A Novel Mechanism for Type-I Superconductivity in Neutron Stars

0.22 MB·English

by  James Charbonneau

A Novel Mechanism for Type-I Superconductivity in Neutron Stars
James Charbonneau and Ariel Zhitnitsky
∗ †
Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada, V6T 1Z1
(Dated: February 4, 2008)
Wesuggestamechanismthatmayresolveaconflictbetweentheprecessionofaneutronstarand
thewidelyacceptedideathatprotonsinthebulkoftheneutronstarformatype-IIsuperconductor.
We will show that if there is a persistent, non-dissipating current running along the magnetic flux
tubes the force between magnetic flux tubes may be attractive, resulting in a type-I, rather than
a type-II, superconductor. If this is the case, the conflict between the observed precession and
the canonical estimation of the Landau-Ginzburg parameter κ > 1/√2 (which suggests type-II
behaviour) will automatically be resolved. We calculate the interaction between two vortices, each
carryinga currentj, and demonstrate thatwhen j > ~c ,where q is thecharge of theCooper pair
2qλ
andλistheMeissnerpenetrationdepth,asuperconductorisalwaystype-I,evenwhenthecannonical
7
Landau-Ginzburg parameter κ indicates type-II behaviour. If this condition is met, the magnetic
0
fieldiscompletely expelled from thesuperconductingregions oftheneutronstar. This leadstothe
0
formation ofthesocalled intermediatestate,wherealternatingdomainsofsuperconductingmatter
2 and normal matter coexist. We further argue that even when the induced current is small j < ~c
2qλ
n the vortex Abrikosov lattice will nevertheless be destroyed due to the helical instability studied
a previously in many condensed matter systems. This would also resolve the apparent contradiction
J with theprecession of the neutron stars. Wealso discuss some instances where anomalous induced

1 currentsmayplayacrucialrole,suchasinneutronstarkicks,pulsarglitches,thetoroidalmagnetic
3 field and the magnetic helicity.

3 PACSnumbers: 97.60.Jd,26.60.+c,74.25.Qt,97.60.Gb
v
8
0 I. INTRODUCTION field( 1012 G)aresufficientforbothsuperfluidandsu-
3 percon∼ducting vortices to form inside the neutron star.
1
This paper is motivated by calculations [1] that show It shouldbe notedthat when a superconductoris placed
0
7 a conflict between the observed magnitude ( 3◦) and in a rotating container it co-rotates with the container
0 frequency ( 1 per year) of neutron star’s pre∼cession [2] at the expense of a small current known as the London
/ and the wid∼ely accepted idea that protons in the bulk current [6].
h
of the neutron star form a type-II superconductor. We Vortices formed in a superfluid will always repel and
p
- begin with a review of this contradiction. form a lattice. Vortices formed in a superconductor can
o eitherattract,inwhichcasethemagneticfieldisexpelled
r froma superconductor,or they can repel eachother and
t
s A. Precession and Superconductivity formatriangularlattice. Thesetwobehaviourslabelthe
a
superconductor. Ifthevorticesattract,thesuperconduc-
:
v tor is called type-I and if they repel it is called type-II.
Inthegenerallyacceptedpicture,theinteriorofaneu-
i
X tron star contains neutrons and a small number of pro- The distinction between the two types of superconduc-
tivity will be very important in this paper.
r tonsandelectrons. Acompilationofneutronandproton
a scattering data [3] implies that the extremely cold (108 The standard way to determine the type of super-
K),dense(1015 g/cm3)natureoftheneutronstarshould conductivity is to calculate the value of the Landau-
Ginzburg parameter [5], κ = λ, where λ is the Meissner
causetheneutronstoform3P Cooperpairsandbecome ξ
2
a superfluid and the protons to form 1S Cooper pairs penetration depth and ξ is the coherence length of the
0 superconductor. If κ>1/√2 then the superconductoris
and become a superconductor.
type-II, otherwise it is type-I. This parameter is weakly
Arotatingsuperfluidcannotformasolidbodytocarry
dependent on density. Typically for nuclear matter in
circulationbut insteadforms vorticesofquantizedcircu-
a neutron star, λ 80 fm and ξ 30 fm [1], which
lation that run in the direction of the angular velocity
∼ ∼
means that κ 2.6. At higher densities it is possible to
[4]. It is also generally accepted that the protons form
∼
haveaLandau-Ginzburgparameterthatindicatestype-I
a type-II superconductor, which means that it supports
behaviour [7], but only in the densest part of the core.
a stable lattice of magnetic flux tubes in the presence of
There still exist large regions of the neutron star where
a magnetic field [5]. The rotation of the neutron star
(1 103 Hz) and the presence of the enormous magnetic type-II behaviour is predicted.
− In a neutron star both condensates are subjected to
angularmomentum andmagnetic flux and formlattices,
but the proton vortices ( 1024 per m2) are much more
∗Electronicaddress: [email protected] numerousthantheneutro∼nvortices( 1010 per m2)and
∼
†Electronicaddress: [email protected] are tangled around them. It is the formation of these
2
lattices that cause the contradiction with the precession B. A New Mechanism for Type-I
of the neutron star. Superconductivity
The existence of precession means that the superfluid
neutronvorticesno longerformalongthe rotationalaxis In this work we suggest a mechanism that leaves the
ofthe star,but alongthe axisthatis the sumofthe pre- Landau-Ginzburg parameter unchanged, but causes the
cession and angular momentum vectors. When the star system to behave quite differently than in the standard
precessesthevorticesnowmovewithrespecttotherota- picture. To be precise, we will show that even when
tion of the star and, in turn, with respect to the proton κ > 1 the system prefers the intermediate state and
√2
vortices which are entangling them. If the precession is thatthe apparentcontradictionbetweenκ 2.6andthe
∼
large enough one of two things must happen; either the observed precession is avoided.
neutron vortices move with the proton vortices or they This is achieved if the system supports a persistent,
pass through each other. non-dissipating current running along the core of a vor-
tex. Suchtopologicalcurrentsappearinmanysystemsas
Requiringtheneutronandprotonvorticestomoveto-
the consequence ofaquantumanomaly. It is wellknown
gether places severe restrictions on the precession. This
thatanomalies,andthetopologicalcurrentstheyinduce,
is the case of ”perfect pinning” that was discussed in [8]
haveimportantand non-trivialimplications; the electro-
using macroscopic dynamics. We will follow the argu-
magnetic decay of neutral pions π0 2γ is a textbook
ments in [1] as we are specifically interested in proton
→
example.
