Superfluidity and Superconductivity in Neutron Stars

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uefliiyadSprodciiyi eto Stars Neutron in Superconductivity and Superfluidity 10.1007 DOI Astr. Astrophys. J. c ninAaeyo Sciences of Academy Indian ihtpcltmeaue fodr10 order of temperatures typical With e words. Key b will stars under neutron matter in of superconductivity phases and superfluidity novel exploring for laboratories mic Abstract. * Belgium. Brussels, 1 Chamel N. orsodn uhr -al [email protected] E-mail: author. Corresponding nttt fAtooyadAtohsc,Université Libr Astrophysics, and Astronomy of Institute / s12036-017-9470-9 ff 21): (2017) sst h ufc n sdsiae in dissipated is and surface the to uses eto tr,tecmatselrrmat fcore-collap of remnants stellar compact the stars, Neutron 1,* 8 eto tr—uefliiysprodciiydnema stars—superfluidity—superconductivity—dense neutron ecp o hnotrheat- outer thin a for (except K 11 − 10 12 frarve about review a (for K 7 ,tehighly the K, oams eoa temperature at zero almost dropped they to mercury 1911, of resistance 8th, electric April the On that observed ex- before. temperatures reached to lower be at them could materials than allowing of thus properties 1908, the plore in first helium the were liquefy Heike collaborators to his 1967. and August Onnes in Kamerlingh pulsars of long discovery the known before were superfluidity and Superconductivity 2.1 superconductors and superfluids Terrestrial 2. discussed. briefly Fi- be will phases given. these of be manifestations observational will possible superfluid the stars nally, of neutron modelling superconducting the and in developments theoretical the of overview an superconductors, and superfluids [12]. high and pressure under laboratories compound hydrogen-sulphide terrestrial of in consisting 2014 in achieved record ots nswt rtcltmeaue fteodrof order the the also of but 11], temperatures 10 10, critical 9, with 8, ones 5, hottest [2, Universe systems the superconducting in and known superfluid largest the be n oansAtneKkfudta h etcapac- heat Keesom the Willem that found 1932, Kok In Antonie hun- Johannes than [14]. and more years for thousand sustained Superconducting dred be hour. actually an can after currents did change and induced any ring, notice magnetically lead not a superconducting a of in experiment time current an electric decay designed the later measure He to exceeds also value. field be critical magnetic super- the some to that if found showed destroyed is Onnes were conductivity 1914, tin [13]). In and Ref. lead superconducting. e.g., later, see discovery, years Two this of account torical 10 eBuels P26 olvr uTimh,B-1050 Triomphe, du Boulevard 226, CP Bruxelles, de e fe eciigtemi rpriso terrestrial of properties main the describing After reyreviewed. briefly e itrclmilestones Historical scmae oamr 0 o h world the for K 203 mere a to compared as K xrm odtos npriua,teocrec of occurrence the particular, In conditions. extreme esproaepoin,aeuiu cos- unique are explosions, supernova se tter. T c ≃ 4 . fra his- an (for K 2 1 Page 2 of 14 J. Astrophys. Astr. (2017) : ity of tin exhibits a discontinuity as it becomes super- Table 1. Properties of various superfluid and superconduct- conducting thus demonstrating that this phase transi- ing systems in order of their critical temperature Tc. Adapted tion is of second order [15]. One year later, Walther from Table 1.1 in Ref. [7]. Meissner and Robert Ochsenfeld made the remarkable 3 observation that when a superconducting material ini- System Density (cm− ) Tc (K) tially placed in a magnetic field is cooled below the Neutron stars 1039 1010 critical temperature, the magnetic flux is expelled from Cuprates ∼ ∼ the sample [16]. This showed that superconductivity and other exotics 1021 1 165 represents a new thermodynamical equilibrium state of Electrons ∼ − matter. In 1935, Lev Vasilievich Shubnikov [17, 18] in ordinary metals 1023 1 25 at the Kharkov Institute of Science and Technology in Helium-4 ∼ 1022 −2.17 ∼ 22 3 Ukraine discovered that some so called ”hard” or type Helium-3 10 2.491 10− ∼ 12 × 6 II superconductors(as opposedto ”soft” or type I super- Fermi alkali gases 10 10− Bose alkali gases ∼ 1015 10 7 ∼ 10 5 conductors) exhibit two critical fields, between which ∼ ∼ − − − the magnetic flux partially penetrates the material. Var- ious superconducting materials were discovered in the following decades. 2.2 Quantum liquids During the 1930’s, several research groups in Lei- Superconductivity and superfluidity are among the most den, Toronto, Moscow, Oxford and Cambridge (United spectacular macroscopic manifestations of quantum me- Kingdom), found that below Tλ 2.17 K, helium-4 (re- chanics. Satyendra Nath Bose and Albert Einstein pre- ≃ ferred to as helium II) does not behave like an ordinary dicted in 1924-1925 that at low enough temperatures liquid (for a review of the historical context, see e.g., an ideal gas of bosons condense into a macroscopic Refs. [19, 20]). In particular, helium II does not boil, as quantum state [28, 29]. The association between Bose- was actually first noticed by Kamerlingh Onnes and his Einstein condensation (BEC) and superfluidity was first collaborators the same day they discovered supercon- advanced by Fritz London [30]. The only known super- ductivity [13]. Helium II can flow without resistance fluid at the time was helium-4, which is a boson. The through very narrow slits and capillaries, almost inde- condensate can behave coherently on a very large scale pendently of the pressure drop. The term “superfluid” and can thus flow without any resistance. It was a key was coined by Pyotr Kapitsa in 1938 by analogy with idea