Solitonic ground states in (color) superconductivity

Transcription

1 Solitonic ground states in (color) superconductivity Michael Buballa (TU Darmstadt), Dominik Nickel (MIT) EMMI workshop Quark-Gluon Plasma meets Cold Atoms - Episode II, August 3-8, 29, Riezlern, Austria

2 Strongly Interacting Matter under Extreme Conditions International Workshop XXXVIII on Gross Properties of Nuclei and Nuclear excitations Waldemar Petersen Haus, Hirschegg, Kleinwalsertal, Austria January 17 23, 21 Hirschegg 21: Topics: QCD phase diagram critical end point and critical fluctations transport properties of strongly interacting matter hadronic and electromagnetic signals heavy ion collisions at the highest energies Strongly Interacting Matter under Extreme Conditions Organizers: P. Braun Munzinger, B. Frimann, K. Langanke, J. Wambach (coordinator) Program advisors: F. Antinori (CERN), A. Drees (Stony Brook), R. Rapp (Texas A & M), D. Rischke (Frankfurt), U. Wiedemann (CERN), Nu Xu (LBNL Berkeley) Invited speakers: A. Andronic (Darmstadt), H. Appelshäuser (Frankfurt), R. Arnaldi (Torino), J. P. Blaizot (Gif sur Yvette), K. Eskola (Jyväskylä), W. Florkowski (Cracow), V. Greco (Catania), H. van Hees (Gießen), T. Hemmick (Stony Brook), C. Höhne (Darmstadt), A. Kisiel (Warsaw), E. Laermann (Bielefeld), D. Miskowiec (Darmstadt), J. Y. Ollitrault (Gif sur Yvette), M. Ploskon (Berkeley), K. Redlich (Wroclaw), B. J. Schaefer (Graz), R. Snellings (Amsterdam), P. Sorensen (Upton), J. Stachel (Heidelberg), G. Usai (Monserrato) Further information: darmstadt.de/hirschegg/ Registration deadline: October 31, 29

3 Motivation We have already heard about: imbalanced Fermi liquids (M. Zwierlein ) asymmetric nuclear matter (X.-G. Huang ) FFLO in one-dimensional optical lattices (R. Bakhtiari ) color superconductivity (A. Schmitt )

4 Motivation We have already heard about: imbalanced Fermi liquids (M. Zwierlein ) asymmetric nuclear matter (X.-G. Huang ) FFLO in one-dimensional optical lattices (R. Bakhtiari ) color superconductivity (A. Schmitt ) This talk is about: LOFF (= FFLO) phases in color superconducting asymmetric quark matter T M. Alford (23) RHIC QGP = color superconducting crystal? gas liq CFL nuclear compact star µ based on: D. Nickel, M.B., Phys. Rev. D79, 549 (29) [arxiv:811.24].

5 Outline 1 Introduction imbalanced pairing in CSC 2 Model NJL model for inhomogeneous pairing 3 Numerical Results one-dimensional solutions 4 Outlook

6 Color superconductivity diquark condensates: q T O q Pauli principle: O = O spin O color O flavor = totally antisymmetric most attractive channel: spin (= antisymmetric) color 3 (= antisymmetric) antisymmetric in flavor pairing between unequal flavors example : q T Cγ 5 τ 2 λ 2 q ( ) (r g g r) (ud du)

7 Neutral quark matter constraints in compact stars: color neutrality: n r = n g = n b electric neutrality: n Q = 2 3 n u 1 3 n d 1 3 n s n e = β equilibrium: µ e = µ d µ u n e n u,d

8 Neutral quark matter constraints in compact stars: color neutrality: (minor n r effect) = n g = n b } electric neutrality: n Q = 2 3 n u 1 3 n d 1 3 n s n e = 2 3 n u 1 3 n d 1 3 n s β equilibrium: µ e = µ d µ u n e n u,d

9 Neutral quark matter constraints in compact stars: color neutrality: (minor n r effect) = n g = n b } electric neutrality: n Q = 2 3 n u 1 3 n d 1 3 n s n e = 2 3 n u 1 3 n d 1 3 n s β equilibrium: µ e = µ d µ u n e n u,d all flavors have different Fermi momenta imbalanced pairing unavoidable (Rajagopal & Schmitt, PRD (26))

