Barcelona, Jun 26, 2018 1 Andreas Schmitt Mathematical Sciences and STAG Research Centre University of Southampton Southampton SO17 1BJ, United Kingdom (Color-)magnetic flux tubes in dense matter A. Haber, A. Schmitt, PRD 95, 116016 (2017); EPJ Web Conf. 137, 09003 (2017) A. Haber, A. Schmitt, JPG 45, 065001 (2018)
Barcelona, Jun 26, 2018 2 Theoretical motivation • What’s the behavior of a multi-component superconductor regarding type-I/type-II? • What (color-)flux tubes are there in a non-abelian superconductor like color-flavor locked (CFL) quark matter? Phenomenological motivation Ω B • Do (color-)flux tubes support grav. waves ellipticities of neutron stars flux → gravitational waves? tubes? K. Glampedakis, D. I. Jones and L. Samuelsson, PRL 109, 081103 (2012) (besides magnetic forces, see also crystalline structures)
Barcelona, Jun 26, 2018 3 Single-component superconductor • Ginzburg-Landau potential U for complex field φ with charge q coupled to gauge field A µ U = B 2 2 + | ( ∇ + iq A ) φ | 2 − µ 2 | φ | 2 + λ | φ | 4
Barcelona, Jun 26, 2018 4 Reminder: type-I/type-II superconductivity flux tube condensate • Ginzburg-Landau parameter ξ κ = λ magnetic field B λ ξ radius Η c2 normal • type-II superconductivity Η c √ flux tubes for κ > 1 / 2: flux tube lattice H Meissner for H c 1 < H < H c 2 Η c1 κ A.A. Abrikosov, Soviet Physics JETP 5, 1174 (1957)
Barcelona, Jun 26, 2018 5 Multi-component superconductors (page 1/2) • two fields with charges q 1 , q 2 (neutron/proton: q 1 = 2 e , q 2 = 0) U = B 2 � i | φ i | 2 + λ i | φ i | 4 � � | ( ∇ + iq i A ) φ i | 2 − µ 2 + 2 h | φ 1 | 2 | φ 2 | 2 2 + i =1 , 2 • fields are coupled indirectly via gauge field (if both q 1 , q 2 � = 0) and directly with coupling h (neutron/proton system: additional derivative coupling A. Haber, A. Schmitt, PRD 95, 116016 (2017) )
Barcelona, Jun 26, 2018 6 Multi-component superconductors (page 2/2) • more fields and multiple (color-)gauge fields 3 U = B 2 2 + B 2 � | ( ∇ + iq i 1 A 1 + iq i 2 A 2 ) φ i | 2 − µ 2 | φ i | 2 + λ | φ i | 4 � � 1 2 2 + i =1 − 2 h ( | φ 1 | 2 | φ 2 | 2 + | φ 1 | 2 | φ 3 | 2 + | φ 2 | 2 | φ 3 | 2 ) • color superconductor: 3 scalar components and 3 gauge fields: 1 electromagnetic and 2 color fields
Barcelona, Jun 26, 2018 7 Two-component superconductors in neutron star cores T c inner crust core core neutron (singlet) H outer crust c2 T c proton H H c1 c κ = 1/2 neutron (triplet) 2 density density • density-dependent κ • type-I/type-II transition in the core? • effect of coupling to superfluid on type-I/type-II transition?