for developing the microscopic theory of super- superconductors [21]. Helium II also flows up over fluidity and superconductivity. Soon afterwards, Las- the sides of a beaker and drip off the bottom (for or- zlo Tisza [31] postulated that a superfluid such as He II dinary liquids, the so called Rollin film is clamped by contains two distinct dynamical components: the con- viscosity). The existence of persistent currents in he- densate, which carries no entropy, coexists with a nor- lium II was experimentally established at the end of the mal viscous fluid. This model explained all phenom- 1950’s and the beginning of 1960’s [22]. The analog of ena observed at the time and predicted thermomechan- the Meissner-Ochsenfeld phenomenon, which was pre- ical effects like “temperature waves”. Although Lan- dicted by Fritz London, was first observed by George dau [32] incorrectly believed that superfluidity is not Hess and William Fairbank at Stanford in June 1967 [23]: related to BEC, he developed the two-fluid model and the angular momentum of helium-4 in a slowly rotating showed in particular that the normal fluid consists of container was found to be reduced as the liquid was “quasiparticles”, which are not real particles but com- cooled below the critical temperature Tλ. plex many-body motions. This two-fluid picture was At the time the first observed pulsars were identi- later adapted to superconductors [33]. fied as neutron stars, several materials had thus been According to the microscopic theory of supercon- found to be superconducting, while helium-4 was the ductivity by John Bardeen, Leon Cooper and Robert unique superfluid known. The superfluidity of helium-3 Schrieffer (BCS) published in 1957 [34], the dynami- was established by Douglas Osheroff, Robert Richard- cal distorsions of the crystal lattice (phonons) in a solid son and David Lee in 1971 [24]. No other superfluids can induce an attractive effective interaction between were discovered during the next two decades until the electrons of opposite spins. Roughly speaking, elec- production of ultracold dilute gases of bosonic atoms trons can thus form pairs and undergo a BEC below in 1995 [25, 26], and of fermionic atoms in 2003 [27]. some critical temperature. A superconductor can thus The main properties of some known superfluids and su- be viewed as a charged superfluid. This picture how- perconductors are summarized in Table 1. ever should not be taken too far. Indeed, electron pairs are very loosely bound and overlap. Their size ξ ∼ ~vF/(kBTc) (usually referred to as the coherencelength), J. Astrophys. Astr. (2017) : Page 3 of 14 with vF the Fermi velocity, kB Boltzmann’s constant, length scales much larger than the intervortex spacing and Tc the critical temperature, is typically much larger dυ, the superfluid flow mimics rigid body rotation such than the lattice spacing. Moreover, electron pairs dis- that appear at temperatures T > Tc. The BEC and the BCS transition are now understood as two different limits of p = mnυκκ, (4) ∇ × the same phenomenon. The pairing mechanism sug- gested that fermionic atoms could also become super- where nυ is the surface density of vortices given by fluid, as was later confirmed by the discovery of super- mΩ = fluid helium-3. Since 2003, various other fermionic su- nυ ~ , (5) perfluids have been found, as mentioned in the previous π section. and the vector κ, whose norm is equalto h/m, is aligned As first discussedby Lars Onsager[35] and Richard with the average angular velocity. Landau’s original Feynman [36], the quantum nature of a superfluid is two-fluid model was further improved in the 1960’s by embedded in the quantization of the flow Hall and Vinen [41, 42], and independently by Bekare- vich and Khalatnikov [43] to account for the presence p dℓ = Nh , (1) of quantized vortices within a coarse-grained average · I hydrodynamic description. The quantization condition (1) also applies to su- where p is the momentum per superfluid particle, h perconductors. But in this case, the momentum (in denotes Planck’s constant, N is any integer, and the CGS units) is given by p mvv + (q/c)A, where m, q, integral is taken over any closed path. It can be im- and v are the mass, electric≡ charge and velocity of su- mediately recognized that this condition is the Bohr- perconducting particles respectively, and A is the elec- Sommerfeld quantization rule. The flow quantization tromagnetic potential vector. Introducing the density follows from the fact that a superfluid is a macroscopic n of superconducting particles and the “supercurrent” quantum system whose momentum is thus given by p = = nqvv, the situation N = 0 as described by Eq. (2) h/λ, where λ is the de Broglie wavelength. Requiring Jleads to the London equation the length of any closed path to be an integral multi- ple of the de Broglie wavelength leads to Eq. (1). The c = B (6) physical origin of this condition has been usually ob- ∇∇×JJ −4πλ2 scured by the introduction of the “superfluid velocity” L Vs = p/m, where m is the mass of the superfluid parti- where B = A is the magnetic field induction, and 2∇ × 2 cles. λL = mc /(4πnq ) is the London penetration depth. In a rotating superfluid, the flow quantization con- Situations with N > 0 are encountered in type II su- p dition (1) leads to the appearence of N quantised vor- perconductors for which λL & ξ. Considering a closed tices. In a region free of vortices, the superflow is char- contour outside a sample of such a superconductor for acterised by the irrotationality condition which = 0 and integrating the momentum p along this contour,J leads to the quantization of the total mag- p = 0 . (2) ∇ × netic flux