Magnetic Fields in Neutron Stars - Institute for Nuclear ... · magnetic ﬁeld (Abrikosov lattice). This belief is based on simple estimations of the coherence length ξand the London

Magnetic Fields in Neutron Stars∗

Ariel ZhitnitskyUniversity of British Columbia

Vancouver

June 8/2004, INT workshop“QCD AND DENSE MATTER: FROM LATTICES

TO STARS”

∗(Based on works with Kirk Buckley and Max Metlitski: PRL, 2004 andPRC, 2004)

Ariel Zhitnitsky Magnetic Fields in Neutron Stars

I. Historical Introduction. Motivation.

1. Conventional picture The extremely denseinterior in neutron stars is mainly composed ofneutrons, with a small amount of protons and electronsin beta equilibrium. The neutrons form 3P2 Cooperpairs and Bose condense to a superfluid state, whilethe protons form 1S0 Cooper pairs and Bose condenseto form a superconductor.

2. Type-II superconductivity It is generallyassumed that the proton superconductor is a type-IIsuperconductor, which means that it supports a stablelattice of magnetic flux tubes in the presence of amagnetic field (Abrikosov lattice). This belief is basedon simple estimations of the coherence length ξ andthe London penetration depth λ which ambiguouslyimply a type-II superconductivity.

3. Landau-Ginzburg parameter

λ =

√mcc2

4πq2np, ξ =

√h̄2

2mcnpa, mc = 2m, q = 2e

1


Typically, the Landau-Ginzburg parameter κ = λ/ξ isintroduced. In a conventional superconductor, if κ <1/√

2 then the superconductor is type-I and vorticesattract. If κ > 1/

√2 then vortices repel each other

and the superconductor is type-II

Borrowed from http://www.lsw.uni-heidelberg.de/~mcamenzi/NS_Mass.html

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4. “Naive” Moral:

=⇒TYPE I SUPERCONDUCTORimplies =⇒ No Field in the Bulk

=⇒TYPE II SUPERCONDUCTORimplies =⇒ Abrikosov Lattice Structure


4. “Naive” Moral:=⇒TYPE I SUPERCONDUCTORimplies No Field in the Bulk=⇒TYPE II SUPERCONDUCTORimplies Abrikosov Lattice Structure

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3


Advanced information on the Nobel Prize in Physics, 7 October 2003

Information Department, P.O. Box 50005, SE-104 05 Stockholm, Sweden Phone: +46 8 673 95 00, Fax: +46 8 15 56 70, E-mail: [email protected], Website: www.kva.se

Superfluids and superconductors: quantum mechanics on a macroscopic scale Superfluidity or superconductivity – which is the preferred term if the fluid is made up of charged particles like electrons – is a fascinating phenomenon that allows us to observe a variety of quantum mechanical effects on the macroscopic scale. Besides being of tremendous interest in themselves and vehicles for developing key concepts and methods in theoretical physics, superfluids have found important applications in modern society. For instance, superconducting magnets are able to create strong enough magnetic fields for the magnetic resonance imaging technique (MRI) to be used for diagnostic purposes in medicine, for illuminating the structure of complicated molecules by nuclear magnetic resonance (NMR), and for confining plasmas in the context of fusion-reactor research. Superconducting magnets are also used for bending the paths of charged particles moving at speeds close to the speed of light into closed orbits in particle accelerators like the Large Hadron Collider (LHC) under construction at CERN.

Discovery of three model superfluids

Two experimental discoveries of superfluids were made early on. The first was made in 1911 by Heike Kamerlingh Onnes (Nobel Prize in 1913), who discovered that the electrical resistance of mercury completely disappeared at liquid helium temperatures. He coined the name “superconductivity” for this phenomenon. The second discovery – that of superfluid 4He – was made in 1938 by Pyotr Kapitsa and independently by J.F. Allen and A.D. Misener (Kapitsa received the 1978 Nobel Prize for his inventions and discoveries in low temperature physics). It is believed that the superfluid transition in 4He is a manifestation of Bose-Einstein condensation, i.e. the tendency of particles – like 4He – that obey Bose-Einstein statistics to condense into the lowest-energy single–particle state at low temperatures (the strong interaction between the helium atoms blurs the picture somewhat). Electrons, however, obey Fermi-Dirac statistics and are prevented by the Pauli principle from having more than one particle in each state. This is why it took almost fifty years to discover the mechanism responsible for superconductivity. The key was provided by John Bardeen, Leon Cooper and Robert Schrieffer, whose 1957 “BCS theory” showed that pairs of electrons with opposite momentum and spin projection form “Cooper pairs”. For this work they received the 1972 Nobel Prize in Physics. In their theory the Cooper pairs are