vorticesas amechanismfor pinning. Becausethe coreof
Aswewilldiscusslater,analogoustopologicalcurrents
the star is superconducting, the proton vortices, which
have even been observed in some condensed matter sys-
carrymagnetic flux,areresistantto being moved[9]and
tems. However, with a few exceptions, the analysis of
thus the neutron vortices are restricted to move slowly.
quantum anomalies has not receivedattention in the lit-
This means that the neutron vortices are pinned to the
erature devoted to dense matter systems in general, and
rotation of the protons and thus are pinned to the rota-
neutron stars in particular. In section II we discuss in
tion of the crust. If this pinning is present the neutron
detailwhy andwhensuchtopologicalcurrentsmayarise
starcanonlyprecessatveryhighfrequencies. Ifthestar
in high density systems.
is to precess more slowly at large amplitudes then it is
If such currents are induced, they drastically change
necessary for the neutron vortices to pass through the
the behaviour of the system. Normally the interaction
proton vortices.
between superconducting vortices has two terms: an at-
The case of ”imperfect pinning” was first discussed in
tractive force which comes from the order field and a
[10] using the concept of vortex drag. Microscopically,
repulsive force which comes from the gauge field. In the
this drag is created by large numbers of neutron vor-
presence of an induced current, a third, attractive force
tices passing through proton vortices [1]. This creates a
will appear and, if the current has sufficient magnitude,
numberofexcitationsandisahighlydissipativeprocess.
the system will behave as a type-I superconductor. This
Both methods find that the precessionis highly damped
situation will be discussed in sections II and IV.
and that there are no modes of large, persistant preces-
sion. Basedonthese estimatesit is concludedthatgiven
the observedprecessionneutronvorticesandprotonflux
C. Relation to Previous Work
tubes cannot coexist in the star [1]. Either the star’s
magnetic field does not penetrate any part of the core
The same problem has been discussed previously in
that is a type-II superconductor or that at least one of
[11],[12]. Inparticular,[11]hasshownthattheexistence
the hadronic fluids is not superfluid. Based on pairing
of a type-I superconductor in a neutronstar resolvesthe
calculations that predict neutron and proton superfluids
conflict addressed in [1]. Reference [11] starts by assum-
coexistinthe outercorethe latter is veryunlikely,so we
ing that the equilibrium structure of a type-I supercon-
will look to the former.
ductor contains alternating superconducting and normal
If the core is a type-I superconductor, the magnetic
domains. By use of a hydrodynamic restriction based
flux could exist in macroscopic regions of normal mat-
on the moment of inertia of the crust and the moment
ter that surround superconducting regions known as an
of inertia of the superfluid it is shown that the alternat-
intermediate state. The magnetic flux would not form
ing domainstructure seenin type-I superconductorswill
proton vortices and there would be nothing to impede
always allow for undamped precession.
the movement of the neutron vortices, thus allowing the
This result can be understood in a different way. The
star to precess with a long period.
arguments of reference [1] relied on the proton vortices
For proton vortices to not form a lattice, a mecha- tanglingaroundtheneutronvortices. Inadomainstruc-
nism must be present to make the interaction between ture there is moreroomforthe neutronvorticesto move
them attractive (type-I behaviour). If it does so even unhindered by the protonvorticesallowing for largeam-
when the cannonical Landau-Ginzburg parameter sug- plitude, high frequency precession.
gests that the superconductor is type-II then the incon- However,the calculationin reference [11] starts by as-
sistency described above will be resolved. suming that type-I superconductivity already exists in
3
neutronstars. Itdoesnotinvestigatehowatype-Isuper- localized inside of vortex or uniformly distributed over a
conductorcouldariseanditis notobvioushowtorecon- large area S. Also, corrections due to non-zero fermion
cile this with standard arguments that suggest κ 2.6, mass and temperature can be explicitly calculated, but
∼
which ambiguously implies that the superconductor is will not be discussed in the present paper.
type-II in a finite volume of the neutron star. It is worth noting that similar non-dissipating cur-
In reference[12] a mechanismhas been suggestedthat rentshave beendiscussedin condensedmatter literature
potentiallyresolvesthe conflictbetweenthe standardes- [19, 20, 21]. In particular, the expression for anomalous
timation of κ and type-I behaviour. The mechanism is supercurrenthasbeenderivedfor3He-Asystembasedon
based on the idea that proton and neutron Cooper pairs the chiral anomaly (see eq.(5.35) and Fig.23 in ref.[19]).
have almost identical interactions,akin to their underly- Some suggestionsof how these effects canbe experimen-
ingisotopicalsymmetry,eventhoughtheFermimomenta tally tested were also presented in ref.[19]. An impor-
and densities for protons µp and neutrons µn are vastly tant feature of the 3He-A system is the existence of the
different. A small difference in interactions was modeled anomalouschiralsymmetry, which is not presentin 4He.
by a smalleffective asymmetryparameterξ 1. In this Therefore this phenomenon exists in 3He-A but not in
≪
caseithasbeenshownthatthetypeofsuperconductivity 4He.
is not governed by the canonical Landau-Ginzburg pa- Anomalous supercurrent may also exist in high T su-
c
rameterκ,but instead,by aneffective Landau-Ginzburg perconductors with d-wave paring or in a graphene sys-
parameterκ=ξκ. Thiseffectiveparametercanbesmall, tem when the relevant degrees of freedom satisfy the
κ<1/√2leadingtotype-Isuperconductivitywhilekeep- masslessDiracequation,withavelocityoforderonehun-
ing canoniecal Landau-Ginzburg parameter large, κ > dredth that of light. In these cases, the chiral symmetry
1e/√2 [50]. isobviouslypresentandthereisagoodchancethatnon-
dissipating, topological currents may exist.