Φ into fluxoids (also referred to as flux tubes or fluxons) Inside a vortex, the superfluidity is destroyed. Because superfluid vortices are essentially of quantum nature, Φ = A dℓ = NΦ , (7) their internal structure cannot be described by a purely · 0 hydrodynamic approach. However, vortices can be ap- I proximately treated as structureless topological defects where Φ0 = hc/ q is the flux quantum. The magnetic at length scales much larger than the vortex core size. flux quantization,| | first envisioned by London, was ex- As shown by Vladimir Konstantinovich Tkachenko [37], perimentally confirmed in 1961 by Bascom Deaver and quantized vortices tend to arrange themselves on a reg- William Fairbank at Stanford University [44], and in- ular triangular array, with a spacing given by dependently by Robert Doll and Martin Näbauer at the Low Temperature institute in Hersching [45]. As pre- h dicted by Aleksei Abrikosov [46], these fluxoids tend to dυ = , (3) arrange themselves into a triangular lattice with a spac- √3mΩ s ing given by where Ω is the angular frequency. Vortex arrays have been observed in superfluid helium [38] and more re- 2hc dΦ = . (8) cently in atomic Bose-Einstein condensates[39, 40]. At s √3 q B | | Page 4 of 14 J. Astrophys. Astr. (2017) :

Averaging at length scales much larger than dυ, the sur- 3.2 Superﬂuid and superconducting phase transitions face density of ﬂuxoids is given by in dense matter

B q B Only one year after the publication of the BCS theory nΦ = = | | , (9) of superconductivity, Bogoliubov was the first to con- Φ hc 0 sider the possibility of superfluid nuclear matter [60]. where B denotes here the average magnetic field strength. In 1959, Arkady Migdal [64] speculated that the inte- The size of a fluxoid (within which the superconductiv- rior of a neutron star might contain a neutron super- ity is destroyed) is of the order of the coherence length fluid, and its critical temperature was estimated by Vi- ξ. The magnetic field carried by a fluxoid extends over taly Ginzburg and David Abramovich Kirzhnits in 1964 a larger distance of the order of the London penetra- using the BCS theory [65]. Proton superconductivity in neutron stars was studied by Richard Wolf in 1966 [61]. tion length λL. The nucleation of a single fluxoid thus occurs at a critical field H Φ /(πλ2 ), and supercon- The possibility of anisotropic neutron superfluidity was c1 ∼ 0 L explored by Hoffberg, Glassgold, Richardson, and Ru- ductivity is destroyed at the critical field H Φ /(πξ2) c2 0 derman [62], and independently by Tamagakiin 1970 [63]. at which point the cores of the fluxoids touch.∼ Neutrons and protons are fermions, and due to the Pauli exclusion principle, they generally tend to avoid themselves. This individualistic behaviour, together with 3. Superstars the strong repulsive nucleon-nucleon interaction at short distance, provide the necessary pressure to counterbal- 3.1 Prelude: internal constitution of a neutron star ance the huge gravitational pull in a neutron star, thereby A few meters below the surface of a neutron star, matter preventing it from collapsing. However at low enough is so compressed by the tremendous gravitational pres- temperatures, nucleons may form pairs [66] similarly sure that atomic nuclei, which are supposedly arranged as electrons in ordinary superconductors as described on a regular crystal lattice, are fully ionized and thus co- by the BCS theory1. These bosonic pairs can there- exist with a quantum gas of electrons. With increasing fore condense, analogous to superfluid helium-3. While depth, nuclei become progressively more neutron-rich. helium-3 becomes a superfluid only below 1 mK, nu- Only in the first few hundred metres below the surface clear superfluidity could be sustainable even at a tem- can the composition be completely determined by ex- perature of several billions degrees in a neutron star due perimentally measured masses of atomic nuclei [47]. to the enormous pressure involved. The nuclear pair- In the deeper layers recourse must be made to theoreti- ing phenomenon is also supported by the properties of cal models (see, e.g., Refs. [48, 49, 50, 51, 52, 53]). At atomic nuclei (see, e.g. Ref. [69]). 11 3 densities of a few 10 g cm− , neutrons start to “drip” Because the nuclear interactions are spin dependent out of nuclei (see, e.g., Ref. [54] for a recent discus- and include non-central tensor components (angular mo- sion). This marks the transition to the inner crust, an in- mentum dependent), different kinds of nucleon-nucleon homogeneous assembly of neutron-proton clusters im- pairs could form at low enough temperatures. The most 2 1 mersed in an ocean of unbound neutrons and highly de- attractive pairing channels are S0 at low densities and 3 generate electrons. According to various calculations, the coupled PF2 channel at higher densities [70]. In the crust dissolves into a uniform mixture of neutrons, principle, different types of pairs may coexist. How- protons and electrons when the density reaches about ever, one or the other are usually found to be energet- 14 3 half the density 2.7 10 g cm− found inside heavy ically favored [71]. Let us mention that nucleons may atomic nuclei (see,∼ e.