Barcelona, Jun 26, 2018 8 Critical magnetic fields in a two-component system • compute flux tube profiles and flux tube interaction → attractive long-distance interaction in type-II regime isolated superconductor superconductor coupled to a superfluid H c2 H c2 H c2 ' H c2 ' H c2 H c flux tubes flux tubes H c flux tubes H c ' H c1 ' H c1 H c1 H c1 H c1 H ν small- approximation complete solution κ (our calculation) (conjecture) numerical calculation supports conjecture A. Haber, D. M¨ uller, preliminary results • first-order phase transition allows for flux tube clusters see also ”type-1.5” superconductors J. Carlstr¨ om, J. Garaud, E. Babaev, PRB 84, 134515 (2011)
Barcelona, Jun 26, 2018 9 Ginzburg-Landau potential for color superconductors U = − Tr[( D µ Φ) † ( D µ Φ)] + a Tr[Φ † Φ] + b Tr[(Φ † Φ) 2 ] + c (Tr[Φ † Φ]) 2 +1 + 1 4 F a µν F µν 4 F µν F µν a φ 0 0 0 0 0 φ 1 ( � r ) 0 0 Φ = 0 φ 0 0 0 0 0 φ 2 ( � r ) 0 0 0 φ 0 0 φ φ 3 ( � r ) 0 0 CFL 2SC ansatz for flux tubes Color-flavor locked (CFL) phase 2SC phase paired: unpaired: s d u s d s d u d s d u u s u d u s
Barcelona, Jun 26, 2018 10 Mixing of gluons and photons in CFL • symmetry breaking pattern of CFL [ SU (3)] c × SU (3) L × SU (3) R × U (1) B → SU (3) c + L + R × Z 2 � �� � � �� � ⊃ [ U (1)] Q ⊃ [ U (1)] ˜ Q • CFL is a superfluid → rotational vortices • Meissner effect for gluons T 1 , . . . , T 7 (”color superconductor”) • all Cooper pairs neutral under ˜ Q = Q + 2 √ 3 T 8 and (differently) charged under orthogonal combination ˜ T 8 – ˜ Q -magnetic field penetrates CFL – Meissner effect for ˜ T 8 -magnetic field (Analogous to gauge field mixing in standard model, [ SU (2)] × [ U (1)] → [ U (1)] Q )
Barcelona, Jun 26, 2018 11 Ginzburg-Landau potential with rotated gauge fields ˜ ˜ B 2 2 + B 2 B 2 � � � � � ∇ + ig � � ∇ − ig � 2 2 g 8 ˜ g 8 ˜ 3 8 � � � � U = 2 + 2 + 2 A 3 + i ˜ A 8 φ 1 + 2 A 3 + i ˜ A 8 φ 2 � � � � � 2 − µ 2 ( | φ 1 | 2 + | φ 2 | 2 + | φ 3 | 2 ) + λ ( | φ 1 | 4 + | φ 2 | 4 + | φ 3 | 4 ) � � �� � g 8 ˜ + ∇ − 2 i ˜ A 8 φ 3 − 2 h ( | φ 1 | 2 | φ 2 | 2 + | φ 1 | 2 | φ 3 | 2 + | φ 2 | 2 | φ 3 | 2 ) • approximations – massless quarks, m s = m d = m u = 0 – purely bosonic approach → neglect effects of magnetic field on Cooper pair constituents – Ginzburg-Landau parameters µ, λ, h : (mostly) use perturbative results and extrapolate down in density (= to large couplings)
Barcelona, Jun 26, 2018 12 Ginzburg-Landau potential with rotated gauge fields ˜ ˜ B 2 2 + B 2 B 2 � � � � � ∇ + ig � � ∇ − ig � 2 2 g 8 ˜ g 8 ˜ 3 8 � � � � U = 2 + 2 + 2 A 3 + i ˜ A 8 φ 1 + 2 A 3 + i ˜ A 8 φ 2 � � � � � 2 − µ 2 ( | φ 1 | 2 + | φ 2 | 2 + | φ 3 | 2 ) + λ ( | φ 1 | 4 + | φ 2 | 4 + | φ 3 | 4 ) � � �� � g 8 ˜ + ∇ − 2 i ˜ A 8 φ 3 − 2 h ( | φ 1 | 2 | φ 2 | 2 + | φ 1 | 2 | φ 3 | 2 + | φ 2 | 2 | φ 3 | 2 ) • ansatz for flux tube solutions φ i ( r, θ ) = ρ i ( r ) e in i θ with winding numbers n 1 , n 2 , n 3 • solve equations of motion for ρ 1 , ρ 2 , ρ 3 , A 3 , ˜ A 8 • boundary conditions: homogeneous CFL far away from flux tube, ρ i = 0 in center if winding n i is nonzero