For now we will restrict our discussion to QCD. Note
II. PERSISTENT NON-DISSIPATING that in case of equal chemical potentials, µ = µ = µ,
R L
TOPOLOGICAL CURRENTS AND VORTICES
the vector current is not induced due to the exact can-
cellationbetweenlefthandedandrighthandedfermions.
Thoughtheideaofnon-dissipatingtopologicalcurrent This wouldbe exactly the case in normalnuclearmatter
in vortices was considered long ago [14] in the context when π meson condensation does not occur or in color
of cosmic strings, we are more interested in the recent superconducting phases, such as CFL at asymptotically
developmentsofsimilarphenomenainhighdensityQCD largechemicalpotentials,whereGoldstonemodesdonot
[15,16,17,18]andcondensedmattersystems[19,20,21]. condense (see recent review [22] and references therein
The most important result can be formulated as follows. on color superconductivity).
ConsiderQCDwithnon-vanishingchemicalpotentials, However, it is known that Goldstone modes are likely
µL and µR, which correspond to two reservoirs of parti- to condense in nuclear matter [23, 24] and will definitely
cles with different chirality. Due to the chiral anomaly condense in color superconducting phases for intermedi-
a number of interesting macroscopic phenomena could ate chemical potential [25], [26], [27], [22]. In the cases
occur: induced non-dissipatingvectorcurrentsonanax- of neutral π0 condensation in nuclear matter [24] and η
ial vortices, induced axial currents on a vector vortices, condensationinthecolorsuperconductingphase[27]the
magnetizationof the axialdomain wall, induced angular vector current will definitly be induced, as shown by the
momentum by vortex loops, to name just a few. simple argument below.
Presently we are interested in the phenomena where The condensation of charged Goldstone mesons
currents are induced in the background of an external (π ,K)inthe systemismuchmorecomplicated,butwe
±
magneticfield. Tobe precise,inthe chirallimitandzero canexpect that the condensationof the chargedpseudo-
temperature (mq = 0,T = 0), each fermion species q scalar Goldstone mesons in the system will lead to dif-
makes an additive contribution to the vacuum expecta- ferent densities for L and R species (and correspond-
tion values for the axial and vector currents, ing to different effective chemical potentials for R and
L modes), in which case the vector current will be also
e(µ +µ )
A R L induced.
j dS = Φ, (1)
hZS · i 4π2 It is not the main goal of this paper to describe all
V e(µL µR) possible phases where axial density (and consequently,
j dS = − Φ.
h · i 4π2 the vector current) can be induced. Rather, we want to
ZS giveasimpleargumentdemonstratingwhythecondensa-
where Φ = d2x Bz(x ) is the total magnetic flux tionofapseudo-scalarGoldstonemodewouldnecessarily
throughthecrossse⊥ctionS⊥,andthefermioncurrentden- lead to a difference in densities for left handed and right
sitiesaredefinRedasjA =q¯γ3γ5q, jV =q¯γ3q. Afewcom- handed species.
ments are in order. Formula (1) has a universal nature, To simplify arguments, we consider a model with just
as it originates from the fundamental quantum anomaly a single flavor. Let us assume that we are in a phase
and it is not sensitive to whether the magnetic field is where the baryon density is non-zero and a neutral η
4
Goldstone mode is condensed. This implies that our glected so E measures free energy per unit length,
ground state can be understood as a coherent superpo-
sition of an infinitely large number of the Goldstone η E = d2x ~2 iqA(x) ψ(x) 2 µ ψ(x)2
mesons. We expect that the ground state of the sys- Z (2m(cid:12)(cid:18)∇− ~c (cid:19) (cid:12) − b| |
tem is not disturbed by adding one extra η meson into (cid:12) (cid:12)
a (cid:12) 1 1(cid:12)
the system. On the other hand, we can relate the ma- + ψ(x)4+(cid:12) ( A(x))2+ (cid:12)j A , (2)
trix element with an extra η meson to the matrix ele- 2| | 8π ∇× c ·
(cid:27)
mentwithouttheη usingthe standardPCACtechnique,
where µ is the chemical potential of the Cooper pairs
hA|O|Aηi ∼ ihA|[O,Q5]|Ai. In the present case the co- andaisbrelatedtothescatteringlengthl,a= 4π~2l. The
efficient of proportionality would not be precisely 1/F m
ψ fielddescribes only the superconducting componentof
(where F is the Goldstone coupling constant) because
the protons. Remember that the fundamental particle
our Goldstones are in the η condensed phase rather
h i here is the proton Cooper pair whose mass is actually
than in a trivial vacuum.
m = 2m and its charge is q = 2e. The current source
p
| |
Taking A to be the ground state and O the baryon thatwasaddedwillbeusedtomodelthecurrentflowing
| i
density operator,one canimmediately see thatif baryon along the core of the vortex. As discussed earlier, this
densitydoesnotvanishinthegroundstate,thentheaxial current can be treated as an external electromagnetic
density [O,Q5] will not vanish also[51]. This implies current.
h i
thatdensitiesforlefthandedandrighthandedspeciesare The Landau-Ginzburg free energy without the extra
different and therefore a vector current will be induced. term for the current source has a symmetry under the
We assume this to be the case in what follows. following gauge transformation,
We should note that the condensation of the pseudo
A(x) A(x)+ ϕ(x) ,
scalar Goldstone mode is not the only mechanism capa- → ∇
ble to produce the asymmetry between R and L modes; ψ(x) e~iqcϕ(x)ψ(x) . (3)
→
any P parity violating processes can do the same job.
