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

QCD with isospin density: pion condensation Gergely Endr˝ odi, Bastian Brandt Goethe University of Frankfurt Lattice ’16, 28. July 2016

Outline • introduction: QCD with isospin • relevant phenomena ◮ deconfinement/chiral symmetry breaking at low µ I տ next talk ◮ pion condensation at high µ I տ this talk • “ λ -extrapolation” ◮ naive method ◮ new method • outlook and summary 1 / 14

Introduction ◮ isospin density n I = n u − n d ◮ n I < 0 → excess of neutrons over protons → excess of π − over π + ◮ applications ◮ neutron stars ◮ heavy-ion collisions ◮ chemical potentials (3-flavor) µ B = 3 ( µ u + µ d ) / 2 µ I = ( µ u − µ d ) / 2 µ S = 0 2 / 14

Introduction ◮ isospin density n I = n u − n d ◮ n I < 0 → excess of neutrons over protons → excess of π − over π + ◮ applications ◮ neutron stars ◮ heavy-ion collisions ◮ chemical potentials (3-flavor) µ B = 3 ( µ u + µ d ) / 2 µ I = ( µ u − µ d ) / 2 µ S = 0 ◮ here: zero baryon number but nonzero isospin µ u = µ I µ d = − µ I 2 / 14

Introduction ◮ QCD at low energies ≈ pions ◮ on the level of charged pions: µ π = 2 µ I at zero temperature µ π < m π vacuum state µ π = m π Bose-Einstein condensation µ π > m π 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) 3 / 14

Setup

Symmetry breaking ◮ QCD with light quark matrix M = / D + m ud 1 + µ I γ 0 τ 3 + i λγ 5 τ 2 ◮ chiral symmetry (flavor-nontrivial) SU ( 2 ) V → U ( 1 ) τ 3 → ∅ 4 / 14

Symmetry breaking ◮ QCD with light quark matrix M = / D + m ud 1 + µ I γ 0 τ 3 + i λγ 5 τ 2 ◮ chiral symmetry (flavor-nontrivial) SU ( 2 ) V → U ( 1 ) τ 3 → ∅ 4 / 14

Symmetry breaking ◮ QCD with light quark matrix M = / D + m ud 1 + µ I γ 0 τ 3 + i λγ 5 τ 2 ◮ chiral symmetry (flavor-nontrivial) SU ( 2 ) V → U ( 1 ) τ 3 → ∅ ◮ spontaneously broken by a pion condensate � ¯ � ψγ 5 τ 1 , 2 ψ ◮ a Goldstone mode appears 4 / 14

Symmetry breaking ◮ QCD with light quark matrix M = / D + m ud 1 + µ I γ 0 τ 3 + i λγ 5 τ 2 ◮ chiral symmetry (flavor-nontrivial) SU ( 2 ) V → U ( 1 ) τ 3 → ∅ ◮ spontaneously broken by a pion condensate � ¯ � ψγ 5 τ 1 , 2 ψ ◮ a Goldstone mode appears ◮ add small explicit breaking 4 / 14

Symmetry breaking ◮ QCD with light quark matrix M = / D + m ud 1 + µ I γ 0 τ 3 + i λγ 5 τ 2 ◮ chiral symmetry (flavor-nontrivial) SU ( 2 ) V → U ( 1 ) τ 3 → ∅ ◮ spontaneously broken by a pion condensate � ¯ � ψγ 5 τ 1 , 2 ψ ◮ a Goldstone mode appears ◮ add small explicit breaking ◮ extrapolate results λ → 0 4 / 14

Simulation details ◮ staggered light quark matrix with η 5 = ( − 1 ) n x + n y + n z + n t � � / D µ + m ud λη 5 M = / − λη 5 D − µ + m ud ◮ γ 5 τ 1 -hermiticity D † η 5 τ 1 / D µ τ 1 η 5 = / µ → determinant is real and positive ◮ first done by [Kogut, Sinclair ’02] ◮ here: N f = 2 + 1 rooted stout-smeared staggered quarks + tree-level Symanzik improved gluons 5 / 14

Pion condensate: definition and renormalization ◮ condensate � π � = T ∂ log Z V ∂λ ◮ additive divergences cancel in λ → 0 � π � lim ◮ multiplicative renormalization Z π = Z − 1 = Z − 1 λ m ud ◮ renormalization + convenient normalization 1 Σ π ≡ m ud · � π � · m 2 π f 2 π ◮ so that in leading-order chiral PT [Son, Stephanov ’00] Σ 2 ψψ ( µ I ) + Σ 2 π ( µ I ) = 1 ¯ 6 / 14

Pion condensate: old method

Pion condensate: old method ◮ traditional method [Kogut, Sinclair ’02] measure full operator at nonzero λ (via noisy estimators) � � Tr M − 1 η 5 τ 2 Σ π ∝ ◮ extrapolation very ‘steep’ 7 / 14