- 1 -

• 1913 1913 KamerlinghKamerlingh OnnesOnnes

•• 1972 1972 BardeenBardeen, Cooper and , Cooper and SchriefferSchrieffer

•• 1973 1973 EsakiEsaki, , GiaeverGiaever and and JosephsonJosephson

•• 1987 1987 BednorzBednorz and Mullerand Muller

•• 1996 Lee, 1996 Lee, OsheroffOsheroff and Richardsonand Richardson

•• 2003 2003 AbrikosovAbrikosov, , GinzburgGinzburg and Leggettand Leggett

Memorable years in the history of Memorable years in the history of superfluiditysuperfluidity and and superconductivity of Fermi systemssuperconductivity of Fermi systems

4


5. Life is much more complicated:

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ond-

mat

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4161

v1

18

Apr

199

7

The Shapes of Flux Domains in the Intermediate Stateof Type-I Superconductors

Alan T. Dorsey∗Department of Physics, University of Florida, Gainesville, FL 32611

Raymond E. Goldstein∗∗Department of Physics and Program in Applied Mathematics

University of Arizona, Tucson, AZ 85721(September 17, 2003)

In the intermediate state of a thin type-I superconductor magnetic flux penetrates in a disorderedset of highly branched and fingered macroscopic domains. To understand these shapes, we study indetail a recently proposed “current-loop” (CL) model [R.E. Goldstein, D.P. Jackson, A.T. Dorsey,Phys. Rev. Lett. 76, 3818 (1996)] that models the intermediate state as a collection of tense currentribbons flowing along the superconducting-normal interfaces and subject to the constraint of globalflux conservation. The validity of this model is tested through a detailed reanalysis of Landau’soriginal conformal mapping treatment of the laminar state, in which the superconductor-normalinterfaces are flared within the slab, and of a closely-related straight-lamina model. A simplifieddynamical model is described that elucidates the nature of possible shape instabilities of flux stripesand stripe arrays, and numerical studies of the highly nonlinear regime of those instabilities demon-strate patterns like those seen experimentally. Of particular interest is the buckling instabilitycommonly seen in the intermediate state. The free-boundary approach further allows for a calcu-lation of the elastic properties of the laminar state, which closely resembles that of smectic liquidcrystals. We suggest several new experiments to explore of flux domain shape instabilities, includingan Eckhaus instability induced by changing the out-of-plane magnetic field, and an analog of theHelfrich-Hurault instability of smectics induced by an in-plane field.

I. INTRODUCTION

A longstanding problem in macroscopic superconduc-tivity is that of understanding the complex patterns offlux penetration observed in the intermediate state of atype-I superconductor. This state is observed when athin superconducting slab is placed in a perpendicularmagnetic field. Unlike type-II superconductors, wherethe field penetration is in the form of tubes each with aquantum of magnetic flux, type-I systems are observed toform intricately branched and fingered macroscopic fluxdomains [1–3]. Thus, instead of establishing a Meissnerphase, in which the magnetic induction B = 0 uniformly,the demagnetizing effects of the large aspect ratio forcethe sample to break up into regions, some of which areuniformly superconducting (with B = 0 inside) and oth-ers that are normal (B != 0). Figure 1 shows a typicalexample of these patterns [3]. The superconducting re-gions appear black, having been decorated with a pow-der (niobium) that is itself superconducting at the sampletemperature and thus migrates to the regions of low mag-netic field. Other imaging techniques include Hall probes[4] and magneto-optics [5]. All reveal similar structures.

The sample in Fig. 1 is at an applied magnetic fieldHa that is very close to the critical field Hc at which

the sample would be completely normal, so the minorityphase is superconducting. Similar patterns are observedwhen Ha/Hc is very small, but now the minority phase

FIG. 1. The intermediate state of a thin slab of indium, inwhich the superconducting regions (black) are decorated withniobium (black). The applied field Ha is close to the criticalfield Hc (h = Ha/Hc = 0.931). Adapted from Haenssler andRinderer [3].