Weassumethattheexternalcurrentjisconserved j=
The crucial point here is not the ability to produce the ∇·
0 in which case (3) obviously remains a symmetry.
asymmetry (which is a common phenomenon in neutron
We will choose a form for our current source, j =
starsduetotheneutrinoemission),buttheabilityofthe
jδ2(x)zˆ, which models a current at r traveling in the
non-dissipating persistent currents to keep this asymme-
zˆ-direction.
try through the entire volume of the star and deliver it
Thefreeenergycanbeminimizedtoyieldthetheequa-
to the surface of the star. The phenomenological signifi-
tions of motion. Minimizing with respect to the vector
cance of this is argued in section V.
potential A yields a Maxwell equation for our system,
As all quarks have non-zero electromagnetic charges,
1 ~q 1
onceavectorcurrentis induced, anelectromagneticcur- 2A ( A) = j j. (4)
Noether
rent will also be induced. Our next step is to derive the 4π ∇ −∇ ∇· −m − c
interaction between two superconducting vortices where The right(cid:0)hand side is writt(cid:1)en in terms of the Noether
an induced electromagnetic current is present in their current
cores. Our ultimate goalis to understand (at least qual-
1 q
ittoattihveesley)intdhueccehdacnugrersewnthsi.chThwiisllisotchcuersuinbjtehcetsoyfstthemendeuxet jNoether = 2i ψ†∇ψ−ψ∇ψ† − ~cA|ψ|2 . (5)
two sections. Minimizing with res(cid:0)pect to the orde(cid:1)r field ψ gives
~2 iq 2
A ψ =a ψ 2ψ µ ψ. (6)
2m ∇− ~c | | − b
(cid:18) (cid:19)
III. STRUCTURE OF A CURRENT CARRYING For determining the structure of the vortex we choose
VORTEX toplaceitattheoriginandwrite theansatzincylidrical
coordinates,
In formulating the problem we assume that the su- µ
ψ = bρ(r)eiφ, (7)
perconductor is due to non-relativistic proton Cooper a
r
pairing though similar results are also valid for phases
where a relatevistic field theory should be used. It does ~qa(r)
not change the qualatative picture described below. We A= φˆ+f(r) zˆ, (8)
c r
startwiththetwodimensionalLandau-Ginzburgfreeen-
ergy with term added to model a current source j. As where n = µb is the density of the superconductor.
0 a
discussed earlier, this current is an induced persistent The function ρ(r) (0,1) describes the profile func-
∈
electromagnetic current and it couples to the gauge field tion of superconducting density, ρ(r) = 0 being no su-
naturally. Dependance along the third direction is ne- perconducting material and ρ(r) = 1 being completely
5
superconducting. If we assume that the current goes where ξ = ~2 is the coherence length.
to zero far from the origin we can use (5) to see that 2mµb
In the neqxt section, when calculating the vortex inter-
lim a(r) = 1 and lim f(r) = 0. The phase of ψ
r r actions,itwillbeusefulto“unwind”thephaseofthevor-
is ch→os∞en to mimic a vorte→x∞with winding number n=1
tex. Thiswillgiveamuchcleanersolutionandisdoneus-
and only depends on the φ coordinate.
ing a gauge transformation (3), where ϕ(x)= ~c φ.
In order to calculate long range interactions between − q
vortices we are interested in solutions the equations of The solutions for the field equations become, (cid:16) (cid:17)
motion as r . To decouple our set of differential
→ ∞ ~c r 2j r
equations it is convenient to define A= c K φˆ c K zˆ, (17)
φ 1 z 0
qλ λ − c λ
ρ(r)=1+σ(r) , (9) (cid:16) (cid:17) (cid:16) (cid:17)
µ √2
b
a(r)=1+rα(r) , (10) ψ = 1 cσK0 r . (18)
r a " − ξ !#
such that σ(r), α(r) 0 as r . Substituting (7)
→ → ∞ We can now move on to calculating the interactions be-
and (8) into equation (4) and linearizing yields the two
tween vortices.
equations,
∂2α 1∂α 1 1
+ + α=0, (11) IV. INTERACTION BETWEEN TWO
∂r2 r ∂r − r2 λ2
(cid:18) (cid:19) CURRENT CARRYING VORTICES
1 4π Superconducting vortices without currents interact
2 f(r)= jδ2(x), (12) through two forces. There is an attractive force caused
∇ − λ2 c
(cid:18) (cid:19) bythesuperconductingorderparameterwantingtohave
onedefectinsteadoftwoandarepulsiveelectromagnetic
whereλ= mc2 istheLondonpenetrationdepth. The
4πq2n force caused by the charges swirling around the vortex.
first equatiqon is the modified Bessel equation of the first Two vortices placed side by side have currents running
order. Wewantasolutionthatgoestozeroasr so inoppositedirectionsontheir nearestsidesanditiswell
→∞
wechoosethesolutiontobeamodifiedBesselfunctionof known that opposite currents repel.
the second kind, α(r)= cλφK1 λr . The second equation Suppose there are two wires, placed parallel to each
is just a statement of the Green’s function, other, carrying current. If the currents run in the same
(cid:0) (cid:1)
direction the wires will be attracted to one another.
( 2 α2)K0(αr)= 2πδ2(x), (13) Now consider that superconducting vortices, instead of
∇ − −
wires, are carrying the current. There are three forces
which implies that f(r) = −2cjczK0 λr . Going back working against each other: the attractive electromag-
through all the substitutions we find that the vector po- netic force from the current, the repulsive electromag-
(cid:0) (cid:1)
tential is netic force from the gauge field, and the attractive force
~c 1 c r 2j r from the order field. If the current were strong enough,
A= + φK φˆ c K zˆ. (14) theattractiveforcewouldbestrongenoughtocompletely
1 z 0
q r λ λ − c λ
(cid:20) (cid:16) (cid:17)(cid:21) (cid:16) (cid:17) cancel the force from the gauge field and the vortices
would always attract and the superconductor would al-
In comparisonthe standard case without j we see that a
ways act like a type-I.
third term has appeared in the interaction. This attrac-
tive componentwillplaythe crucialrolein whatfollows.
Asimilarprocedurefollowsforthesolutiontotheorder
A. Calculations
field. Substituting (7) and (10) into (6) and linearizing
yields
The philosophy behind calculating the interaction be-
1 ∂ ∂σ 4mµ tween vortices is to find the energy of the entire system
b
r = σ. (15)
r∂r ∂r ~2 and then subtract off the energy of the individual vorti-
(cid:18) (cid:19) cies as originally outlined in [29]. The technique we will
This is a modified Bessel equation of the zeroth order usewasintroducedin[30]andhasbeenusedtocalculate
which has a solution, σ(x) = c K √2r . Substituting vortex interactions in models with two order parameters
σ 0 ξ [12], [31]. The same philosophy as in [29] is used but the
this back we find that, (cid:16) (cid:17)
actual calculation becomes much less cumbersome. We
will reduce the theory to a non-interacting, linear one
µ √2
ψ = b 1 c K r eiφ, (16) andthenmodelthevorticiesaspointsources. Theinter-
σ 0
r a " − ξ !# action energy is then calculated from this linear theory.