× g. Ref. [5] for a review about also form quartets such as α-particles, which can them- neutron-star crusts). Near the crust-core interface, nu- selves condense at low enough temperatures (see, e.g. clear clusters with very unusual shapes such as elon- Ref. [72]). Most microscopic calculations have been gated rods or slabs may exist (see, e.g., Section 3.3 of carried out in pure neutron matter using diagrammatic, Ref. [5], see also Ref. [55]). These so-called “nuclear pastas” could account for half of the crustal mass, and 1The high temperatures 107 K prevailing in neutron stars interi- play a crucial role for the dynamical evolution of the ∼ star and its cooling [56, 57]. The composition of the ors prevent the formation of electron pairs recalling that the highest innermost part of neutron-star cores remains highly un- critical temperatures of terrestrial superconductors do not exceed 200 K. In particular, iron expected to be present in the outermost certain: apart from nucleons and leptons, it may also layers∼ of a neutron star was found to be superconducting in 2001, contain hyperons, meson condensates, and deconfined but with a critical temperature Tc 2 K [67]. See also Ref. [68]. 2 2S≃+1 quarks (see, e.g. Ref. [3]; see also Refs. [58, 59]). A given channel is denoted by LJ , where J is the total angular momentum, L is the orbital angular momentum, and S the spin of nucleon pair. J. Astrophys. Astr. (2017) : Page 5 of 14 variational, and more recently Monte Carlo methods the presence of protons, leptons, and possibly other par- (see, e.g., Refs. [70, 71] for a review). At concentra- ticles in neutron-star cores. Few microscopic calcula- 3 tions below 0.16 fm− , as encountered in the inner tions have been performed so far in beta-stable matter crust and in∼ the outer core of a neutron star, neutrons (see, e.g., Ref. [87]). Because the proton concentra- 1 are expected to become superfluid by forming S0 pairs, tion in the outer core of a neutron star is very low, pro- 10 1 with critical temperatures of about 10 K at most (see, tons are expected to become superconducting in the S0 e.g. Refs. [70, 73, 74, 75]). At neutron concentrations channel. However, the corresponding critical tempera- 3 3 above 0.16 fm− , pairing in the coupled PF2 chan- tures are very poorly known due to the strong influence nel becomes∼ favored but the maximum critical temper- of the surrounding neutrons [88]. Neutron-proton pair- ature remains very uncertain, predictions ranging from ing could also in principle occur, but is usually disfa- 108 K to 109 K (see, e.g. Refs.[74, 75, 76, 77]). vored by the very low proton content of neutron stars This∼ lack of knowledge∼ of neutron superfluid properties (see, e.g., Ref. [89]). Other more speculative possi- mainly stems from the highly nonlinear character of the bilities include hyperon-hyperon and hyperon-nucleon pairing phenomenon, as well as from the fact that the pairing (see, e.g. Ref. [59] and references therein). The nuclear interactions are not known from first principles core of a neutron star might also contain quarks in var- (see, e.g., Ref. [78] for a recent review). ious color superconducting phases [90]. Another complication arises from the fact that neu- According to cooling simulations, the temperature tron stars are not only made of neutrons. The pres- in a neutron star is predicted to drop below the esti- ence of nuclear clusters in the crust of a neutron star mated critical temperatures of nuclear superfluid phases may change substantially the neutron superfluid prop- after 10 102 years. The interior of a neutron star is erties. Unfortunately, microscopic calculations of inho- thus thought∼ − to contain at least three different kinds of mogeneous crustal matter employing realistic nuclear superfluids and superconductors [10]: (i) an isotropic 1 interactions are not feasible. State-of-the-art calcula- neutron superfluid (with S0 pairing) permeating the in- tions are based on the nuclear energy density functional ner region of the crust and the outer core, (ii) an anisotropic 3 theory, which allows for a consistent and unified de- neutron superfluid (with PF2 pairing) in the outer core, 1 scription of atomic nuclei, infinite homogeneous nu- and (iii) an isotropic proton superconductor (with S0 clear matter and neutron stars (see, e.g. Ref. [79] and pairing) in the outer core. The neutron superfluids in references therein). The main limitation of this approach the crust and in the outer core are not expected to be is that the exact form of the energy density functional separated by a normal region. is not known. In practice, phenomenological function- als fitted to selected nuclear data must therefore be em- 3.3 Role of a high magnetic field ployed. The superfluid in neutron-star crusts, which Most neutron stars that have been discovered so far are bears similarities with terrestrial multiband supercon- radio pulsars with typical surface magnetic fields of or- ductors, was first studied within the band theory of solids der 1012 G (as compared to 10 1 G for the Earth’s in Ref. [80]. However, this approach is computation- − magnetic field), but various other∼ kinds of neutron stars ally very expensive, and has been so far limited to the have been revealed with the development of the X-ray deepest layers of the crust. For this reason, most cal- and gamma-ray astronomy [91]. In particular, a small culations of neutron superfluidity in neutron-star crusts class of very highly magnetised neutron stars thus dubbed (see, e.g. Ref. [81]) have been performed using an ap- magnetars by Robert Duncan & Christopher Thomp- proximation introduced by Eugene Wigner and Fred- son in 1992 [92] (see e.g. Ref. [93] for a review) have erick Seitz in 1933 in the context of electrons in met- been identified in the form of soft-gamma ray repeaters als [82]: the Wigner-Seitz or Voronoi cell of the lattice (SGRs) and anomalous x-ray pulsars (AXPs). Tremen- (a truncated octahedron in case of a body-centred cubic dous magnetic fields up to about 2 1015 G have been lattice) is replaced by a sphere of equal volume. How- measured at the surface of these