1 (Superconducting region is black, normal phase isbright)

5


6


7


8


(Superconducting region is bright, normal phaseis dark)

9


II. Recent Development

1. Paper by I. H. Stairs, A. G. Lyne, and S. L.Shemar, Nature 406, 484 (2000).

The periodic timing behavior of PSRB182811 and correlated changes in beam profilehave been interpreted as due to precession with aperiod of 1 yr and an amplitude of ∼ 3o

2. Paper by B.Link, Phys.Rev.Lett. 91 (2003)101101

Abstract: “I show that the standard picture of theneutron star core containing coexisting neutron andproton superfluids, with the proton component forminga type II superconductor threaded by flux tubes, isinconsistent with observations of long-period (∼ 1yr) precession in isolated pulsars. I conclude thateither the two superfluids coexist nowhere in the stellarcore, or the core is a type I superconductor ratherthan type II. Either possibility would have interestingimplications for neutron star cooling and theories ofspin jumps (glitches).”

10


3. The summary of the Link’s paper:a). The estimates show that a neutron star corecontaining coexisting neutron vortices and proton fluxtubes cannot precess with a period of ∼1 yr.b). The fraction of the neutron components momentof inertia that is pinned against flux tubes must be� 10−8. Hence, observations require that neutronvortices and proton flux tubes coexist nowhere in thestar.c). Either the stars magnetic field does not penetrateany part of the core as the Abrikosov vorticescorresponding to a type II superconductor,d). or: at least one of the hadronic fluids is notsuperfluid. (This latter possibility appears unlikely inthe face of pairing calculations which predict coexistingneutron and proton superfluids in the outer core.)e). If the core is a type I superconductor, at leastin those regions containing vortices, the magnetic fluxcould exist in macroscopic normal regions that surroundsuperconducting regions that carry no flux. In this case,the magnetic field would not represent the impedimentto the motion of vortices that flux tubes do, and thestar could precess with a long period.

11


III. Type I instead of Type II

1. Main goalMotivated by the previous papers, we suggest a possiblescenario which would lead to the type I behavior in spiteof the fact that λ, ξ remain the same as in the “naive”estimates suggesting type II behavior.2. Main idea. Analogies with other fieldsa). There are many situations where the standardpicture will be qualitatively modified. For example,if there is a second component (such as a neutroncomponent in our specific case), it may be energeticallyfavorable for the cores of vortices to be filled with anonzero condensate of this second component, as itwas originally suggested in the cosmological context byWitten (cosmic strings).b). There are numerous examples of physical systemswhere such phenomena occurs: superconductingcosmic strings in cosmology, magnetic flux tubesin the high Tc superconductors (antiferromagneticcondensate fills the vortex core), Bose-Einsteincondensates (two component BEC systems), superfluid3He(A and B), and high baryon density quark matter.c). Our main assumption is: such a nontrivial vortex

12


structure occurs due to the strong interaction betweentwo superfluids ( n and p) in two component system.d). If it happens, the vortex-vortex interactionpattern changes ( repulsion ⇒ attraction). Thisautomatically changes the type II ⇒ type I behavior.3. Possible Resolution of the Paradoxa).The vortex-vortex (attracting) interaction due to thenontrivial core structure would resolve the apparentdiscrepancy (B.Link) between the observation oflong period precession (I.Stairs) and the typicalparameters of the neutron stars which naively suggesttype-II superconductivity in neutron stars.b). Specifically, on a macroscopic distancescale, the magnetic flux must be embedded inthe superconductor. This would mean that thesuperconductor is in an intermediate state as opposedto the vortex state of the type-II superconductor.c). The superconducting domains will then exhibit theMeissner effect, while the normal domains will carrythe required magnetic flux.d). Source for a nontrivial vortex core structure mayhave a different nature (Aurel Bulgac).