Barcelona, Jun 26, 2018 13 � • usually: vortex → baryon circulation Γ = d ℓ · v � flux tube → magnetic flux Φ = d ℓ · A • CFL line defects can have both! ˜ Γ ∝ n 1 + n 2 + n 3 , Φ 3 ∝ n 1 − n 2 , Φ 8 ∝ n 1 + n 2 − 2 n 3 ( n 1 , n 2 , n 3 ) Γ [ π/ 3 µ q ] Φ 3 [ π/g ] ˜ CFL line defects Φ 8 [ π/ ˜ g 8 ] Global vortex ( n, n, n ) − n 0 0 Forbes, Zhitnitsky (2002) − n 2 n ”Semi-superfluid” vortex (0 , 0 , n ) 0 3 3 Balachandran, Digal, Matsuura (2006) Magnetic flux tube T 112 ( n, n, − 2 n ) 0 0 − 2 n Iida (2005) Magnetic flux tube T 101 ( n, 0 , − n ) 0 − n − n Haber, Schmitt (2018)
Barcelona, Jun 26, 2018 14 • vortices (Γ � = 0): ”topological” since π 1 [ U (1)] = Z – global vortex decays into 3 semi-superfluid vortices M. G. Alford, S. K. Mallavarapu, T. Vachaspati and A. Windisch, PRC 93, 045801 (2016) • flux tubes (Γ = 0): ”non-topological” since π 1 [ SU (3)] = 0 – stabilized through external magnetic field → see rest of the talk ( n 1 , n 2 , n 3 ) Γ [ π/ 3 µ q ] Φ 3 [ π/g ] ˜ CFL line defects Φ 8 [ π/ ˜ g 8 ] Global vortex ( n, n, n ) − n 0 0 Forbes, Zhitnitsky (2002) − n 2 n ”Semi-superfluid” vortex (0 , 0 , n ) 0 3 3 Balachandran, Digal, Matsuura (2006) Magnetic flux tube T 112 ( n, n, − 2 n ) 0 0 − 2 n Iida (2005) Magnetic flux tube T 101 ( n, 0 , − n ) 0 − n − n Haber, Schmitt (2018)
Barcelona, Jun 26, 2018 15 Flux tube profiles 1.4 1.4 T 112 T 101 f 2 1.2 1.2 f 1 = f 2 1.0 1.0 f 3 f 1 f 3 0.8 0.8 0.6 0.6 ˜ B 0.4 0.4 8 ˜ B B 3 0.2 0.2 8 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 R R • flux tube with ”2SC core” • flux tube with ”unpaired core” A. Haber, A. Schmitt, JPG 45, 065001 (2018) K. Iida, PRD 71, 054011 (2005) • additional B 3 field (cost in free energy) • non-vanishing condensate in core (gain in free energy)
Barcelona, Jun 26, 2018 16 Phase structure without flux tubes 3.0 2.5 unpaired 2SC 2.0 H [ μ 2 / λ 1 / 2 ] 1.5 1.0 0.5 CFL h / λ =- 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 strong coupling constant g • in weak coupling: h/λ = − 0 . 5 • CFL superseded by 2SC except for small values of strong coupling constant g (even though here m s ≃ 0!) • in neutron stars µ q ≃ 400 MeV ⇒ g ≃ 3 . 5
Barcelona, Jun 26, 2018 17 Phase diagram including flux tubes H c2 unpaired 15 H c2 H c H [ μ 2 / λ 1 / 2 ] H c1 ( S 1 ) H 2SC 10 c1 ( D ) H c H c1 ( T 112 ) 5 CFL H c1 ( T 101 ) 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 T c / μ q • type-II regime for sufficiently large T c /µ q (neutron stars: T c /µ q ∼ (10 − 100) / (400 − 500) ∼ 0 . 1) • type-I/type-II transition complicated (multi-component structure!)
Barcelona, Jun 26, 2018 18 Phase diagram including flux tubes H c2 unpaired 15 H c2 H c H [ μ 2 / λ 1 / 2 ] H c1 ( S 1 ) H 2SC 10 c1 ( D ) H c H c1 ( T 112 ) 5 CFL H c1 ( T 101 ) 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 T c / μ q • CFL flux tubes with 2SC core ( T 101 ) preferred • critical fields H ∼ 10 19 G ≫ H NS , however: creation of flux tubes through cooling into superconducting phase? • 2SC domain walls ( D ) preferred over ordinary 2SC flux tubes ( T 1 ) for sufficiently large T c /µ q
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