stars× from both spin- ever, this approximation can only be reliably applied in down and spectroscopic studies [94, 95, 96], and vari- the shallowest region of the crust due to the appearance ous observations suggest the existence of even higher of spurious shell effects [83]. Such calculations have internal fields [97, 98, 99, 100, 101, 102, 103]. Al- shown that the phase diagram of the neutron superfluid though only 23 such stars are currently known [94], in the crust is more complicated than that in pure neu- recent observations indicate that ordinary pulsars can tron matter; in particular, the formation of neutron pairs also be endowed with very high magnetic fields of or- can be enhanced with increasing temperature (see, e.g., der 1014 G [104]. According to numerical simulations, Refs. [84, 85, 86]). Microscopic calculations in pure neutron stars may potentially be endowed with internal neutron matter at densities above the crust-core bound- magnetic fields as high as 1018 G (see, e.g. Refs. [105, ary are not directly applicable to neutron stars due to 106] and references therein). Page 6 of 14 J. Astrophys. Astr. (2017) :

The presence of a high magnetic field in the interior fused with those introduced in microscopic many-body of a neutron star may have a large impact on the su- theories (see, e.g. Ref. [119]). Because of the strong perfluid and superconducting phase transitions. Proton interactions between neutrons and protons, entrainment superconductivity is predicted to disappear at a critical effects in neutron-star cores cannot be ignored (see, e.g. field of order 1016 1017 G [107]. Becausespins tend to Refs. [119, 120, 121, 122, 123] for recent estimates). In be aligned in a magnetic− field, the formation of neutron the neutron-rich core of neutron stars, we typically have 1 ♯ ♯ pairs in the S0 channel is disfavored in a highly mag- m⋆ m m , and m⋆ m (0.5 1)m , where m n ∼ n ∼ n p ∼ p ∼ − p n netized environment, as briefly mentioned by Kirzhnits and mp denote the “bare” neutron and proton masses re- 1 in 1970 [108]. It has been recently shown that S0 pair- spectively. As shown by Brandon Carter in 1975 [124], ing in pure neutron matter is destroyed if the magnetic at the global scale of the star, general relativity induces field strength exceeds 1017 G [109]. Moreover, the additional couplings between the fluids due to Lense- ∼ magnetic field may also shift the onset of the neutron- Thirring effects, which tend to counteract entrainment. drip transition in dense matter to higher or lower densi- As recently found in Ref. [125], frame-dragging effects ties due to Landau quantization of electron motion thus can be as important as entrainment. changing the spatial extent of the superfluid region in An elegant variational formalism to derive the hy- magnetar crusts [110, 111, 112, 113]. drodynamic equations of any relativistic (super)fluid mixtures was developed by Carter and collaborators (see, 3.4 Dynamics of superfluid and superconducting neu- e.g., Refs. [126, 127, 128, 129]). This formalism re- tron stars lies on an action principle in which the basic variables are the number densities and currents of the different The minimal model of superfluid neutron stars consists fluids. The equations of motion can be derived by con- of at least two distinct interpenetrating dynamical com- sidering variations of the fluid particle trajectories. Dis- ponents [114]: (i) a plasma of electrically charged par- sipative processes (e.g. viscosity in non-superfluid con- ticles (electrons, nuclei in the crust and protons in the stituents, superfluid vortex drag, ordinary resistivity be- core) that are essentially locked together by the interior tween non-superfluid constituents, nuclear reactions) can magnetic field, and (ii) a neutron superfluid. Whether be treated within the same framework. The convective protons in the core are superconducting or not, they co- formalism developed by Carter was later adapted to the move with the other electrically charged particles (see, comparatively more intrincate Newtonian theory within e.g. Ref. [8]). a 4-dimensionally covariant framework [130, 131, 132] The traditional heuristic approach to superfluid hy- (see, e.g. Refs. [133, 134, 135] and references therein drodynamics blurring the distinction between velocity for a review of other approaches using a 3+1 space- and momentum makes it difficult to adapt and extend time decomposition). This fully covariant approach not Landau’s original two-fluid model to the relativistic con- only facilitates the comparison with the relativistic the- text, as required for a realistic description of neutron ory (see, e.g., Refs. [152, 121]), but more importantly stars (see, e.g. Ref. [115]). In particular, in superfluid lead to the discovery of new conservation laws in super- mixtures such as helium-3 and helium-4 [116], or neu- fluid systems such as the conservation of generalised trons and protons in the core of neutron stars [117, 118], helicy currents. the different superfluids are generally mutually coupled As pointed out by Ginzburg and Kirzhnits in 1964 [65], by entrainment effects whereby the true velocity v and X the interior of a rotating neutron star is expected to be the momentum p of a fluid X are not aligned: X threaded by a very large number of neutron superfluid 1 XY vortices (for a discussion of the vortex structure in S0 pX = vY , (10) 3 K and PF2 neutron superfluids, see, e.g. Ref. [8]). In- XY troducing the spin period P in units of 10 ms, P 10 ≡ where XY is a symmetric matrix determined by the P/(10 ms), the surface density of vortices (5) is of the interactionsK between the constituent particles. In the order