13


IV. Vortex Structure

1. Effective Landau-Ginsburg free energy

F =∫d2r

(h̄2

2mc|(∇− iq

h̄cA)ψ1|2 + |∇ψ2|2

)

+(

B2

8π− µ1|ψ1|2 − µ2|ψ2|2

)+(a11

2|ψ1|4 +

a22

2|ψ2|4 + a12|ψ1|2|ψ2|2

)where mc = 2m and q = 2e, µi is the Bose chemical

potential of the ith component and aij = 4πh̄2lij/mc.

The Bose chemical potentials µ1, µ2 determined bythe Cooper pair densities n1 ≡ |ψ1|2 and n2 ≡ |ψ2|2and coefficients aij.

n1 =a22µ1 − a12µ2

a11a22 − a212

, n2 =a11µ2 − a12µ1

a11a22 − a212

.

14


2. Few Important Remarks:

a) Original symmetry is : U(1) × U(1)corresponding to the conservation of two species ψ1

and ψ2 ;

b) In the limit µi = µ, aij = a the symmetry isU(2). The vacuum manifold is given by the 3-sphere|ψ1|2 + |ψ2|2 = µ/a ;

c) A very small U(2) violating change in thechemical potentials µ1 and µ2 that violates the U(2)symmetry produces a very large asymmetry of protonand neutron Cooper pair densities n1, n2 by selectinga particular vacuum state on the original degeneratemanifold.

d) In particular, if aij = a, but the chemicalpotentials are slightly different, µ1 = µ2 − δµ, δµ > 0then neutrons condense, n2 = µ2/a, n1 = 0, which

corresponds a very large asymmetry of proton andneutron Cooper pair densities n2/n1 = ∞.

15


3. Main Assumption:We assume that free energy F is mainly U(2)symmetric, and the large asymmetry of protonand neutron Cooper pair densities (known to realizein nature) can be achieved in the effective lagrangianapproach by small explicit U(2) violation: µ1 = µ−δµ,µ2 = µ + δµ, where δµ/µ � 1, and a11 = a22 = a,a12 = a− δa, where δa/a� 1.

n1 ≈µ

2a− δµ

δa, n2 ≈

µ

2a+δµ

δa, 0 < n2/n1 <∞

a) Technical motivation for the assumption: We knew(based on the previous experience ) that in this casea nontrivial vortex structure occurs. It leads to theunusual vortex-vortex interaction features.b) More physical (rather than technical) explanation:The original isotopical symmetry could not disappearwithout a trace, it must be hidden somewhere.... Inparticular, in a similar problem in QCD with Nc =2 (fundamental quarks) or Nc = 3(adjoint quarks),the fermi surfaces for different flavors could be verydifferent. However, the original symmetry can be usedto calculate the different Cooper pair condensates.

16


4. Equations of Motion• The Landau-Ginzburg equations of motion followingfrom the free energy are:

h̄2

2mc(∇− iq

h̄cA)2ψ1 = −µ1ψ1 + a|ψ1|3 + (a− δa)|ψ2|2ψ1,

h̄2

2mc∇2ψ2 = −µ2ψ2 + a|ψ2|3 + (a− δa)|ψ1|2ψ2,

∇× (∇×A)4π

=−iqh̄2mcc

(ψ∗1(∇−

iq

h̄cA)ψ1 − h.c

)• In previous calculations it has been assumed that theneutron order parameter ψ2 will remain at its vacuumexpectation value |ψ2| = const. in the vicinity of theproton vortex ( this is not the case in many similarsystems. )• So, anticipating a non-trivial behavior of the neutronfield ψ2, we adopt the following ansatz for the fieldsdescribing a proton vortex with a unit winding number:

ψ1 =√n1 f(r) eiθ, ψ2 =

√n2 g(r), A =

h̄c

q

a(r)r

θ̂

17


• We wish to find the asymptotic behavior of fieldsψ1, ψ2 and A far from the proton vortex core, asthis will determine whether distant vortices repel orattract each other. The asymptotic behavior can befound analytically

f(r) = 1 + F (r), g(r) = 1 +G(r), a(r) = 1− rS(r),

so that far away from the vortex core, F,G, rS � 1and F,G, S → 0 as r → ∞. This allows us tolinearize far from the vortex core the equations ofmotion corresponding to the free energy to obtain:

(∂2

∂r2+

1r

∂

∂r)(FG

)= M

(FG

)where matrix M mixing the fields F and G is,

M =4mc

h̄2

(a a− δaa− δa a

)·(

n1 00 n2

)

S′′ +1rS′ − 1

r2S =

1λ2S ⇒ S =

CA

λK1(r/λ)

18


5. Standard picture:

In previous works the influence of the neutroncondensate on the proton vortex was neglected, whichformally amounts to setting the off-diagonal term M12

to 0. In that case,

F = CFK0(√

2r/ξ), G = 0.

