(Color-)magnetic flux tubes in dense matter A. Haber, A. Schmitt, - - PowerPoint PPT Presentation

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(Color-)magnetic flux tubes in dense matter A. Haber, A. Schmitt, - - PowerPoint PPT Presentation

Oct 03, 2023 103 likes •309 views

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);

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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)
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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

  • Do (color-)flux tubes support

ellipticities of neutron stars → gravitational waves?

  • K. Glampedakis, D. I. Jones and L. Samuelsson,

PRL 109, 081103 (2012)

B Ω

  • grav. waves

flux tubes?

(besides magnetic forces, see also crystalline structures)

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Barcelona, Jun 26, 2018 3

Single-component superconductor

  • Ginzburg-Landau potential U for complex field φ with charge q

coupled to gauge field Aµ

U = B2 2 + |(∇ + iqA)φ|2 − µ2|φ|2 + λ|φ|4

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Barcelona, Jun 26, 2018 4

Reminder: type-I/type-II superconductivity

radius magnetic field B condensate ξ λ

flux tube

  • Ginzburg-Landau parameter

κ = λ ξ

  • type-II superconductivity

for κ > 1/ √ 2: flux tube lattice for Hc1 < H < Hc2

A.A. Abrikosov, Soviet Physics JETP 5, 1174 (1957)

H κ

Ηc1 Ηc2 Ηc flux tubes Meissner normal

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Barcelona, Jun 26, 2018 5

Multi-component superconductors (page 1/2)

  • two fields with charges q1, q2 (neutron/proton: q1 = 2e, q2 = 0)

U = B2 2 +

  • i=1,2
  • |(∇ + iqiA)φi|2 − µ2

i|φi|2 + λi|φi|4

+ 2h|φ1|2|φ2|2

  • fields are coupled indirectly via gauge field (if both q1, q2 = 0)

and directly with coupling h (neutron/proton system: additional derivative coupling A. Haber, A. Schmitt, PRD 95, 116016 (2017))

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Multi-component superconductors (page 2/2)

  • more fields and multiple (color-)gauge fields

U = B2

1

2 + B2

2

2 +

3

  • i=1
  • |(∇ + iqi1A1 + iqi2A2) φi|2 − µ2|φi|2 + λ|φi|4

−2h(|φ1|2|φ2|2 + |φ1|2|φ3|2 + |φ2|2|φ3|2)

  • color superconductor: 3 scalar components

and 3 gauge fields: 1 electromagnetic and 2 color fields

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Two-component superconductors in neutron star cores

  • uter crust

density Tc inner crust core

neutron (singlet) proton

density Tc

c1 c2

H H H

c 2

κ = 1/2

neutron (triplet)

core

  • density-dependent κ
  • type-I/type-II transition in the core?
  • effect of coupling to superfluid on type-I/type-II transition?
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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

Hc1 Hc2 Hc Hc1 Hc2 Hc Hc1 ' Hc1 Hc2 Hc2 ' Hc Hc1 ' Hc2 '

small- approximation complete solution

flux tubes flux tubes flux tubes

(our calculation) (conjecture)

H κ

ν

isolated superconductor superconductor coupled to a superfluid

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¨

  • m, J. Garaud, E. Babaev, PRB 84, 134515 (2011)
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Barcelona, Jun 26, 2018 9

Ginzburg-Landau potential for color superconductors

U = −Tr[(DµΦ)†(DµΦ)] + aTr[Φ†Φ] + bTr[(Φ†Φ)2] + c(Tr[Φ†Φ])2 +1 4F a

µνF µν a

+ 1 4FµνF µν Φ =   φ 0 0 0 φ 0 0 0 φ     0 0 0 0 0 0 0 0 φ     φ1( r) φ2( r) φ3( r)   CFL 2SC ansatz for flux tubes

d d s u s u d s u

d d d u u u s s s unpaired: paired:

Color-flavor locked (CFL) phase 2SC phase

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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)]Q

×U(1)B → SU(3)c+L+R

  • ⊃[U(1)] ˜

Q

×Z2

  • CFL is a superfluid → rotational vortices
  • Meissner effect for gluons T1, . . . , T7 (”color superconductor”)
  • all Cooper pairs neutral under ˜

Q = Q + 2

√ 3T8

and (differently) charged under orthogonal combination ˜ T8 – ˜ Q-magnetic field penetrates CFL – Meissner effect for ˜ T8-magnetic field

(Analogous to gauge field mixing in standard model, [SU(2)] × [U(1)] → [U(1)]Q)

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Ginzburg-Landau potential with rotated gauge fields

U = ˜ B2 2 + B2

3

2 + ˜ B2

8

2 +

  • ∇ + ig

2A3 + i˜ g8 ˜ A8

  • φ1
  • 2

+

  • ∇ − ig

2A3 + i˜ g8 ˜ A8

  • φ2
  • 2

+

  • ∇ − 2i˜

g8 ˜ A8

  • φ3
  • 2 − µ2(|φ1|2 + |φ2|2 + |φ3|2) + λ(|φ1|4 + |φ2|4 + |φ3|4)

−2h(|φ1|2|φ2|2 + |φ1|2|φ3|2 + |φ2|2|φ3|2)

  • approximations

– massless quarks, ms = md = mu = 0 – purely bosonic approach → neglect effects of magnetic field