6
To make the calculation easier it is useful to use a written[52],
gauge tranformation to remove the phase in ψ(x). This
is described in the previous section and yields the form ~2µ √2
ψ = µab(1−σ). To linearize the theory we expandinρ Eint = Z d2x(−m ab2πδ2(|x−x1|)K0 ξ |x−x2|!
and A and keep only quadratic terms to get
p ~2c2 ∂δ2(x x ) x x
1 2
− K | − |
E = d2x µb ~2 ( σ)2+2µ2bσ −2q2λ ∂r 1(cid:18) λ (cid:19)
free
a 2m ∇ a 2j j x x
Z (cid:26) 1 2δ2(x x )K | − 2| ,
1 A2 − c2 − 1 0 λ
+ ( A)2+ . (19) (cid:18) (cid:19)(cid:27)
8π ∇× λ2 1 ~c 2 4q2λ2j j d
(cid:18) (cid:19)(cid:27) 1 2
= 1 K
2 qλ − ~2c4 0 λ
We now add source terms to model the vortices, (cid:18) (cid:19) (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)
√2d
K , (27)
Esource = d2x τσ+J A , (20) − 0 ξ !#
{ · }
Z
where d= x x . To evaluate the second term in the
where τ and J are the sources for the fields σ and A. | 1− 2|
integral we made use of equations (23) and (13).
Minimizing this we get the equations of motion,
If j and j are set to zero we obtain the interaction
1 2
2 m a between gauge vortices without current. The only new
∇2− ξ2 σ = ~2µ τ, (21) pieceintheinteractionisthatwhichcomesdirectlyfrom
(cid:18) (cid:19) b thecurrent. Ifj andj runinthesamedirectionthereis
1 2
an attractive force and if they run in opposite directions
1 there is a repulsive force. This is the expected result if
2 A=4πJ . (22) we considered parallel wires carrying current.
∇ − λ2
(cid:18) (cid:19) The interaction energy (27) determines whether the
We want to solve for the sources J and τ such that σ vortices attract or repel and whether we see type-I or
and A have the same asymptotic solutions we obtained type-II behaviour in the superconductor. If we set j1 =
earlierin(17)and(18). Using(13),thederivativeof (13) j2 = j then there are two cases to explore; one when
with respect to the radialcomponent r, and the identity j < ~c2 and one when j > ~c2. In the first case the
2qλ 2qλ
first term of (27) is positive and we obtain the canon-
d
K (αr)= αK (αr), (23) ical behaviour for a superconductor where the Landau-
0 1
dr − Ginzburgparameterdecideswhetherthesystemexhibits
type-I or type-II behaviour.
we can solve for the sources,
~2µ
τ = b2πδ2(x), (24) B. Discussions
−m a
First, let us consider the regime when currentis large,
~c∂δ2(x) j
J = φˆ+ δ2(x) zˆ. (25) ~c2
2q ∂r c j > , (28)
2qλ
The interaction energy is found by substituting J =
J +J , A = A +A , τ = τ +τ and σ = σ +σ and the first term in (27) becomes negative. This means
1 2 1 2 1 2 1 2
that there is no longer a repulsive term present in the
intothe totalenergyE =E +E andsubtracting
free source
interaction and the vortices will always attract. This
of the energies of the vortices, leaving only cross terms.
is the main result of this paper. If the condition (28) is
Thesubscripts1and2refertotwoseparatevorticesand
met,allthecomponentsoftheinteractionbecomeattrac-
positions x and x respectively. Using the equations of
1 2
tive. While a naive calculation of the Landau-Ginzburg
motion we get left over cross terms that are interpreted
parameter suggests it is a type-II superconductor, it ac-
as the interaction energy;
tually behaves like type-I, and the conflict is resolved.
In this case an intermediate state will be formed, sug-
E = d2x τ σ +J A . (26)
int 1 2 1 2 gestingthatalternatingdomainsofsuperconductingand
{ · }
Z normal matter coexist in this regime. This state would
Thoughitisnotapparent,theinteractionenergyissym- be the lowest energy state on the phase diagram with
metric in the exchange of the subscripts 1 and 2. The given B,j satisfying condition (28). This implies that
apparent asymmetry arises when the equations of mo- type-II vortices don’t form at any time in the neutron
tion for either subscript 1 or 2 are substituted in. Us- star’s life, even during the short period of cooling when
ing (8), (9), (24) and (25) the interaction energy can be the transition to a superconducting state takes place.
7
Oneshouldremarkthatduetothestaticnatureofthe that may replace the Abrikosov lattice. We shall re-
problem the result (28) persists for relativistic systems, fer to the absence of the Abrikosov lattice (which is a
which may be relevant for study color super conducting consequenceoftype-IIsuperconductivity)asatype-Isu-
phases. The equations of motion used to derive (28) re- perconductor that supports the intermediate state even
~2 though many other phases may result from current in-
mainunchangedifwereplace ψ φ andredefine
2m → rel duced vortex instabilities. Therefore, the conflict be-
the corresponding coupling coqnstants.
tween the precession of a neutron star and the standard
Nowletusconsideramorerealisticcasewhenthecur-
estimationofthe Landau-Ginzburgparameterlikely will
rent is small,
be resolved even when induced currents are small.