two-fluid model, entrainment can be equivalently for- 5 1 2 nυ 6 10 P− cm− . (11) mulated in terms of “effective masses”. Considering the ∼ × 10 neutron-proton mixture in the core of neutron stars, the The average intervortex spacing (3) p = ⋆v neutron momentum can thus be expressedas pn mn vn 1/2 3 ⋆ nn d n 10 P cm , (12) in the proton rest frame (vp = 0), with m = . υ −υ − 10 p n K ∼ ∼ Alternatively, a different kind of effective mass can be is much larger thanp the size of the vortex core (see, e.g. ♯ nn np pn pp introduced, namely mn = / , such Ref. [136]). Neutron superfluid vortices can pin to nu- ♯ K − K K K that pn = mnvn in the proton momentum rest frame clear inhomogeneities in the crust. However, the pin- (pp = 0). These effectives masses should not be con- ning strength remains uncertain (see, e.g. Ref. [137] J. Astrophys. Astr. (2017) : Page 7 of 14 and referencestherein; see also Section 8.3.5 of Ref. [5]). and references therein). In recent years, simulations of Protons in the core of a neutron star are expected to large collections ( 102 104) of vortices have been car- become superconducting at low enough temperatures. ried out, thus providing∼ − some insight on collective be- Contrary to superfluid neutrons, superconducting pro- haviors, such as vortex avalanches(see, e.g., Ref. [150]). tons do not form vortices. As shown by Baym, Pethick, However, these simulations have been restricted so far and Pines in 1969 [107, 138], the expulsion of the mag- to Bose condensates. The extent to which the results netic flux accompanying the transition takes place on can be extrapolated to neutron stars remains to be de- a very long time scale 106 years due to the very termined. Such large-scale simulations also require mi- high electrical conductivity∼ of the dense stellar matter. croscopic parameters determined by the local dynamics The superconducting transition thus occurs at constant of individual vortices (see e.g. Ref. [151]). magnetic flux. The proton superconductor is usually The variational formulation of multifluid hydrody- thought to be of type II [107] (but see also Refs. [139, namics was extended for studying the magnetoelasto- 90] and references therein), in which case, the mag- hydrodynamics of neutron star crusts, allowing for a netic flux penetrates the neutron star core by forming consistent treatment of the elasticity of the crust, su- fluxoids, with a surface density (9) of order perfluidity and the presence of a strong magnetic field, both within the Newtonian theory [152, 153] and in the 18 2 nΦ 5 10 B12 cm− , (13) fully relativistic context [154]. In particular, these for- ∼ × mulations can account for the entrainment of the neu- where the magnetic field strength B is expressedas B12 tron superfluid by the crustal lattice [155], a non-dissipative B/(1012 G). This surface density corresponds to a spac-≡ effect arising from Bragg scattering of unbound neu- ing trons first studied in Refs. [156, 157, 158] using the band theory of solids. More recent systematic calcula- 1/2 10 1 tions based on a more realistic description of the crust dΦ n−Φ 5 10− B12− cm . (14) ∼ ∼ × have confirmed that these entrainment effects can be Since the magnetic fluxp is frozen in the stellar core, very strong [159]. These results are at variance with fluxoids can form even if the magnetic field is lower those obtained from hydrodynamical studies [160, 161, 15 than the critical field Hc1 10 G [107]. Proton su- 162, 163, 164, 165]. However, as discussedin Ref. [165], perconductivity is destroyed∼ at the higher critical field these approaches are only valid if the neutron superfluid 16 Hc2 10 G [107]. Due to entrainment effects, neu- coherence length is much smaller than the typical size tron∼ superfluid vortices carry a magnetic flux as well, of the spatial inhomogeneities, a condition that is usu- given by [117, 140] ally not fulfilled in most region of the inner crust. The neglect of neutron pairing in the quantum calculations ♯ of Ref. [159] has been recently questioned [70, 165]. mp Φ=Φ0 1 , (15) Although detailed numerical calculations are still lack- mp −  ing, the analytical study of Ref. [166] suggests that neu-     tron pairing is unlikely to have a large impact on the   where Φ0 = hc/(2 e). Electrons scattering off the mag- entrainment coupling. netic field of the vortex lines leads to a strong fric- tional coupling between the core neutron superfluid and the electrically charged particles [140]. Neutron su- 4. Observational manifestations perfluid vortices could also interact with proton fluxoids [8, 141, 142, 143], and this may have important 4.1 Pulsar frequency glitches implications for the evolution of the star [2, 8, 144, 145, 146, 147]. For typical neutron star parameters Pulsars are neutron stars spinning very rapidly with ex- (P = 10 ms, B = 1012 G, radius R = 10 km), the num- tremely stable periods P ranging from milliseconds to bers of neutron superfluid vortices and proton fluxoids seconds, with delays P˙ dP/dt that in some cases do 2 18 2 30 21≡ 18 are of order nυπR 10 and nΦπR 10 , respec- not do not exceed 10− , as compared to 10− for the tively. Such large numbers∼ justify a smooth-averaged∼ most accurate atomic clocks [167]. Nevertheless, irreg- hydrodynamical description of neutron stars. However, ularities have been detected in long-term pulsar timing this averaging still requires the understanding of the un- observations (see, e.g., Ref. [168]). In particular, some derlying vortex dynamics (see, e.g. Ref. [11]). A more pulsars have been found to suddenly spin up. These elaborate treatment accounting for the macroscopic anisotropy“glitches” in their rotational frequency Ω, ranging from 9 5 / ∆Ω/Ω 10− to 10− , are generally followed by a induced by the underlying presence of vortices and or ∼ ∼ flux tubes was developed by Carter based on a Kalb- long relaxation lasting from days to years, and some- Ramond type formulation [148] (see also Ref. [149] times accompanied by an abrupt change of the spin- Page 8 of 14 J. Astrophys. Astr. (2017) :

