QCD with isospin density: pion condensation Gergely Endr odi, - - PowerPoint PPT Presentation

QCD with isospin density: pion condensation Gergely Endr˝

• di, Bastian Brandt

Goethe University of Frankfurt Lattice ’16, 28. July 2016

Outline

• introduction: QCD with isospin
• relevant phenomena

◮ deconfinement/chiral symmetry breaking at low µI

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◮ pion condensation at high µI

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• “λ-extrapolation”

◮ naive method ◮ new method

• outlook and summary

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Introduction

◮ isospin density nI = nu − nd ◮ nI < 0 → excess of neutrons over protons

→ excess of π− over π+

◮ applications

◮ neutron stars ◮ heavy-ion collisions

◮ chemical potentials (3-flavor)

µB = 3(µu + µd)/2 µI = (µu − µd)/2 µS = 0

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Introduction

◮ isospin density nI = nu − nd ◮ nI < 0 → excess of neutrons over protons

→ excess of π− over π+

◮ applications

◮ neutron stars ◮ heavy-ion collisions

◮ chemical potentials (3-flavor)

µB = 3(µu + µd)/2 µI = (µu − µd)/2 µS = 0

◮ here: zero baryon number but nonzero isospin

µu = µI µd = −µI

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Introduction

◮ QCD at low energies ≈ pions ◮ on the level of charged pions: µπ = 2µI

at zero temperature µπ < mπ vacuum state µπ = mπ Bose-Einstein condensation µπ > mπ undefined

◮ on the level of quarks: lattice simulations

◮ no sign problem ◮ conceptual analogies to baryon density

(Silver Blaze, hadron creation, saturation)

◮ technical similarities

(proliferation of low eigenvalues)

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Setup

Symmetry breaking

◮ QCD with light quark matrix

M = / D + mud1 + µIγ0τ3 + iλγ5τ2

◮ chiral symmetry (flavor-nontrivial)

SU(2)V → U(1)τ3 → ∅

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Symmetry breaking

◮ QCD with light quark matrix

M = / D + mud1 + µIγ0τ3 + iλγ5τ2

◮ chiral symmetry (flavor-nontrivial)

SU(2)V → U(1)τ3 → ∅

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Symmetry breaking

◮ QCD with light quark matrix

M = / D + mud1 + µIγ0τ3 + iλγ5τ2

◮ chiral symmetry (flavor-nontrivial)

SU(2)V → U(1)τ3 → ∅

◮ spontaneously broken by a pion

condensate

¯

ψγ5τ1,2ψ

• ◮ a Goldstone mode appears

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Symmetry breaking

◮ QCD with light quark matrix

M = / D + mud1 + µIγ0τ3 + iλγ5τ2

◮ chiral symmetry (flavor-nontrivial)

SU(2)V → U(1)τ3 → ∅

◮ spontaneously broken by a pion

condensate

¯

ψγ5τ1,2ψ

• ◮ a Goldstone mode appears

◮ add small explicit breaking

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Symmetry breaking

◮ QCD with light quark matrix

M = / D + mud1 + µIγ0τ3 + iλγ5τ2

◮ chiral symmetry (flavor-nontrivial)

SU(2)V → U(1)τ3 → ∅

◮ spontaneously broken by a pion

condensate

¯

ψγ5τ1,2ψ

• ◮ a Goldstone mode appears

◮ add small explicit breaking ◮ extrapolate results λ → 0

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Simulation details

◮ staggered light quark matrix with η5 = (−1)nx+ny+nz+nt

M =

• /

Dµ + mud λη5 −λη5 / D−µ + mud

• ◮ γ5τ1-hermiticity

η5τ1 / Dµτ1η5 = / D†

µ

→ determinant is real and positive

◮ first done by [Kogut, Sinclair ’02] ◮ here: Nf = 2 + 1 rooted stout-smeared staggered quarks +

tree-level Symanzik improved gluons

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Pion condensate: definition and renormalization

◮ condensate

π = T V ∂ log Z ∂λ

◮ additive divergences cancel in

lim

λ→0 π ◮ multiplicative renormalization

Zπ = Z −1

λ

= Z −1

mud ◮ renormalization + convenient normalization

Σπ ≡ mud · π · 1 m2

πf 2 π ◮ so that in leading-order chiral PT [Son, Stephanov ’00]