10 BCS pairing Cooper instability: Fermi gas free energy E - µ p F

11 BCS pairing Cooper instability: Fermi gas + attraction condensation of Cooper pairs free energy E - µ p F

12 BCS pairing Cooper instability: Fermi gas + attraction condensation of Cooper pairs reorganisation of the Fermi surface gaps free energy p F

13 BCS pairing Cooper instability: Fermi gas + attraction condensation of Cooper pairs reorganisation of the Fermi surface gaps Cooper pairs in BCS theory: close to the Fermi surface opposite momenta free energy p F

14 BCS pairing Cooper instability: Fermi gas + attraction condensation of Cooper pairs reorganisation of the Fermi surface gaps Cooper pairs in BCS theory: close to the Fermi surface opposite momenta free energy p F works only if p a F = pb F

15 Imbalanced pairing unequal Fermi momenta: p a,b F = p F ± δp F

16 Imbalanced pairing unequal Fermi momenta: p a,b F = p F ± δp F option 1: still BCS-like pairing

17 Imbalanced pairing unequal Fermi momenta: p a,b F = p F ± δp F option 1: still BCS-like pairing first equalize Fermi momenta

18 Imbalanced pairing unequal Fermi momenta: p a,b F = p F ± δp F option 1: still BCS-like pairing first equalize Fermi momenta

19 Imbalanced pairing unequal Fermi momenta: p a,b F = p F ± δp F option 1: still BCS-like pairing first equalize Fermi momenta then pair

20 Imbalanced pairing unequal Fermi momenta: p a,b F = p F ± δp F option 1: still BCS-like pairing first equalize Fermi momenta then pair

21 Imbalanced pairing unequal Fermi momenta: p a,b F = p F ± δp F option 1: still BCS-like pairing first equalize Fermi momenta then pair favored if δp F 2 (Chandrasekhar, Clogston (1962))

22 Imbalanced pairing unequal Fermi momenta: p a,b F = p F ± δp F option 1: still BCS-like pairing first equalize Fermi momenta then pair favored if δp F 2 (Chandrasekhar, Clogston (1962)) option 2: phase separation (globally neutral mixed phases) Coulomb and surface effects? pattern sizes?

23 Imbalanced pairing unequal Fermi momenta: p a,b F = p F ± δp F option 1: still BCS-like pairing first equalize Fermi momenta then pair favored if δp F 2 (Chandrasekhar, Clogston (1962)) option 2: phase separation (globally neutral mixed phases) Coulomb and surface effects? pattern sizes? option 3: gapless phases (= Sarma phases = breached pairing) chromomagnetic instabilities

24 Inhomogeneous phases option 4: pairs with nonzero total momentum p a F pb F no problem

25 Inhomogeneous phases option 4: pairs with nonzero total momentum p a F pb F no problem FF (Fulde, Ferrell, 1964): single plane wave q( x)q( x) e 2i q x for fixed q disfavored by phase space

26 Inhomogeneous phases option 4: pairs with nonzero total momentum p a F pb F no problem FF (Fulde, Ferrell, 1964): single plane wave q( x)q( x) e 2i q x for fixed q disfavored by phase space LO (Larkin, Ovchinnikov, 1964): multiple plane waves (e.g., cos(2 q x)) Ginzburg-Landau expansion

27 Inhomogeneous phases in color superconductivity CSC literature so far: single plane wave (Alford, Bowers, Rajagopal, 21)

28 Inhomogeneous phases in color superconductivity CSC literature so far:.4 X a.2 single plane wave (Alford, Bowers, Rajagopal, 21) superposition of several plane waves Y a.2.4 (e.g., Bowers, Rajagopal, 22; Casalbuoni et al., ) different directions, but equal wave lengths mostly Ginzburg-Landau.4.2 (Rajagopal, Sharma, 26) Z a

29 Inhomogeneous phases in color superconductivity CSC literature so far:.4 X a.2 single plane wave (Alford, Bowers, Rajagopal, 21) superposition of several plane waves Y a.2.4 (e.g., Bowers, Rajagopal, 22; Casalbuoni et al., ) different directions, but equal wave lengths mostly Ginzburg-Landau.4.2 (Rajagopal, Sharma, 26) Z a aim of our work: NJL model for inhomogeneous pairing beyond Ginzburg-Landau superimpose different wave lengths