It is estimated that λ ∼ 80 fm and ξ ∼ 30 fm, whichleads to κ = λ/ξ ∼ 3 i.e. κ > 1/

√2. Therefore,

distant vortices repel each other leading to

type-II behavior.

This is the standard picture of the protonsuperconductor in neutron stars that is widelyaccepted in the astrophysics community.

19


6. Beyond the standard picture:

• The standard procedure described above isinherently flawed since the system exhibits anapproximate U(2) symmetry, This makes the mixingmatrix M nearly degenerate.

The general solution is:(FG

)=∑

i=1,2

CiK0(√νir) vi

where νi and vi are the eigenvalues and eigenvectorsof matrix M, and Ci are constants.

• In the limit γ = n1/n2 � 1 and ε = 2δa/a � 1one can estimate the eigenvalues and eigenvectors ofthe matrix M as:

ν1 ' 2εξ2, v1 '

(−1γ

),

ν2 ' 2γξ2

, v2 '(

11

)20


• The physical meaning of solution: there aretwo modes in our two component system. Thefirst mode describes fluctuations of relative density(concentration) of two components and the secondmode describes fluctuations of overall density of twocomponents.

• Notice that ν1 � ν2, and hence the overalldensity mode has a much smaller correlationlength than the concentration mode. Therefore,far from the vortex core, the contribution of the overalldensity mode can be neglected, and one can write:

(FG

)(r →∞) ' C1K0(

√2εr/ξ) ·

(−1γ

)(1)

• The most important result: a typical scaleis of order ξ/

√ε - the correlation length of the

concentration mode. Since ε � 1, this distancescale can be much larger than the proton correlationlength ξ, which is typically assumed to be the radiusof the proton vortex core.

21

Ariel Zhitnitsky Magnetic Fields in Neutron Stars5

distant vortices repel each other leading to type-II be-havior. This is the standard picture of the proton super-conductor in neutron stars that is widely accepted in theastrophysics community.

However, the standard procedure described above isinherently flawed since the system exhibits an approxi-mate U(2) symmetry, and therefore the couplings aij areapproximately equal a11 ≈ a22 ≈ a12. This makes themixing matrix M nearly degenerate. The general solu-tion to Eq. (17) is:(

FG

)=

∑i=1,2

CiK0(√

νir) vi (22)

where νi and vi are the eigenvalues and eigenvectorsof matrix M, and Ci are constants to be calculated bymatching to the solution of the original nonlinear equa-tions of motion. We would like to introduce the parame-ter ε (which measures the asymmetry between the protonand neutron Cooper pairs) defined in the following way:

ε = (a11a22 − a212)/a2

ij $ 2δa

a. (23)

We should remark here that our qualitative results donot depend on the value of γ ≡ n1/n2. Indeed, no matterwhat n1 and n2 are, the mixing matrix is still singularin the limit ε → 0. Hence, we still get one eigenvaluewhich vanishes when ε → 0. So, the only crucial as-sumption is ε ' 1. However, to simplify our formulafor the eigenvalues in what follows we assume a specificvalue of γ ≡ n1/n2 ' 1. It simply allows our resultsto be expressed in a more transparent way. In the limitγ = n1/n2 ' 1 and ε = 2δa/a ' 1 one can estimate theeigenvalues and eigenvectors of the matrix M as:

ν1 $ 2ε

ξ2, v1 $

(−1γ

), (24)

ν2 $ 2

γξ2, v2 $

(11

). (25)

The physical meaning of solution (22) is simple: there aretwo modes in our two component system. The first modedescribes fluctuations of relative density (concentration)of two components and the second mode describes fluctu-ations of overall density of two components. Notice thatν1 ' ν2, and hence the overall density mode has a muchsmaller correlation length than the concentration mode.Therefore, far from the vortex core, the contribution ofthe overall density mode can be neglected, and one canwrite: (