~c2
j < . (29)
2qλ
V. CONCLUSION AND SPECULATIONS
In this case we cannot make any precise statements
within our framework. However, based on experience If currents are induced in vortices then we have found
in similar situations in condensed matter systems one a mechanism that reconciles the condradiction between
should expect very dramatic changes to the vortex lat- the precession of neutron stars and the standard pre-
tice when there are currents directed along the external sumptionthatthereistype-IIsuperconductivityinsidea
magnetic field [32],[33],[34],[35],[36],[37]. neutronstar. A sufficientlystrongcurrentrunningalong
Inthis literature,the presenceoflongitudinalcurrents the core of the vortex and satisfying inequality (28) al-
is shown to cause a vortex to develop a spiral-vortex in- lows the vorticesto attract evenif the Landau-Ginzburg
stability. This instability can be delayed for small cur- parameter indicates they should repel.
rents or even stabilized due to the impurities. The les- A neutron star would rather form the domain struc-
son fromthese condensed matter systems is that when a ture seen in type-I superconductors rather than the vor-
current aligns with the magnetic field of the vortex the texlatticestructureseenintype-IIsuperconductors,thus
properties of the vortex lattice are completely changed resolvingthepuzzle. Wealsoarguedthatevensmallcur-
or destroyed. rentsalongmagneticfieldcancompletelychange/destroy
We expect similar behaviour in regions of the neutron the structure of the Abrikosov lattice.
star where both the Landau-Ginzburg parameter sug- A pertinent questionis whether these currents canac-
gests type-II behaviour and longitudinal currents are in- tually be induced in neutron stars. The answer depends
duced. While many features of the system are still to be crucially on the details of the specific phase realized in
explored, the main massage for the presentstudy that it the core of a neutron star. As formulated in section III,
is very likely (similarly to CM studies mentioned above) the electromagneticcurrentswill be inducedif the Gold-
that even small currents (1) can completely destroy the sone modes condense in the presence of a background
vortex lattice by replacing it with a new still unknown magnetic field.
structure. Ifweassumethisisthecasethentherearemanyques-
It is not our purpose to discuss the rich physics re- tions to be considered. How would the magnetic field be
lated to vortex instabilities resulting from longitudinal distributed? What is the fate of these currents?
currents, but rather stress that the resulting state will A) If the currentis large,it is expected that the mag-
definitelybe nottherigidAbrikosovlattice. Itisunclear netic field could exist in macroscopically large regions
whatstructurewillreplacetheAbrikosovlatticebutitis where there are alternating domains of superconducting
reasonable to believe that superconductivity would per- (type-I) matter and normal matter - the so called inter-
sistinthis new regime;the energyscalesassociatedwith mediate state. It has been estimated long ago [9] that it
currentsaremuchsmallerthanthesuperconductinggap. takes a very long time to expel a typical magnetic flux
It is possible (but not necessary) that the intermedi- from the neutron star core. Therefore, if the magnetic
atestatetypicalfortype-Isuperconductivitywilldevelop field existed before the neutron star became a type-I su-
and alternating domains of superconducting and normal perconductor (before it sufficiently cooled down), it is
matterwouldcoexist. Thesizeandshapeofthedomains likely that the magnetic field will remain there.
areknowntobeverysensitivetomanythings: geometry, The intermediate state is characterized by alternating
initialconditions,themethodofpreparationofasample, domains of superconducting and normal matter where
boundaryconditions,surfaceeffects. Asisknown,thein- thesuperconductingdomainsexhibittheMeissnereffect,
termediate state is not in thermodynamic equilibrium in while the normal domains carry the required magnetic
thestrictthermodynamicalsense,butratherdepends on flux. The pattern of these domains is strongly related
the history of the system. It is also possible that other to the geometry of the problem. The simplest geometry,
states, such as Bragg glass phase [36] would develop, or originally considered by Landau [38], is a laminar struc-
vortex -lattice melting transition would take place [37]. ture of alternating superconducting and normal layers.
The exact state is not essential at the moment. What While precise calculations are required for understand-
is essential is that the Abrikosov lattice is destroyed by ing of the magnetic structure in this case [53], one can
longitudinal currents. There are many alternative states give some simple estimation of the size of the domains
8
using the calculations Landau presented for a different the surface of the star may it result in producing the
geometry. His formula [38] suggests that the typical size proper motion of the entire star.
of a domain is Duetotheirtopologicalnature,thecurrent(1)maybe
capableofdeliveringtherequiredasymmetryproducedin
a 10√R∆, (30) theinteriorofthestartothesurfacewithoutdissipation.
∼
Even in a strongly interacting theory, the current (1)
where R is a typical external size identified with a neu- is persistent and non-dissipating. In an environment as
tron star core (R 10 km), while ∆ is the typical width unfriendly asthe dense quark/nuclearmatter in neutron
∼
ofthe domainwallseparatingnormalandsuperconduct- starsthereisstillnodissipationduetore-scattering,and
ing states. We estimate ∆ λ as the largest micro- can be effectively used to deliver information across the
∼
scopical scale of the problem. Numerically, a 10 1 cm bulk of the star.
−
∼
which implies that a typical domain can accommodate When the current makes a U-turn on the surface, a
104 neutron vortices separated by a distance 10−3 largeamountofmomentum(duetophotonemission)can
∼ ∼
cm. be transfered to the star. Therefore, this is a unique op-
B)Whatisthefateofthesecurrents? Itisquitepossi- portunity to use our topological currents (1) for deliver-
blethatthecurrentwilltravelinsidethesuperconducting ing the asymmetry produced in the bulk ofthe star (e.g.
regioninonlyonedirection. Thecurrentisconserved,so dueto theGoldstonecondensation)tosolvethe problem
it must make a U-turn and start travelling in the oppo- of neutron star kicks [41, 42].
site direction either along the crust of the neutron star One should notice that the currents may not satisfy
or through regions of normal matter in the intermedi- constraint eq. (28) for the explanation of neutron star
ate state. This is similar to the case of 3He-A system kicks. Indeed, relatively small current is still capable of
discussed in ref.[19] (see Fig.23 from that reference). transfering momentum because the U-turn mechanism
C) If the current on the way back travels through the remains operative.
normal matter of a different domain (or the crust) then This asymmetry mechanism is different from most in
large current loops with typical sizes comparable to the that it does not occur during the supernova, but over
neutron star radius would be created. These large cur- a long period of time. The momentum transfered from
rentloopswouldinduceacoherenttoroidalmagneticfield eachemmitedphotonissmallbutifgivenlongenoughis
that, when combined with the poloidal field present in a sufficient to accelerate the neutron star to the observed
neutron star, would create a non-zero magnetic helicity. proper velocities.
