mass

Coarse lattice, one lattice spacing

But! Lattice → Around Tc condensate can decreaseBali et al. PRD (2012)

Physical quark masses

Continuum limit

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Dependence of the condensate on the magnetic field

Bali et al. PRD (2012)

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Goal of the present work

Understand the physics of what is happening at Tc

Two competing mechanisms at Tc

Back reaction of light quarks on the gluon field

Delicate effect

Dependence on quark mass

Continuum limit

Role of the Polyakov loop

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What happens if B is switched on around Tc?

Quark condensate:

〈ψψ〉(B)=1Z

∫

dA e−S(A) det[D(A,B)+m]︸ ︷︷ ︸

“sea”

Tr[

(D(A,B)+m)−1]

︸ ︷︷ ︸

“valence”

Polyakov loop

〈P〉=1Z

∫

dA e−S(A)det[D(A,B)+m] P(A)

Why does the condensate depend on B?

Spectrum of D(A,B) in given gauge backgound changes → “valence”

Typical gauge field A changes → “sea”

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Change of Dirac spectral density with BT = 142 MeV, Nt = 6, generated with B = 0

→ spectral density around zero increases with B

→ 〈ψψ〉 with small quark mass increases (for m → 0 ρ(0) ≈ 〈ψψ〉 )

→ magnetic catalysis

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“Sea” and “valence” contribution to the condensate

∆Σ=〈ψψ(B,T )〉− 〈ψψ(0,T )〉

〈ψψ(0,0)〉

valence

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“Sea” and “valence” contribution to the condensate

∆Σ=〈ψψ(B,T )〉− 〈ψψ(0,T )〉

〈ψψ(0,0)〉

valence sea

→ at Tc sea and valence are competing

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How does B influence the gauge fields?

Back reaction of quarks on the gluon field

What happens to the gauge field if B is switched on?

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How does B influence the gauge fields?

Back reaction of quarks on the gluon field

What happens to the gauge field if B is switched on?

Polyakov loop increases (gets more ordered)

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Why does the Polyakov loop increase with B?

Quark action:

Sq =− logdet(D+m) =− ∑Imλi>0

log(

λ 2i +m2

)

m small→ fluctuations of Sq dominated by small eigenvalues

Quark action suppresses small Dirac eigenvalues

Few small eigenvalues ⇔ large Polyakov loop

T ≪ Tc → P ≈ 0 → many small modes (sχSB)

T ≫ Tc → P ≈ 1 → λ1 ∝ T (lowest Matsubara mode)

Quark action prefers large Polyakov loop;switching on B enhances this effect

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Change in action versus the Polyakov loopScatter plot of 104 ×4 lattice configurations around Tc , generated with B = 0

∆Sq = logdet(D(0,A)+m)− logdet(D(B,A)+m)

85 90 95 100 1050.1

0.2

0.3

0.4

0.5

0.6

simple average (B=0)reweighted to B

∆Sq → exponentially suppressed by B

P

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The Polyakov loop and inverse catalysis

Condensate:ψψ = tr(D+m)−1 ≈ ∑

Imλi>0

mλ 2

i +m2

m small → dominated by small eigenvalues

P large → fewer small eigenvalues

→ smaller condensate

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Polyakov loop versus quark condensateScatter plot of 104 ×4 lattice configurations around Tc

0.1 0.2 0.3 0.4 0.5 0.60

100

200

300

400

ψψ

P

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How does inverse catalysis work?

Magnetic field → more small Dirac eigenvalues

Valence

More small modes → larger condensate

Sea

→ quark action enhances Polyakov loop

→ fewer small Dirac eigenvalues

→ quark condensate decreases

“Sea” and “valence” compete

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Why inverse catalysis around Tc?

Around Tc “sea” suppression wins. Why?

Tc cross-over — order parameter Polyakov loop

At Tc Polyakov loop effective potential flat

Small magnetic field contribution has large ordering effect

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Conclusions

Both catalysis and inverse catalysisdepend on small quark modes

Inverse catalysis:suppression of small Dirac ev’s by B in the quark det

Contribution of B in det to the P-loop effective potential

Small modes → sensitive to quark mass

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