Σ2

¯ ψψ(µI) + Σ2 π(µI) = 1

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Pion condensate: old method

Pion condensate: old method

◮ traditional method [Kogut, Sinclair ’02]

measure full operator at nonzero λ (via noisy estimators) Σπ ∝

• TrM−1η5τ2
• ◮ extrapolation very ‘steep’

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Pion condensate: old method

◮ traditional method [Kogut, Sinclair ’02]

measure full operator at nonzero λ (via noisy estimators) Σπ ∝

• TrM−1η5τ2
• ◮ extrapolation very ‘steep’

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Pion condensate: old method

◮ traditional method [Kogut, Sinclair ’02]

measure full operator at nonzero λ (via noisy estimators) Σπ ∝

• TrM−1η5τ2
• ◮ extrapolation very ‘steep’

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Pion condensate: new method

Singular value representation

◮ pion condensate

π = iTr(M−1η5τ2) = Tr 2λ ( / Dµ + m)†( / Dµ + m) + λ2

◮ singular values

( / Dµ + m)†( / Dµ + m) ψi = ξ2

i ψi ◮ spectral representation

π = T V

• i

2λ ξ2

i + λ2 =

• dξ ρ(ξ)

2λ ξ2 + λ2

λ→0

− − − → πρ(0) first derived in [Kanazawa, Wettig, Yamamoto ’11]

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Singular value representation

◮ pion condensate

π = iTr(M−1η5τ2) = Tr 2λ ( / Dµ + m)†( / Dµ + m) + λ2

◮ singular values

( / Dµ + m)†( / Dµ + m) ψi = ξ2

i ψi ◮ spectral representation

π = T V

• i

2λ ξ2

i + λ2 =

• dξ ρ(ξ)

2λ ξ2 + λ2

λ→0

− − − → πρ(0) first derived in [Kanazawa, Wettig, Yamamoto ’11]

◮ compare to Banks-Casher-relation at µI = 0

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Singular value density

◮ spectral densities at λ/mud = 0.17

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Density at zero

◮ scaling with λ is improved drastically

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Density at zero

◮ scaling with λ is improved drastically

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Density at zero

◮ scaling with λ is improved drastically ◮ leading-order reweighting

πrew = πWλ / Wλ Wλ = exp[−λV4π + O(λ2)]

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Comparison between old and new methods

◮ extrapolation in λ gets almost completely flat

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Phase boundary

◮ interpolate ρ(0) as function of µI to find phase boundary

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Phase boundary

◮ interpolate ρ(0) as function of µI to find phase boundary

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Phase boundary

◮ interpolate ρ(0) as function of µI to find phase boundary

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Phase boundary

◮ interpolate ρ(0) as function of µI to find phase boundary

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Phase boundary

◮ interpolate ρ(0) as function of µI to find phase boundary

<u d>=0 γ5 <u d>=0 γ5 π < >=0 mπ T |µ |

I

A

◮ compare to expectations from χPT [Son, Stephanov ’00]

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Phase boundary

◮ interpolate ρ(0) as function of µI to find phase boundary

<u d>=0 γ5 <u d>=0 γ5 π < >=0 mπ T |µ |

I

A

◮ compare to expectations from χPT [Son, Stephanov ’00] ◮ no pion condensate above T ≈ 160 MeV

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Outlook

◮ order of transition? ◮ deconfinement/chiral symmetry breaking transition? ◮ asymptotic-µI limit? ◮ BCS phase at large µI? ◮ analogies to two-color QCD [Holicki, Thu] ◮ test Taylor-expansion in µI

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Outlook

◮ order of transition? ◮ deconfinement/chiral symmetry breaking transition? ◮ asymptotic-µI limit? ◮ BCS phase at large µI? ◮ analogies to two-color QCD [Holicki, Thu] ◮ test Taylor-expansion in µI ◮ stay for next talk [Brandt, Thu]

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Summary

◮ QCD with isospin chemical potentials via lattice simulations

at the physical point

◮ determine pion condensate via

Banks-Casher-type relation flat extrapolation in pion source

◮ phase boundary surprisingly flat

for intermediate µI

◮ chance to test effective theories

and low-energy models

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Backup

Order of the transition – fits

◮ attempt a fit around µI = mπ/2 via

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Order of the transition – fits

◮ attempt a fit around µI = mπ/2 via

◮ chiral perturbation theory [Splittorff et al ’02, Endr˝

• di ’14]

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Order of the transition – fits

◮ attempt a fit around µI = mπ/2 via

◮ chiral perturbation theory [Splittorff et al ’02, Endr˝

• di ’14]

◮ O(2) scaling [Ejiri et al ’09]

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