Such a toroidal magnetic field is apparently necessary E) A different, but likely related phenomena, is the
todescribethetemperaturedistributionofthecrust[39]. recentobservationofpulsarjets[44]whichareapparently
It has been also argued long ago that a toroidal compo- relatedto neutronstar kicks[45, 46]. It has beenargued
nent in the magnetic field of a neutron star is necessary that spin axes and proper motion directions of the Crab
for stablility of the poloidal magnetic field [40]. and Vela pulsars are aligned. Such a correlation would
D) The existence of a current making a U-turn near follownaturallyifwesupposethatthekickiscausedbya
the surface of the star may be a key to the understand- non-dissipatingcurrent,assuggestedinD).Thecurrent,
ing of the long standing problem of neutron star kicks andthusthepropermotion,isalignedwiththemagnetic
[41, 42]. As is known, pulsars exhibit rapid proper mo- field, which itself is correlated with the axis of rotation.
tion characterized by a mean birth velocity of 450 90 Aswementionedabove,theU-turnmechanismisnec-
±
km/s. Their velocities range from 100 to 1600 km/s [41] essarily accompanied by the photon emission (which de-
with about 15% of all pulsars having speeds over 1000 livers the momenta required for the neutron star kick).
km/s[42]. Pulsars are born in supernova explosions so It would be very tempting to identify the observed in-
common theories naturally to look for an explanation in ner jets [44] with the photons emitted when the current
the internal dynamics of the supernova. However,three- makes the U-turn and starts travelling in the opposite
dimensional numerical simulations [43] show that even direction along the crust. In this sense the mechanism
themostextreme,asymmetricexplosionsdonotproduce forthekickissimilartotheelectromagneticrocketeffect
pulsarvelocitiesgreaterthan200km/s. Therefore,adif- suggested previously[46].
ferent explanation is needed. F) What else could happen with vortices near the
Theoriginofthesevelocitieshasbeenthesubjectofin- crust? Reference [47] presents calculations in which vor-
tensestudy. Manyofthetheoriesinvolveanassymetryin tices near the crust are grabbed and bundled together
the star’s structure, and indeed, many mechanisims are bythe Kelvin-Helmoltz [KH]wavescreatedby the insta-
capable “in principle” of producing the required asym- bility that arises when there is shear stress between two
metry. In the presence of an external magnetic field, the fluids. Thisbundlingofvorticescouldtwistthemsothey
neutrinos produced in the star are automatically asym- no longer line up in an array, but instead form vortex
metric with respect to the direction of B~. However, the loops (vortons). These vortex loops lie in a plane per-
most common problem with the suggested mechanisms pendicular to the angular momentum rather than than
is the difficulty of delivering the producedasymmetry to along it.
thesurfaceofthestar. Onlywhentheassymetryreaches Such surface KH instability may explain pulsar
9
glitches[48]. Spiralvortexinstability[32]observedincon- remain to be explored.
densedmattersystemsmayalsohavesomerelationtothe
formation of these vortons and to glitches. The helical
structure of the vortex could expand to the surface and Acknowledgments
transfer its angular momentum to the crust.
Thevortexloopsmadefromsuperfluidvorticesarenot
TheauthorswouldliketothankJeremyHeylforuseful
typically stable (similar to cosmic strings [14]) but the
discussions and Lars Bildsten for the discussions during
presence of a current in the core of a superfluid vortex
his visit to Vancouver. The authors would also like to
and an external magnetic field could make these loops
thank Bennett Link, Maxim Lyutikov and Armen Se-
stable. If vortons are stable then they could attract to
drakian for their many useful comments. This work was
each other and form a column. Since they attract very
supportedinpartby the NationalScience andEngineer-
closetoeachotherthisstructurewouldceasetolooklike
ing Research Council of Canada.
abunchofindividualvorticesbutlikeasinglecylindrical
vortex sheet which carries a surface current similar to a
solenoid. Vortex sheets have been studied in the context
of superfluidity by Landau and Lifshitz [49]. References
Theseandmanyotherconsequencesofthispicturestill
[1] B. Link, Phys. Rev. Lett. 91, 101101 (2003); B. Link [22] M. Alford and K. Rajagopal, arXiv:hep-ph/0606157.
A&A458, 881 (2006) [arXiv:astro-ph/0608319] [23] D. B. Kaplan and A. E. Nelson, Phys. Lett. B 175, 57
[2] I.H. Stairs, A. G. Lyne,and S. L. Shemar, Nature 406, (1986).
484 (2000); [24] K. Takahashi and T. Tatsumi, Phys. Rev. C 63, 015205
T. V. Shabanova, A. G. Lyne, and U. O. Urama, Astro- (2001).
phys.J. 552, 321 (2001). [25] T. Schafer, Phys. Rev. Lett. 85, 5531 (2000)
[3] A.Sedrakian and J. W. Clark, arXiv:nucl-th/0607028. [arXiv:nucl-th/0007021].
[4] R.P.Feynman,Prog. Low Temp. Phys., 1, 17 (1955). [26] D.B. Kaplan and S. Reddy, Phys. Rev. D 65, 054042
[5] Lifshitz and Pitaevskii, Statistical Physics Part 2, Else- (2002) [arXiv:hep-ph/0107265].
vierButterworth-Heinemann, 1998 [27] A. Kryjevski, D. B. Kaplan and T. Schafer, Phys. Rev.
[6] J. A. Sauls, NAToASISeries C 262, 457 (1989) D 71, 034004 (2005) [arXiv:hep-ph/0404290].
[7] D.M.Sedrakina,A.DSedrakianandG.F.Zharkov,Mon. [28] M.A.MetlitskiandA.R.Zhitnitsky,Nucl.Phys.B731,
Not.R. Astron. Soc., 290, 203 (1997) 309 (2005) [arXiv:hep-ph/0508004]; Phys. Lett. B 633,
[8] J.Shaham, Astrophys.J., 214, 251 (1977) 721 (2006) [arXiv:hep-ph/0510162].
[9] G.Baym, C.Pethick,D.Pines, Nature 224, 673 (1969). [29] L. Kramer, Phys.Rev.B 3, 3821 (1971).
[10] A.Sedrakian,I.WassermanandJ.M.Cordes,AstroPhys. [30] J. M. Speight, Phys.Rev. D 55, 3830 (1997).
J., 524, 341 (1999) [31] R.MacKenzie, M.A.VachonandU.F.Wichoski,Phys.
[11] A. Sedrakian, Phys. Rev. D 71, 083003 (2005) Rev. D 67, 105024 (2003).