30 Model Model Lagrangian with NJL-type qq interaction: L = q (i / + ˆµγ ) q + L int L int = H ( q iγ 5 τ A λ A q c )( q c iγ 5 τ A λ A q), A,A =2,5,7 q c = C q T

31 Model Model Lagrangian with NJL-type qq interaction: L = q (i / + ˆµγ ) q + L int L int = H ( q iγ 5 τ A λ A q c )( q c iγ 5 τ A λ A q), A,A =2,5,7 q c = C q T bosonize: ϕ AA (x) = 2H q c (x) γ 5 τ A λ A q(x) L int = 1 { ( q γ 5 τ A λ A q c ) ϕ AA + h.c. 1 } 2 2H ϕ AA ϕ AA A,A

32 Model Model Lagrangian with NJL-type qq interaction: L = q (i / + ˆµγ ) q + L int L int = H ( q iγ 5 τ A λ A q c )( q c iγ 5 τ A λ A q), A,A =2,5,7 q c = C q T bosonize: ϕ AA (x) = 2H q c (x) γ 5 τ A λ A q(x) L int = 1 { ( q γ 5 τ A λ A q c ) ϕ AA + h.c. 1 } 2 2H ϕ AA ϕ AA A,A mean-field approximation: ϕ AA (x) = A (x) δ AA, ϕ AA (x) = A(x) δ AA A (x) classical fields retain space-time dependence!

33 Mean-field model Lagrangian: L MF (x) = Ψ(x) S 1 (x) Ψ(x) 1 4H Nambu-Gor kov bispinors: A (x) 2 ) A Ψ(x) = 1 2 ( q(x) q c (x) inverse dressed propagator: ( ) i / + ˆµγ S 1 ˆ (x) γ5 (x) = ˆ, ˆ (x) := (x) γ 5 i / ˆµγ A (x) τ A λ A A

34 Mean-field model Lagrangian: L MF (x) = Ψ(x) S 1 (x) Ψ(x) 1 4H Nambu-Gor kov bispinors: A (x) 2 ) A Ψ(x) = 1 2 ( q(x) q c (x) inverse dressed propagator: ( ) i / + ˆµγ S 1 ˆ (x) γ5 (x) = ˆ, ˆ (x) := (x) γ 5 i / ˆµγ A (x) τ A λ A Thermodynamic potential: Ω MF (T, µ) = 1 T S 1 Tr ln 2 V T + T V A [, 1 T ] V d A 4 x A(x) 2 4H Tr ln S 1 nontrivial because of x-dependent gap functions!

35 Crystalline ansatz gap functions: time-independent, periodic in space ˆ (x) ˆ ( x) = ˆ ( x + a i ), i = 1, 2, 3

36 Crystalline ansatz gap functions: time-independent, periodic in space ˆ (x) ˆ ( x) = ˆ ( x + a i ), i = 1, 2, 3 Fourier decomposition: ˆ (x) = ( ) ˆ qk e iqk x, q k =, q k q k q k a i 2π Z

37 Crystalline ansatz gap functions: time-independent, periodic in space ˆ (x) ˆ ( x) = ˆ ( x + a i ), i = 1, 2, 3 Fourier decomposition: ˆ (x) = ( ) ˆ qk e iqk x, q k =, q k q k q k a i 2π Z inverse propagator: (p/ n + ˆµγ ) δ pm,p n ˆ qk δ qk,p m p n γ 5 Sp 1 q m,p n = k ˆ qk δ qk,p n p m γ 5 (p/ n ˆµγ ) δ pm,p n q k

38 Crystalline ansatz gap functions: time-independent, periodic in space ˆ (x) ˆ ( x) = ˆ ( x + a i ), i = 1, 2, 3 Fourier decomposition: ˆ (x) = ( ) ˆ qk e iqk x, q k =, q k q k q k a i 2π Z inverse propagator: (p/ n + ˆµγ ) δ pm,p n ˆ qk δ qk,p m p n γ 5 Sp 1 q m,p n = k ˆ qk δ qk,p n p m γ 5 (p/ n ˆµγ ) δ pm,p n q k condensates couple different momenta! q k qk p m p n