FG

)(r → ∞) $ C1K0(

√2εr/ξ) ·

(−1γ

)(26)

The most important result of the above discussion is thatthe distance scale over which the proton and neutron con-densates tend to their vacuum expectation values near aproton vortex is of order ξ/

√ε - the correlation length of

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30

f,G

/γ,a

r̃

FIG. 1: In this figure we show the functions f(r̃), G(r̃)/γ, anda(r̃) (defined in Eqs. (13,16)) as a function of the dimension-less radial coordinate r̃ = r/ξ. The dotted line correspondsto a(r̃), the solid line approaching 1 at large r̃ corresponds tof(r̃), and the solid line approaching 0 at large r̃ correspondsto G(r̃)/γ.

the concentration mode. Since ε ' 1, this distance scalecan be much larger than the proton correlation length ξ,which is typically assumed to be the radius of the protonvortex core. The appearance of the concentration modeis not surprising since we presented some arguments (be-fore calculations) supporting the picture that the neutroncondensate will increase its magnitude slightly in the vor-tex core, while the proton condensate will decrease itsmagnitude to 0 in the core center. We note that in thelimit ε → 0 the size of the proton vortex core becomesinfinitely large, and the vortex is thereby destroyed. Thisis in accordance with the topological arguments, whichstate that if the U(2) symmetry were exact with ε ≡ 0,and it is spontaneously broken to U(1), there will be nostable vortices in the system.

We have also verified the above results numerically bysolving the equations of motions (10,11,12) with a par-ticular choice of the approximately U(2) symmetric in-teraction potential V . Our numerical results support theanalytical calculations given above. Namely, we find thatthe magnitude of the neutron condensate is slightly in-creased in the vortex core, the radius of the magnetic fluxtube is of order λ and the radius of the proton vortex coreis of order ξ/

√ε. In Fig. 1 we show the numerical solu-

tion of the profiles of the proton vortex (f(r̃)), neutroncondensate (G(r̃)/γ), and a(r̃) (related to the gauge fieldthrough Eq. (16)) as a function of the dimensionless ra-dial coordinate r̃ = r/ξ, where ξ is the coherence length(14). We have used κ = 3, n1/n2 = 0.05, and ε = 0.02 inthis numerical solution.

III. VORTEX-VORTEX INTERACTION

Now that we know the approximate solution for theproton vortex, we will proceed to look at the interac-tion between two widely separated proton vortices. If

Here we show the numerical solution of the profilesof the proton vortex (f(r̃)), neutron condensate(G(r̃)/γ), and a(r̃) (electromagnetic field) as afunction of the dimensionless radial coordinate r̃ =r/ξ, where ξ is the coherence length. We have usedκ = 3, n1/n2 = 0.05, and ε = 0.02 in this numericalsolution.

22


V. Vortex - Vortex Interaction

1. Few general remarks

•If the interaction between two vortices isrepulsive, it is energetically favorable for thesuperconductor to organize an Abrikosov vortex latticewith each vortex carrying a single magnetic fluxquantum. This is classic type-II behavior. If theinteraction between two vortices is attractive, it isenergetically favorable for n vortices to coalesce andform a vortex of winding number n. This is type-Ibehavior.

• The Landau-Ginzburg parameter κ = λ/ξ isintroduced. In a conventional superconductor, if κ <1/√

2 then the superconductor is type-I and vorticesattract. If κ > 1/

√2 then vortices repel each other

and the superconductor is type-II.

• The typical value for a neutron star is κ ∼ 3, soone could naively expect that the proton superfluid isa type-II superconductor.

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• The case we consider here has one new element,ε which was not present in the standard type-I/IIclassification. However, we expect that analogousclassification ( when one compare λ and actual vortexcore) should remain in effect. In such an analysis ξshould be replaced by the actual size of the protonvortices

ξ ⇒ δ ' ξ/√ε.

• Therefore, we will define a new Landau-Ginzburgparameter for our case,

κnp ≡ λδ =

√ελξ .

• We expect type-I behavior with vorticesattracting each other if κnp � 1 and type-

II behavior if κnp � 1. For relatively small εsuch an argument would immediately suggest that forthe typical parameters of the neutron stars type-Isuperconductivity is realized (rather than the naivelyassumed type-II superconductivity).