[arXiv:astro-ph/0408467]. [32] J. Clem, Phys.Rev.Lett. 38, 1425 (1977)
[12] K. B. W. Buckley, M. A. Metlitski and A. R. Zhit- [33] E.H. Brandt, Phys. Rev.B 25, 5756 (1982).
nitsky, Phys. Rev. Lett. 92, 151102 (2004), [34] G. Marsh, Phys. Rev.B 49, 450 (1994).
[arXiv:astro-ph/0308148]; Phys. Rev. C 69, 055803 [35] Y. Genenko,Phys. Rev.B 53, 11757 (1996).
(2004), [arXiv:hep-ph/0403230]. [36] M. Kohandel and M. Kardar, Phys. Rev. B 61, 11729
[13] M. Alford, G. Good and S. Reddy, Phys. Rev. C 72, (2000).
055801 (2005) [arXiv:nucl-th/0505025]. [37] Saveliev et al, Phys. Rev.B 67, 180509(R) (2003).
[14] E. Witten, Nucl.Phys. B 249, 557 (1985). [38] L. D. Landau, Sov. Phys. JETP 7, 371 (1937);
[15] D.T.SonandA.R.Zhitnitsky,Phys.Rev.D70,074018 Zh.Eksp.Teor.Fiz. 13, 377 (1943).
(2004) [arXiv:hep-ph/0405216]. [39] U. Geppert, M. Kukerand D. Page, astro-ph/0512530
[16] M. A. Metlitski, Phys. Lett. B 612, 137 (2005) [40] P. Markley and R.J. Tayler, MNRAS163 (1973) 77
[arXiv:hep-ph/0501144]. [41] A.G.LyneandD.R.Lorimer, Nature,369,127(1994);
[17] M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D 72, J. E. Gunn and J. P. Ostriker, Astrophys. J. 160, 979
045011 (2005) [arXiv:hep-ph/0505072]. (1970);
[18] G.M.NewmanandD.T.Son,Phys.Rev.D73,045006 A. G. Lyne, M. J. Salter, Mon. Not. R. astr. Soc. 261,
(2006) [arXiv:hep-ph/0510049]. 113 (1993),
[19] G. E. Volovik, Rev.Mod. Phys. 59 (1987), 533. J.M.CordesandD.F.Chernoff,Astrophys.J.505,315
[20] G. E. Volovik, Phys. Rept. 351, 195 (2001) (1998),
[arXiv:gr-qc/0005091]; B.M.S.HansenandE.S.Phinney,Mon.Not.R.Astron.
The Universe in a Helium Droplet. Oxford University Soc. 291, 569 (1997);
Press, Oxford,(2003). C. Fryer, A. Burrows, and W. Benz, Astrophys. J. 496,
[21] A. Y. Alekseev, V. V. Cheianov and J. Frohlich, 333 (1998).
arXiv:cond-mat/9803346. [42] Z. Arzoumanian, D. F. Chernoff and J. M. Cordes, As-
10
trophys.J. 568, 289 (2002). uesofdifferentoperatorsindensematter(suchasbaryon
[43] C. L. Fryer,Astrophys.J. 601, L175 (2004). density,diquarkcondensate,chiralcondensate,pioncon-
[44] G.G.Pavlov et al, 2000, AAS meeting 196, 37.04; densate, etc) have been calculated a number of times,
G.G.Pavlov et al, Astrophys.J. 554, L189 (2001) in a number of models (see e.g. review [22]), the cal-
[45] C. Wang, D. Lai and J. Han, Astrophys. J. 639, 1007 culation of axial density has not received much atten-
(2006) [arXiv:astro-ph/0509484]. tion. In fact, while axial density is generically non-zero
[46] D. Lai, D. F. Chernoff and J. M. Cordes, Astrophys. J. when a Goldstone mode is condensed, as argued above,
549, 1111 (2001) [arXiv:astro-ph/0007272]. it has been calculated only recently in QCD(Nc = 2)
[47] C. Peralta, A. Melatos, M. Giacobello and A. Ool, As- and in QCD(Nc = 3) with isospin chemical potential
trophys.J. 635, 1224 (2005) [arxiv:astro-ph/0509416]. µI = 0 [28]. In these cases the exact expressions can be
6
[48] A. Mastrano and A. Melatos, Mon. Not. Roy. Astron. obtainedfortheaxialdensityintheregimeµI ΛQCD.
≪
Soc. 361, 927 (2005) [arXiv:astro-ph/0505529]. An important lesson that [28] teaches us is that the ax-
[49] L.D. Landau and E.M. Lifshitz, Dokl. Akad. Nauk. ial density does not vanish in spite of the fact that the
U.S.S.R.100 (1955), 669. corresponding axial chemical potential was not explic-
[50] The basic assumption of ref. [12] has been criticized itly introduced. Rather, it was generated dynamically.
in [13]. It is not the goal to discuss the approach de- Thepresenceoftheaxialdensityunambiguouslyimplies
veloped in [13] in the present paper, but a short re- thatRandLhandedmodeshavedifferentdensities,and
mark is warrented. If one uses the technique from [13] therefore, the vectorcurrent (1) will be induced.
for QCD(Nc=2) (which can be solved exactly for µi [52] WearethankfultoMaximLyutikovwhopointedoutthe
≪
ΛQCD)oneshouldanticipatesimilarresults,namelythat missing factor ”c” in the expressions (27, 28).
vastlydifferentdensitiesfordifferentflavourswouldlead [53] It is clear that the corresponding calculations would re-
to very different scattering lengths for different flavours. quireanunderstandingofthenon-equilibriumdynamics.
TheexactsolutionforQCD(Nc=2)teachesusdifferently: Indeed,theresultingpictureofthesystemwouldbevery
thatdifferentflavoursinteractinthesamewayasaresult different if magnetic field is turnedon before thesystem
oftheunderlyingflavoursymmetrydespitethepossiblil- becomes superconducting or after. This unambiguously
ityofvastlydifferentdensitiesfordifferentflavours.More implies thatthecorresponding state isnot in thermody-
studies are required before the basic assumption of ref. namic equilibrium.
[12] can beshown to beincorrect.
[51] It is interesting to note that while the expectation val-