Pion condensate: old method ◮ traditional method [Kogut, Sinclair ’02] measure full operator at nonzero λ (via noisy estimators) � � Tr M − 1 η 5 τ 2 Σ π ∝ ◮ extrapolation very ‘steep’ 7 / 14

Pion condensate: old method ◮ traditional method [Kogut, Sinclair ’02] measure full operator at nonzero λ (via noisy estimators) � � Tr M − 1 η 5 τ 2 Σ π ∝ ◮ extrapolation very ‘steep’ 7 / 14

Pion condensate: new method

Singular value representation ◮ pion condensate 2 λ π = i Tr ( M − 1 η 5 τ 2 ) = Tr ( / D µ + m ) † ( / D µ + m ) + λ 2 ◮ singular values D µ + m ) † ( / D µ + m ) ψ i = ξ 2 ( / i ψ i ◮ spectral representation 2 λ 2 λ π = T � λ → 0 � i + λ 2 = d ξ ρ ( ξ ) − − − → πρ ( 0 ) ξ 2 + λ 2 ξ 2 V i first derived in [Kanazawa, Wettig, Yamamoto ’11] 8 / 14

Singular value representation ◮ pion condensate 2 λ π = i Tr ( M − 1 η 5 τ 2 ) = Tr ( / D µ + m ) † ( / D µ + m ) + λ 2 ◮ singular values D µ + m ) † ( / D µ + m ) ψ i = ξ 2 ( / i ψ i ◮ spectral representation 2 λ 2 λ π = T � λ → 0 � i + λ 2 = d ξ ρ ( ξ ) − − − → πρ ( 0 ) ξ 2 + λ 2 ξ 2 V i first derived in [Kanazawa, Wettig, Yamamoto ’11] ◮ compare to Banks-Casher-relation at µ I = 0 8 / 14

Singular value density ◮ spectral densities at λ/ m ud = 0 . 17 9 / 14

Density at zero ◮ scaling with λ is improved drastically 10 / 14

Density at zero ◮ scaling with λ is improved drastically 10 / 14

Density at zero ◮ scaling with λ is improved drastically ◮ leading-order reweighting W λ = exp [ − λ V 4 π + O ( λ 2 )] � π � rew = � π W λ � / � W λ � 10 / 14

Comparison between old and new methods ◮ extrapolation in λ gets almost completely flat 11 / 14

Phase boundary ◮ interpolate ρ ( 0 ) as function of µ I to find phase boundary 12 / 14

Phase boundary ◮ interpolate ρ ( 0 ) as function of µ I to find phase boundary 12 / 14

Phase boundary ◮ interpolate ρ ( 0 ) as function of µ I to find phase boundary 12 / 14

Phase boundary ◮ interpolate ρ ( 0 ) as function of µ I to find phase boundary 12 / 14

Phase boundary ◮ interpolate ρ ( 0 ) as function of µ I to find phase boundary T <u d>=0 γ 5 A <u d>=0 γ 5 < >=0 π m π |µ | I ◮ compare to expectations from χ PT [Son, Stephanov ’00] 12 / 14

Phase boundary ◮ interpolate ρ ( 0 ) as function of µ I to find phase boundary T <u d>=0 γ 5 A <u d>=0 γ 5 < >=0 π m π |µ | I ◮ compare to expectations from χ PT [Son, Stephanov ’00] ◮ no pion condensate above T ≈ 160 MeV 12 / 14

Outlook ◮ order of transition? ◮ deconfinement/chiral symmetry breaking transition? ◮ asymptotic- µ I limit? ◮ BCS phase at large µ I ? ◮ analogies to two-color QCD [Holicki, Thu] ◮ test Taylor-expansion in µ I 13 / 14

Outlook ◮ order of transition? ◮ deconfinement/chiral symmetry breaking transition? ◮ asymptotic- µ I limit? ◮ BCS phase at large µ I ? ◮ analogies to two-color QCD [Holicki, Thu] ◮ test Taylor-expansion in µ I ◮ stay for next talk [Brandt, Thu] 13 / 14

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 µ I ◮ chance to test effective theories and low-energy models 14 / 14

Backup

Order of the transition – fits ◮ attempt a fit around µ I = m π / 2 via 1 / 1

Order of the transition – fits ◮ attempt a fit around µ I = m π / 2 via ◮ chiral perturbation theory [Splittorff et al ’02, Endr˝ odi ’14] 1 / 1

Order of the transition – fits ◮ attempt a fit around µ I = m π / 2 via ◮ chiral perturbation theory [Splittorff et al ’02, Endr˝ odi ’14] ◮ O ( 2 ) scaling [Ejiri et al ’09] 1 / 1