39 Crystalline ansatz gap functions: time-independent, periodic in space ˆ (x) ˆ ( x) = ˆ ( x + a i ), i = 1, 2, 3 Fourier decomposition: ˆ (x) = ( ) ˆ qk e iqk x, q k =, q k q k q k a i 2π Z inverse propagator: (p/ n + ˆµγ ) δ pm,p n ˆ qk δ qk,p m p n γ 5 Sp 1 q m,p n = k ˆ qk δ qk,p n p m γ 5 (p/ n ˆµγ ) δ pm,p n q k condensates couple different momenta! q k qk p m p n

40 Crystalline ansatz gap functions: time-independent, periodic in space ˆ (x) ˆ ( x) = ˆ ( x + a i ), i = 1, 2, 3 Fourier decomposition: ˆ (x) = ( ) ˆ qk e iqk x, q k =, q k q k q k a i 2π Z inverse propagator: (p/ n + ˆµγ ) δ pm,p n ˆ qk δ qk,p m p n γ 5 Sp 1 q m,p n = k ˆ qk δ qk,p n p m γ 5 (p/ n ˆµγ ) δ pm,p n q k condensates couple different momenta! q k diagonal in energy Matsubara sum as usual qk p m p n

41 Hamiltonian The inverse propagator can be put into the form S 1 p m,p n = γ (iω pn H pm, p n ) δ ωpm,ω pn H = effective Hamilton operator ω n independent not diagonal in 3-momentum hermitean can be diagonalized (in principle)

42 Hamiltonian The inverse propagator can be put into the form S 1 p m,p n = γ (iω pn H pm, p n ) δ ωpm,ω pn H = effective Hamilton operator ω n independent not diagonal in 3-momentum hermitean can be diagonalized (in principle) thermodynamic potential: Ω MF = 1 [ 4V Eλ + 2T ln ( 1 + 2e E λ/t )] + A E λ : λ eigenvalues of H q k A,qk 2 4H

43 Thermodynamic potential remaining problem: diagonalize H (γ p/ n ˆµ) δ pm, p n ˆ qk γ γ 5 δ qk, p m p n q H pm, p n = k q k ˆ qk γ γ 5 δ qk, p n p m (γ p/ n + ˆµ) δ pm, p n

44 Thermodynamic potential remaining problem: diagonalize H (γ p/ n ˆµ) δ pm, p n ˆ qk γ γ 5 δ qk, p m p n q H pm, p n = k q k ˆ qk γ γ 5 δ qk, p n p m (γ p/ n + ˆµ) δ pm, p n q k form a discrete reciprocal lattice H is block diagonal in momentum space (one block H( k) for each vector k in the Brioullin zone

45 Thermodynamic potential remaining problem: diagonalize H (γ p/ n ˆµ) δ pm, p n ˆ qk γ γ 5 δ qk, p m p n q H pm, p n = k q k ˆ qk γ γ 5 δ qk, p n p m (γ p/ n + ˆµ) δ pm, p n q k form a discrete reciprocal lattice H is block diagonal in momentum space (one block H( k) for each vector k in the Brioullin zone We finally obtain: Ω MF = 1 4 d 3 k (2π) 3 B.Z. λ [ ( )] E λ ( k) + 2T ln 1 + 2e E λ( k)/t + A q k A,qk 2 4H E λ ( k): eigenvalues of H( k).

46 Simplified model consider only 2SC-like pairing strange quarks and blue quarks decouple

47 Simplified model consider only 2SC-like pairing strange quarks and blue quarks decouple chemical potentials: µ u = µ + δµ, µ d = µ δµ

48 Simplified model consider only 2SC-like pairing strange quarks and blue quarks decouple chemical potentials: µ u = µ + δµ, µ d = µ δµ high-density approximations to simplify Dirac structure

49 Simplified model consider only 2SC-like pairing strange quarks and blue quarks decouple chemical potentials: µ u = µ + δµ, µ d = µ δµ high-density approximations to simplify Dirac structure remaining diagonalization problem: ( ) (pm µ δµ) δ pm, p (H,δµ ) pm, p n = n pm p n (p m µ + δµ) δ pm, p n p n p m

50 Regularization unregularized expression for Ω MF divergent needs to be regularized 3-momentum cutoff? inhom. phases: H depends on two momenta! cut off both of them, e.g., p m, p n Λ? strong regularization artifacts: large q suppressed, e.g., q < 2Λ violates model independent low-energy results Pauli-Villars-like scheme: F(E λ ) 2 F(E λ,j ), E λ,j ( k) = j= E 2 λ ( k) + jλ 2

