QCD matter with isospin-asymmetry. Gergely Endrődi. Goethe University of Frankfurt in collaboration with Bastian Brandt, Sebastian Schmalzbauer

QCD matter with isospin-asymmetry Gergely Endrődi Goethe University of Frankfurt in collaboration with Bastian Brandt, Sebastian Schmalzbauer SIGN 2017 22. March 2017

Outline introduction: QCD with isospin relevant phenomena pion condensation chiral symmetry breaking λ-extrapolation naive method singular value representation leading-order reweighting results summary 1 / 25

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 / 25

Introduction QCD at low energies pions on the level of charged pions: µ π = 2µ 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 / 25

Setup

Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 4 / 25

Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 spontaneously broken by a pion condensate ψγ 5 τ 1,2 ψ a Goldstone mode appears 4 / 25

Simulation details nx +ny +nz +nt staggered light quark matrix with η 5 = ( 1) ( ) D / M = µ + m λη 5 λη 5 /D µ + m we have γ 5 τ 1 -hermiticity η 5 τ 1 Mτ 1 η 5 = M determinant is real and positive det M = det( /D µ + m 2 + λ 2 ) pioneering studies [Kogut, Sinclair 02] [de Forcrand, Stephanov, Wenger 07] with unimproved action here: N f = 2 + 1 rooted stout-smeared staggered quarks + tree-level Symanzik improved gluons 5 / 25

Condensates: definition and renormalization ψψ = T V log Z m π = T V log Z λ multiplicative renormalization Z π = Z 1 λ = Z 1 m = Z ψψ convenient normalization Σ ψψ m ψψ 1 m 2 πf 2 π 1 Σ π m π mπf 2 π 2 so that in leading-order chiral PT [Son, Stephanov 00] Σ 2 ψψ (µ I ) + Σ 2 π(µ I ) = 1 6 / 25

Condensates: old method

Pion condensate: old method traditional method [Kogut, Sinclair 02] measure full operator at nonzero λ (via noisy estimators) Σ π TrM 1 η 5 τ 2 extrapolation very steep 7 / 25

Chiral condensate: old method traditional method [Kogut, Sinclair 02] measure full operator at nonzero λ (via noisy estimators) Σ ψψ TrM 1 extrapolation very steep 8 / 25

Pion condensate: new method

Singular value representation pion condensate π = λ log det( /D µ + m 2 + λ 2 2λ ) = Tr /D µ + m 2 + λ 2 singular values /D µ + m 2 ψ i = ξi 2 ψ i spectral representation π = T V i 2λ ξi 2 + λ 2 = dξ ρ(ξ) 2λ ξ 2 + λ 2 λ 0 πρ(0) first derived in [Kanazawa, Wettig, Yamamoto 11] 9 / 25

Singular value density spectral densities at λ/m = 0.17 10 / 25

Singular value density spectral densities at λ/m = 0.17 read off improved pion condensate 10 / 25

Reweighting reweighting factor but λ is small, so expand in it: so we obtain R = det( /D µ + m 2 ) det( /D µ + m 2 + λ 2 ) O rew = OR LO R LO R LO = e λv 4π + higher orders in λ 0.22 0.20 0.18 λ = 0. 0075 full LOE R vs. R LO on small lattices for ψψ ψψ 0.16 0.14 0.12 0.10 0.08 0.06 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 µ 11 / 25

Density at zero scaling with λ is improved drastically 12 / 25

Comparison between old and new methods extrapolation in λ gets almost completely flat 13 / 25

New method for other observables

Singular value representation chiral condensate ψψ = singular values m log det( /D µ + m 2 +λ 2 ) = Tr ( /D µ + m) + ( /D µ + m) /D µ + m 2 + λ 2 spectral representation /D µ + m 2 ψ i = ξ 2 i ψ i ψψ = T V i 2 Re ψ i /D µ + m ψ i ξ 2 i + λ 2 14 / 25