  • n Cooper pair constituents

– Ginzburg-Landau parameters µ, λ, h: (mostly) use perturbative results and extrapolate down in density (= to large couplings)

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Barcelona, Jun 26, 2018 12

Ginzburg-Landau potential with rotated gauge fields

U = ˜ B2 2 + B2

3

2 + ˜ B2

8

2 +

  • ∇ + ig

2A3 + i˜ g8 ˜ A8

  • φ1
  • 2

+

  • ∇ − ig

2A3 + i˜ g8 ˜ A8

  • φ2
  • 2

+

  • ∇ − 2i˜

g8 ˜ A8

  • φ3
  • 2 − µ2(|φ1|2 + |φ2|2 + |φ3|2) + λ(|φ1|4 + |φ2|4 + |φ3|4)

−2h(|φ1|2|φ2|2 + |φ1|2|φ3|2 + |φ2|2|φ3|2)

  • ansatz for flux tube solutions

φi(r, θ) = ρi(r) einiθ with winding numbers n1, n2, n3

  • solve equations of motion for ρ1, ρ2, ρ3, A3, ˜

A8

  • boundary conditions: homogeneous CFL far away from flux tube,

ρi = 0 in center if winding ni is nonzero

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  • usually: vortex → baryon circulation Γ =
  • dℓ · v

flux tube → magnetic flux Φ =

  • dℓ · A
  • CFL line defects can have both!

Γ ∝ n1 + n2 + n3 , Φ3 ∝ n1 − n2 , ˜ Φ8 ∝ n1 + n2 − 2n3

CFL line defects (n1, n2, n3) Γ [π/3µq] Φ3 [π/g] ˜ Φ8 [π/˜ g8] Global vortex (n, n, n) −n

Forbes, Zhitnitsky (2002)

”Semi-superfluid” vortex (0, 0, n) −n 3 2n 3

Balachandran, Digal, Matsuura (2006)

Magnetic flux tube T112 (n, n, −2n) −2n

Iida (2005)

Magnetic flux tube T101 (n, 0, −n) −n −n

Haber, Schmitt (2018)

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  • vortices (Γ = 0): ”topological” since π1[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 π1[SU(3)] = 0

– stabilized through external magnetic field → see rest of the talk

CFL line defects (n1, n2, n3) Γ [π/3µq] Φ3 [π/g] ˜ Φ8 [π/˜ g8] Global vortex (n, n, n) −n

Forbes, Zhitnitsky (2002)

”Semi-superfluid” vortex (0, 0, n) −n 3 2n 3

Balachandran, Digal, Matsuura (2006)

Magnetic flux tube T112 (n, n, −2n) −2n

Iida (2005)

Magnetic flux tube T101 (n, 0, −n) −n −n

Haber, Schmitt (2018)

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Flux tube profiles

f1=f2 f3 B ˜

8

T112

2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

R f1 f2 f3 B ˜

8

B3 T101

2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

R

  • flux tube with ”unpaired core”
  • K. Iida, PRD 71, 054011 (2005)
  • flux tube with ”2SC core”
  • A. Haber, A. Schmitt, JPG 45, 065001 (2018)
  • additional B3 field

(cost in free energy)

  • non-vanishing condensate in

core (gain in free energy)

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Phase structure without flux tubes

CFL 2SC unpaired h/λ=-0.5

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

strong coupling constant g H [μ2/λ1/2]

  • in weak coupling: h/λ = −0.5
  • CFL superseded by 2SC except for small values of strong coupling

constant g (even though here ms ≃ 0!)

  • in neutron stars µq ≃ 400 MeV ⇒ g ≃ 3.5
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Phase diagram including flux tubes

CFL 2SC unpaired

Hc2 Hc2 Hc Hc Hc1 ( S1 ) H

c1

( D ) Hc1(T112) Hc1(T101)

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 5 10 15

Tc/μq H [μ2/λ1/2]

  • type-II regime for sufficiently large Tc/µq

(neutron stars: Tc/µq ∼ (10 − 100)/(400 − 500) ∼ 0.1)

  • type-I/type-II transition complicated (multi-component structure!)
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Barcelona, Jun 26, 2018 18

Phase diagram including flux tubes

CFL 2SC unpaired

Hc2 Hc2 Hc Hc Hc1 ( S1 ) H

c1

( D ) Hc1(T112) Hc1(T101)

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 5 10 15

Tc/μq H [μ2/λ1/2]

  • CFL flux tubes with 2SC core (T101) preferred
  • critical fields H ∼ 1019 G ≫ HNS, however:

creation of flux tubes through cooling into superconducting phase?

  • 2SC domain walls (D) preferred over ordinary 2SC flux tubes (T1)

for sufficiently large Tc/µq

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Summary

  • multi-component superconductors have a nontrivial type-I/type-II

transition

  • dense quark matter is a multi-component superconductor and has

various possible line defects

  • CFL flux tubes (without baryon circulation) are not protected by

topology, but can be stabilized by a magnetic field

  • we have found new magnetic defects in CFL and 2SC:

CFL tubes with 2SC core and 2SC domain walls

  • defects in superconducting/superfluid nuclear and quark matter

are relevant for neutron star observables, e.g., gravitational waves