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2. Explicit calculation of the vortex-vortexinteraction

• Define: (F,G,A) = (F1+F2, G1+G2,A1+A2)be the exact fields produced by two vortices at locationsr1 and r2. When r is far from the cores of both vortices,Fi, Gi,Ai are small for both i = 1, 2 and are known tobe:

Fi ' −Gi/γ ' −C1K0(√

2ε|r− ri|/ξ),

Ai 'h̄c

qλCAK1(|r− ri|/λ) θ̂

• To calculate the interaction energy of the twovortices, we divide the space into two cells T1 andT2, which contain the centers of vortices 1 and 2respectively. The vortex-vortex interaction energy isthen:

Fint = F [F1+F2, G1+G2,A1+A2]−2F [F1, G1,A1]

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• We assume the vortex separation to be large.Since this boundary is far away from either vortexcenter, we can use the asymptotic expressions for thefields (Fi, Gi,Ai) to explicitly calculate the integrals.

Fint ' 2∮

T

dS · (h̄2n1

mcF2∇F1 +

h̄2n2

mcG2∇G1

+14π

A2 × (∇×A1)) (2)

Here the integral is over the boundary of cell T .

• Substituting asymptotic solutions into the above,we find the vortex-vortex interaction energy per unitlength to be:

U(d) = 2πh̄2n1mc

(C2

AK0(d/λ)− C21K0(

√2εd/ξ)

)

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3. Interpretation of the result

• As we anticipated the relevant parameter is anew Landau-Ginzburg parameter for two componentsystem,

κnp ≡ λδ =

√ελξ .

• Ifκnp > 1/√

2 then vortices repel each otherand the superconductor is type-II. This corresponds tothe first (conventional) term in the interaction energy

U(d) ' 2πh̄2n1

mc(K0(d/λ)) ∼ exp (−d

λ)

• If κnp < 1/√

2 then the superconductoris type-I and vortices attract. This corresponds to thesecond term in the interaction energy

U(d) ' −2πh̄2n1

mc

(K0(

√2εd/ξ)

)∼ − exp (−

√2εdξ

)

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VI. Critical Magnetic Fields

1.Motivation. General Comments

• We realized that two component system mayhave type I rather than type II behavior. Therefore,the standard estimation of critical field which destroyssuperconductivity must be also reconsidered;

• Comparison Hc with Hc2 will support ourclaim that our system indeed exhibits the type Isuperconductivity rather than type II .

• Usually one calculates the critical magnetic fieldsHc and Hc2. These are the physically meaningfulfields above which the superconductivity is destroyedin type-I and type-II superconductors respectively. IfHc > Hc2 then the superconductor is type-I, otherwise,the superconductor is type-II.

2. Calculation of Hc

• Hc is defined as the point at which the Gibbs freeenergy of the normal phase is equal to the Gibbs freeenergy of the superconducting phase.

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• The Gibbs free energy in the presence of anexternal magnetic field H is:

g(H,T ) = f(B, T )− BH

4π

where H is the external magnetic field, B is themagnetic induction, T is the temperature.• For the superconducting state: 〈|ψ1|2〉 = n1,〈|ψ2|2〉 = n2, and B = 0 (Meissner effect), the Gibbsfree energy is

gs(H,T ) = −µ2

2a− (δµ)2

δa− δa

( µ2a

)2

.

• For the normal state, we have 〈|ψ1|2〉 = 0, 〈|ψ2|2〉 =(µ+ δµ)/a, and B = H. The Gibbs free energy is:

gn(H,T ) = −H2

8π− µ2

2a− µδµ

a

Hc is defined as the point at which gs(Hc) = gn(Hc),i.e

Hc =√

8πδa(µ

2a− δµ

δa

)→ n1

√8πδa,

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2. Calculation of Hc2

• In order to calculate Hc2, we follow the standardprocedure and linearize the equations of motion for ψ1

about the normal state with 〈|ψ1|2〉 = 0 and 〈|ψ2|2〉 =(µ+ δµ)/a. The linearized equation of motion reads,

h̄2

2mc

(−i∇− q

h̄cA)2

ψ1 = ωψ1,

where ω ≡ (µ + δµ)δaa − 2δµ. This is simply a

Schrodinger equation for a particle in a magneticfield, with an energy of ω. This is a standardquantum mechanics problem. The first Landau levelis the ground state energy of ε0(H) = h̄|q|H/2mcc.Therefore, if ω < ε0, then only the trivial solution withψ1 = 0 is possible. The critical field Hc2 is defined asthe point at which ω = ε0(Hc2).