Singular value representation spectral representation at λ = 0 ψψ = T V N i=1 2 Re ψ i /D µ + m ψ i ξ 2 i convergence not visible for N 150 15 / 25

Improvement work instead with the difference δ ψψ ψψ(λ = 0) ψψ(λ) = 2T [ N 1 Re ψ i /D V µ +m ψ i ξ 2 i=1 i ] 1 ξi 2 + λ 2 ψψ(0) = ψψ(λ) + δ ψψ convergence already for small N 16 / 25

Improvement work instead with the difference δ ψψ ψψ(λ = 0) ψψ(λ) = 2T [ N 1 Re ψ i /D V µ +m ψ i ξ 2 i=1 i ] 1 ξi 2 + λ 2 ψψ(0) = ψψ(λ) + δ ψψ noisy estimators convergence already for small N 16 / 25

Improvement work instead with the difference δ ψψ ψψ(λ = 0) ψψ(λ) = 2T [ N 1 Re ψ i /D V µ +m ψ i ξ 2 i=1 i ] 1 ξi 2 + λ 2 ψψ(0) = ψψ(λ) + δ ψψ noisy estimators singular values convergence already for small N 16 / 25

Comparison between old and new methods extrapolation in λ gets almost completely flat 17 / 25

Improvement δ ni the same strategy for the isospin density n I = (log Z)/ µ I n I (0) n I (λ) = 2T [ ] N Re ψ i ( /D V µ +m) /D 1 1 µ ψ i ξi 2 + λ 2 i=1 ξ 2 i n I (0) = n I (λ) + δ ni noisy estimators singular values convergence already for small N 18 / 25

Comparison between old and new methods extrapolation in λ gets almost completely flat 19 / 25

Applications for the results

Transition temperature additively renormalized chiral condensate Σ ψψ (T ) Σ ψψ (T = µ I = 0) define T c using the temperature at a constant value (valid at low µ I ) ψψ R [GeV4 ] mr 0-3e-05-6e-05-9e-05-0.00012 preliminary µ I = 0 MeV µ I = 17 MeV µ I = 34 MeV T [MeV] 180 170 160 150 140 130 120 110 crossover (T C, µ I,C ) P (T C, µ I,C ) C Pion condensation preliminary 120 140 160 180 T [MeV] 100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 µ I /m π 20 / 25

Check Taylor-expansion isospin density via Taylor-expansion at µ I = 0 n I (µ I ) = χ I 2 µ I + χ I 4 µ 3 I +... using χ I 2,4 from [BMWc, 1112.4416] ni /T 3 3.5 3 2.5 2 1.5 1 Taylor exp. O(µ I) O(µ 3 I ) simulation 6 24 3 T = 124 MeV m π/2 preliminary ni /T 3 7 6 5 4 3 2 Taylor exp. O(µ I) O(µ 3 I ) simulation preliminary m π/2 6 24 3 0.5 1 T = 176 MeV 0 0 20 40 60 80 100 120 140 µ I [MeV] 0 0 50 100 150 200 250 300 350 µ I [MeV] low T : breakdown of expansion at µ I = m π /2 high T : pin down validity range of LO and NLO expansion 21 / 25

Order of the transition finite volume scaling of pion condensate 22 / 25

Order of the transition fits fit transition region using chiral perturbation theory [Splittorff et al 02, Endrődi 14] 23 / 25

Order of the transition fits fit using O(2) scaling [Ejiri et al 09] 24 / 25

Summary improve observables using singular values of /D µ + m flat λ-extrapolation 7 6 Taylor exp. O(µI) O(µ 3 I ) simulation direct check of Taylor-expansion convergence ni /T 3 5 4 3 2 1 preliminary mπ/2 6 24 3 T = 176 MeV 0 0 50 100 150 200 250 300 350 µi [MeV] phase transition of second order 25 / 25