51 Numerical results external parameters: T, µ, δµ here: T =, µ = 4 MeV only δµ is varied

52 Numerical results external parameters: T, µ, δµ here: T =, µ = 4 MeV only δµ is varied model parameters: H, Λ Λ = 4 MeV H BCS = 8 MeV

53 Numerical results external parameters: T, µ, δµ here: T =, µ = 4 MeV only δµ is varied model parameters: H, Λ Λ = 4 MeV H BCS = 8 MeV crystal structure: general problem (too) difficult consider one-dimensional modulations (in 3+1 D): (z) = k k e 2ikqz further restriction: (z) = real k = k

54 step 1: minimize Ω at fixed q

55 step 1: minimize Ω at fixed q example: δµ =.7 BCS (z) [MeV] 5 q=.1 BCS q=.2 BCS q=.5 BCS q= 1. BCS z [π/ BCS ]

56 step 1: minimize Ω at fixed q example: δµ =.7 BCS (z) [MeV] 5 q=.1 BCS q=.2 BCS q=.5 BCS q= 1. BCS q.5 BCS : sinusoidal z [π/ BCS ]

57 step 1: minimize Ω at fixed q example: δµ =.7 BCS (z) [MeV] 5-5 q=.1 BCS q=.2 BCS q=.5 BCS q= 1. BCS q.5 BCS : sinusoidal q.5 BCS : soliton lattice shape of transition region almost independent of q z [π/ BCS ] amplitude BCS

58 step 1: minimize Ω at fixed q example: δµ =.7 BCS (z) [MeV] 5-5 q=.1 BCS q=.2 BCS q=.5 BCS q= 1. BCS q.5 BCS : sinusoidal q.5 BCS : soliton lattice shape of transition region almost independent of q z [π/ BCS ] amplitude BCS parametrization of the gap function: (z) = A sn ( κ(z z ); ν ) (works extremely well)

59 step 2: minimize Ω w.r.t. q

60 step 2: minimize Ω w.r.t. q 1 preferred q:.8 q/ BCS.6.4 q 1D q FF δµ/ BCS inhom. - BCS: q 2nd order (FF - BCS: 1st order)!

61 step 2: minimize Ω w.r.t. q preferred q: amplitude: q/ BCS.6.4 q 1D q FF qmax () [MeV] 6 4 1D FF δµ/ BCS δµ/ BCS inhom. - BCS: q 2nd order (FF - BCS: 1st order)! inhom. - normal: 1st order (FF - normal: 2nd order)!

62 step 2: minimize Ω w.r.t. q preferred q: amplitude: q/ BCS.6.4 q 1D q FF qmax () [MeV] 6 4 1D FF δµ/ BCS δµ/ BCS inhom. - BCS: q 2nd order (FF - BCS: 1st order)! inhom. - normal: 1st order (FF - normal: 2nd order)! inhom. FF

63 Gap functions putting everything together: amplitude: 1 (z) [MeV] δµ/ BCS =.695 qmax () [MeV] D FF δµ/ BCS preferred q: z [fm] q/ BCS q 1D q FF δµ/ BCS

64 Gap functions putting everything together: amplitude: 1 (z) [MeV] δµ/ BCS = qmax () [MeV] D FF δµ/ BCS preferred q: z [fm] q/ BCS q 1D q FF δµ/ BCS

65 Gap functions putting everything together: amplitude: 1 (z) [MeV] δµ/ BCS =.696 qmax () [MeV] D FF δµ/ BCS preferred q: z [fm] q/ BCS q 1D q FF δµ/ BCS

66 Gap functions putting everything together: amplitude: 1 (z) [MeV] δµ/ BCS =.7 qmax () [MeV] D FF δµ/ BCS preferred q: z [fm] q/ BCS q 1D q FF δµ/ BCS

67 Gap functions putting everything together: amplitude: 1 (z) [MeV] δµ/ BCS =.71 qmax () [MeV] D FF δµ/ BCS preferred q: z [fm] q/ BCS q 1D q FF δµ/ BCS

68 Gap functions putting everything together: amplitude: 1 (z) [MeV] δµ/ BCS =.72 qmax () [MeV] D FF δµ/ BCS preferred q: z [fm] q/ BCS q 1D q FF δµ/ BCS