Hc2 =2mcc

h̄|q|[(µ+ δµ)

δa

a− 2δµ] ' 4mcc

h̄|q|δa n1

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3. Numerical estimates

• If Hc < Hc2 this means that it isenergetically favorable for microscopic regions ofthe superconducting state to be nucleated as His decreased. This is type-II behavior, and thisnucleation manifests itself in the form of an vortexlattice.

• If Hc > Hc2, then it is energetically favorablefor macroscopic regions of the superconducting stateto be present as H is decreased. This is a type-Isuperconductor and the superconducting state persistseverywhere in the material when H < Hc.

Hc2

Hc'√

2mcc√πh̄

√δa

q=√

2 κnp,

in agreement with the results obtained from the vortex-vortex interaction calculation.

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• Numerical estimate:

Hc =ϕ0

2πλξ

√δa

a, ϕ0 =

2πh̄cq

= 2× 107G · cm2,

where ϕ0 is the quantum of the fundamental flux. If wesubstitute λ = 80 fm and ξ = 30 fm (typical values)in the expression for the critical magnetic field Hc isestimated to be the Hc ' 1014 G.

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VII. Intermediate State

1. Overview

• If the core is indeed a type-I superconductor,the magnetic field must be expelled from thesuperconducting region (Meissner effect).

• On the other hand: it takes a very long timeto expel a typical magnetic flux from the neutronstar core (∼ 109 years, G.Baym, C.Pethick, D.Pines,1969). Therefore, if the magnetic field existed beforethe neutron star became a type-I superconductor, it islikely that magnetic field will remain there.

2. Resolution of the puzzle:

• The magnetic field could exist in macroscopicallylarge regions where there are alternating domains ofsuperconducting (type-I) matter and normal matter(Intermediate state). In this case, neutron starscould have long period precession.

• The well-know example of this phenomenon is dueto Landau, 1937 who argued that in this case a domain

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structure can be formed, similar to ferromagneticsystems, see commandment 5 below,

a =√

[ ∆df(h)], h ≡ H/Hc, f(h→ 0) ' h2

π ln 0.56h

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• Specifically, on a macroscopic distance scale,the magnetic flux must be embedded in thesuperconductor. This would mean that thesuperconductor is in an intermediate state.

• Another argument suggesting the same outcomefollows from the fact that topology (

∫d3x ~A · ~B

magnetic helicity, linking number) is frozen in theenvironment with high conductivity; therefore, themagnetic field must remain in the bulk of the neutronstar.

3. Intermediate state. General properties.

• The intermediate state is characterized byalternating domains of superconducting and normalmatter. The superconducting domains will then exhibitthe Meissner effect, while the normal domains will carrythe required magnetic flux.

• The pattern of these domains is usually stronglyrelated to the geometry of the problem.

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• The simplest geometry, originally consideredby Landau is a laminar structure of alternatingsuperconducting and normal layers.

4. Lessons from condensed matter physics

• It is known that the domain morphology is not athermodynamic state function; it depends on the pathin field-temperature space through which the samplehas been brought to a given point.

• For instance, cooling in zero field below Tc andthen applying the field, tends to produce patterns inwhich normal domains are embedded in a matrix ofsuperconductor.

• When the same point in T −H space is reachedby cooling below Tc in a fixed field the normal domainsconnect to the sample edges.

• These observations suggest that the patterns arenot true ground states of the system– the sample isnot in the thermodynamic equilibrium.

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VIII. Conclusion

The consequences of this picture still remain to beexplored. In particular, the standard theory of glitches(based on the ideas of type II superconductivity) hasto be reconsidered.

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Magnetic Fields in Neutron Stars - Institute for Nuclear ... · magnetic ﬁeld (Abrikosov lattice). This belief is based on simple estimations of the coherence length ξand the London