6 2 down rate from ∆Ω˙ /Ω˙ 10− up to 10− . At 4.2 Thermal relaxation of transiently accreting neu- the time of this| writing,| ∼ 482 glitches have∼ been de- tron stars during quiescence tected in 168 pulsars [169]. Since these phenomena In a low-mass X-ray binary, a neutron star accretes mat- have not been observed in any other celestial bodies, ter from a companionstar during severalyears or decades, they must reflect specific properties of neutron stars driving the neutron-star crust out of its thermal equi- (for a recent review, see, e.g. Ref. [170]). In partic- librium with the core. After the accretion stops, the ular, giant pulsar frequency glitches ∆Ω/Ω 10 6 − heated crust relaxes towards equilibrium (see, e.g., Sec- 10 5 as detected in the emblematic Vela pulsar∼ are usu-− − tion 12.7 of Ref. [5], see also Ref. [192]). The thermal ally attributed to sudden transfers of angular momen- relaxation has been already monitored in a few systems tum from a more rapidly rotating superfluid compo- (see, e.g., Ref. [193] and references therein). The ther- nent to the rest of star whose rotation frequency is di- mal relaxation time depends on the properties of the rectly observed (for a short historical review of theo- crust, especially the heat capacity. In turn, the onset of retical developments, see, e.g. Ref. [171] and refer- neutron superfluidity leads to a strong reduction of the ences therein). The role of superfluidity is corrobo- heat capacity at temperatures T T thus delaying the rated by the very long relaxation times [107] and by c thermal relaxation of the crust (see,≪ e.g., Ref. [194]). If experiments with superfluid helium [172]. The stan- neutrons were not superfluid, they could store so much dard scenario of giant pulsar glitches is the following. heat that the thermal relaxation would last longer than The inner crust of a neutron star is permeated by a neu- what is observed [195, 196]. On the other hand, the tron superfluid that is weakly coupled to the electrically thermal relaxation of these systems is not completely charged particles by mutual friction forces (in a semi- understood. For instance, additional heat sources of un- nal work, Alpar, Langer and Sauls [140] argued that the known origin are needed in order to reproduce the ob- core neutron superfluid is strongly coupled to the core, servations [193, 196, 197, 198, 199, 200, 201]. These and therefore does not participate to the glitch). The discrepancies may also originate from a lack of under- superfluid thus follows the spin-down of the star via the standing of superfluid properties [199]. In particular, motion of vortices away from the rotation axis unless the low-energy collective excitations of the neutron su- vortices are pinned to the crust [173]. In such a case, a perfluid were found to be strongly mixed with the vi- lag between the superfluid and the rest of the star will brations of the crystal lattice, and this can change sub- build up, inducing a Magnus force acting on the vor- stantially the thermal properties of the crust [202, 203]. tices. At some point, the vortices will suddenly unpin, the superfluid will spin down and, by the conservation 4.3 Rapid cooling of Cassiopeia A of angular momentum the crust will spin up. During the subsequent relaxation, vortices progressively repin Cassiopeia A is the remnant of a star that exploded 330 until the next glitch [174]. This scenario is supported years ago at a distance of about 11000 light years from by the analysis of the glitch data, suggesting that the us. It owes its name to its location in the constellation superfluid represents only a few per cent of the angular Cassiopeia. The neutron star is not only the youngest momentum reservoir of the star [175, 176, 177]. On the known, thermally emitting, isolated neutron star in our other hand, this interpretation has been recently chal- Galaxy, but it is also the first isolated neutron star for lenged by the 2007 glitch detected in PSR J1119 6127, which the cooling has been directly observed. Ten-year and by the 2010 glitch in PSR B2334+61 [178,− 179, monitoring of this object seems to indicate that its tem- 180]. More importantly, it has been also shown that the perature has has decreased by a few percent since its neutron superfluid in the crust of a neutron star does discovery in 1999 [204] (but see also the analysis of not contain enough angular momentum to explain giant Refs. [205, 206] suggesting that the temperature de- glitches due to the previously ignored effects of Bragg cline is not statistically significant). If confirmed, this scattering [181, 182, 183, 184]. This suggests that the cooling rate would be substantially faster than that ex- core superfluid plays a more important role than previ- pected from nonsuperfluid neutron-star cooling theo- ously thought [185, 186]. In particular, the core super- ries. It is thought that the onset of neutron superflu- fluid could be decoupled from the rest of the star due to idity opens a new channel for neutrino emission from the pinning of neutron vortices to proton fluxoids [146, the continuous breaking and formation of neutron pairs. 