69 Gap functions putting everything together: amplitude: 1 (z) [MeV] δµ/ BCS =.74 qmax () [MeV] D FF δµ/ BCS preferred q: z [fm] q/ BCS q 1D q FF δµ/ BCS

70 Gap functions putting everything together: amplitude: 1 (z) [MeV] δµ/ BCS =.76 qmax () [MeV] D FF δµ/ BCS preferred q: z [fm] q/ BCS q 1D q FF δµ/ BCS

71 Gap functions putting everything together: amplitude: 1 (z) [MeV] δµ/ BCS =.78 qmax () [MeV] D FF δµ/ BCS preferred q: z [fm] q/ BCS q 1D q FF δµ/ BCS

72 Gap functions putting everything together: amplitude: 1 (z) [MeV] δµ/ BCS =.79 qmax () [MeV] D FF δµ/ BCS preferred q: z [fm] q/ BCS q 1D q FF δµ/ BCS

73 General one-dimensional solutions free-energy gain: 1e+7 δω [MeV 4 ] -1e+7-2e+7-3e+7 δω 1D δω FF δω BCS δµ/ BCS

74 General one-dimensional solutions free-energy gain: 1e+7 δω [MeV 4 ] -1e+7-2e+7-3e+7 δω 1D δω FF δω BCS LO window 2 FF window δµ/ BCS

75 Quasiparticle spectra anisotropic dispersion relations: E λ ( k) = E λ ( k, k z )

76 Quasiparticle spectra anisotropic dispersion relations: E λ ( k) = E λ ( k, k z ) typical examples at fixed k :

77 Band structure superposition of the eigenvalue spectra of all k z : sinusoidal (q =.9 BCS ) soliton lattice (q =.2 BCS ) E(k) [MeV] E(k) [MeV] k [MeV] k [MeV] almost gapped regions between low- and high-lying modes low-lying modes related to solitons: q E

78 Outlook I to be done: globally neutral two-flavor quark matter in beta equilibrium density profiles (example: A. Bulgac, M.M. Forbes, PRL (28), Density Functional Theory calculation for polarized Fermi systems at unitarity) n(z)/n Delta(z) three flavors (including masses) finite temperature, phase diagram two- and three-dimensional crystals -5 5 z/l

79 Chiral phase transition (D. Nickel, arxiv: , arxiv: ) constituent quark mass M(z) with 1 dimensional modulations

80 Chiral phase transition (D. Nickel, arxiv: , arxiv: ) constituent quark mass M(z) with 1 dimensional modulations analytically known solutions for 1+1 D can be lifted to 3+1 D

81 Chiral phase transition (D. Nickel, arxiv: , arxiv: ) constituent quark mass M(z) with 1 dimensional modulations analytically known solutions for 1+1 D can be lifted to 3+1 D phase diagram: 15 T MeV Μ q MeV

82 Chiral phase transition (D. Nickel, arxiv: , arxiv: ) constituent quark mass M(z) with 1 dimensional modulations analytically known solutions for 1+1 D can be lifted to 3+1 D phase diagram: T MeV Μ q MeV

83 Chiral phase transition (D. Nickel, arxiv: , arxiv: ) constituent quark mass M(z) with 1 dimensional modulations analytically known solutions for 1+1 D can be lifted to 3+1 D phase diagram: T MeV Μ q MeV 1st-order line covered by the inhomogeneous phase

84 Chiral phase transition (D. Nickel, arxiv: , arxiv: ) constituent quark mass M(z) with 1 dimensional modulations analytically known solutions for 1+1 D can be lifted to 3+1 D phase diagram: M2 (z), wave vector q T MeV Μ q MeV M z MeV q MeV Μ q MeV 1st-order line covered by the inhomogeneous phase

85 Chiral phase transition (D. Nickel, arxiv: , arxiv: ) constituent quark mass M(z) with 1 dimensional modulations analytically known solutions for 1+1 D can be lifted to 3+1 D phase diagram: M2 (z), wave vector q T MeV Μ q MeV M z MeV q MeV Μ q MeV 1st-order line covered by the inhomogeneous phase all phase boundaries 2nd order

86 Outlook II to be done: include Polyakov loop three flavors two- and three-dimensional crystals combine with color superconductivity