187]. So far, most global numerical simulations of pul- This process, which is most effective for temperatures sar glitches have been performed within the Newtonian slightly below the critical temperature of the superfluid theory (see, e.g. Refs. [188, 189, 190, 191]). How- transition, enhances the cooling of the star during sev- ever, a recent study shows that general relativity could eral decades. As a consequence, observations of Cas- significantly affect the dynamical evolution of neutron siopeia A put stringent constraints on the critical tem- stars [125]. peratures of the neutron superfluid and proton super- J. Astrophys. Astr. (2017) : Page 9 of 14 conductor in neutron-star cores [207, 208, 209]. How- 5. Conclusion ever, this interpretation has been questioned and alter- native scenarios have been proposed [210, 211, 212, The existence of superfluid and superconducting phases 213, 214, 215, 216, 217] (most of which still requiring in the dense matter constituting the interior of neutron superfluidity and/or superconductivity in neutron stars). stars has been corroborated both by theoretical developments and by astrophysical observations. In particular, 1 4.4 Pulsar timing noise and rotational evolution neutron stars are expected to contain a S0 neutron superfluid permeating the inner region of the crust and the Apart from pulsar frequency glitches, superfluidity and outer core, a 3PF neutron superfluid in the outer core, superconductivity may leave their imprint on other tim- 2 and a 1S proton superconductor in the outer core. Still, ing irregularities. In particular, pulsar timing noise (see, 0 many aspects of these phenomena need to be better un- e.g. Ref. [168]) could be the manifestation of super- derstood. Due to the highly nonlinear character of the fluid turbulence although other mechanisms are likely pairing mechanism giving rise to nuclear superfluidity to play a role (see, e.g. Ref. [218] and referencestherein). and superconductivity, the associated critical temper- Interpreting the long-period ( 100 1000 days) oscil- atures remain very uncertain, especially for the 3PF lations in the timing residuals∼ of some− pulsars such as 2 channel. The dynamics of these phase transitions as the PSR B1828 11 (see, e.g. Ref. [219]) as evidence of star cools down, and the possible formation of topo- free precession,− it has been argued that either the neu- logical defects need to be explored. Although the for- tron superfluid does not coexist with the proton super- malism for describing the relativistic smooth-averaged conductor in the core of a neutron star, or the proton magnetoelastohydrodynamics of superfluid and super- superconductor is type I so as to avoid pinning of neu- conducting systems already exists, modelling the global tron superfluid vortices to proton fluxoids [220, 221]. evolution of neutron stars in full general relativity still However, this conclusion seems premature in view of remains very challenging. To a large extent, the diffi- the complexity of the neutron-star dynamics [222, 223]. culty lies in the many different scales involved, from Alternatively, these oscillations might be related to the the kilometer size of the star down to the size of indi- propagation of Tkachenko waves in the vortex lattice vidual neutron vortices and proton fluxoids at the scale (see, e.g. Ref.[224] and references therein). The pres- of tens or hundred fermis. Studies of neutron-star dy- ence of superfluids and superconductors in the interior namics using the Newtonian theory provide valuable of a neutron star may also be revealed from the long- qualitative insight, and should thus be pursued. The term rotational evolution of pulsars by measuring the presence of other particles such as hyperons or decon- braking index n =ΩΩ¨ /Ω˙ 2. Deviations from the canon- fined quarks in the inner core of neutron stars adds to ical value n = 3 as predicted by a rotating magnetic the complexity. The occurrence of exotic superfluid and dipole model in vacuum can be explained by the decou- superconducting phases remains highly speculative due pling of the neutron superfluid in the core of a neutron to the lack of knowledge of dense matter. On the other star (due to pinning to proton fluxoids for instance) [225, hand, astrophysical observations offer a unique oppor- 226]. However, a similar rotational evolution could be tunity to probe the phase diagram of matter under ex- mimicked by other mechanisms without invoking su- treme conditions that are inaccessible in terrestrial lab- perfluidity (see, e.g. Ref. [227] for a recent review). oratories. 4.5 Quasi-periodic oscillations in soft gamma-ray repeaters Acknowledgement Quasi-periodic oscillations (QPOs) in the hard X-ray emission were detected in the tails of giant flares from This work has been supported by the Fondsde la Recherche SGR 1806-20, SGR 1900+14, and SGR 0526-66, with Scientifique - FNRS (Belgium) under grant n◦ CDR frequencies ranging from 18 Hz to 1800 Hz (see, e.g. J.0187.16, and by the European Cooperation in Science Ref. [228] for a recent review). As anticipated by Dun- and Technology (COST) action MP1304 NewCompStar. can [229], these QPOs are thought to be the signatures of global magneto-elastic seismic vibrations of the star. If this interpretation is confirmed, the analysis of these References QPOs could thus provide valuable information on the interior of a neutron star. In particular, the identifica- [1] Deshpande, A. A., Ramachandran, R., Srinivasan, G. tion of the modes could potentially shed light on the ex- 1995, J. Astrophys. Astr.,16, 69. istence of superfluid and superconducting phases (see, [2] Srinivasan,G. 1997,in Stellar Remnants, EdsG. Meynet e.g. Ref. [230]). and D. Schaerer, Springer-Verlag, p. 97. Page 10 of 14 J. Astrophys. Astr. (2017) :

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