Crystalline color superconductors: A review
Abstract
Nonhomogenous superconductors and nonhomogenous superfluids appear in a variety of contexts which include quark matter at extreme densities, fermionic systems of cold atoms, typeII cuprates and organic superconductors. In the present review we shall focus on the properties of quark matter at high baryonic density which can exist in the interior of compact stars. The conditions that are realized in this stellar objects tend to disfavor standard symmetric BCS pairing and may be in favor of a nonhomogenous color superconducting phase. We discuss in details the properties of nonhomogenous color superconductors and in particular of crystalline color superconductors. We also review the possible astrophysical signatures associated with the presence of nonhomogenous color superconducting phases within the core of compact stars.
We dedicate this paper to the memory of Giuseppe Nardulli for his important collaboration to the general subject of this review.
Contents
 I Introduction

II The twoflavor nonhomogenous phases
 II.1 Mismatched Fermi spheres
 II.2 Gapless 2SC phase of QCD
 II.3 The twoflavor crystalline color superconducting phase
 II.4 Dispersion laws and specific heats
 II.5 Smearing procedure
 II.6 Chromomagnetic stability of the twoflavor crystalline phase
 II.7 Solitonic ground state
 II.8 An aside: Condensed matter systems
 III The threeflavor nonhomogenous phases
 IV Astrophysics
I Introduction
Ideas about color superconducting (CSC) matter date back to more than 30 years ago Collins and Perry (1975); Barrois (1977); Frautschi (1978); Bailin and Love (1984), but only recently this phenomenon has received a great deal of consideration (for recent reviews see Rajagopal and Wilczek (2000); Hsu (2000); Hong (2001); Alford (2001); Nardulli (2002); Schafer (2003b); Rischke (2004); Alford et al. (2008)). Color superconductivity is the quark matter analogous of the standard electromagnetic superconductivity and is believed to be the ground state of hadronic matter at sufficiently large baryonic densities. At very high density the naive expectation, due to asymptotic freedom, is that quarks form a Fermi sphere of almost free fermions. However, Bardeen, Cooper and Schrieffer (BCS) Cooper (1956); Bardeen et al. (1957b, a) have shown that the Fermi surfaces of free fermions are unstable in presence of an attractive, arbitrary small, interaction between fermions. In QuantumChromodynamics (QCD) the attractive interaction between quarks can be due to instanton exchange Schafer and Shuryak (1998), at intermediate densities, or to gluon exchange in the color channel, at higher densities. Therefore, one expects that at high densities quarks form a coherent state of Cooper pairs.
It should be noticed that the mentioned old papers Collins and Perry (1975); Barrois (1977); Frautschi (1978); Bailin and Love (1984) were based on the existence of the attractive color channel and on analogies with ordinary superconductors. The main result of these analyses was that quarks form Cooper pairs with a gap of order a few MeV. In more recent times two papers Rapp et al. (1998) and Alford et al. (1998), have brought this result to question. These authors considered diquark condensation arising from instantonmediated interactions and although their approximations are not under rigorous quantitative control, the result was that gaps can be as large as MeV. However, as pointed out by Hsu Hsu (2000): “this is an unfair characterization of Bailin and Love’s results Bailin and Love (1984). A value of the strong coupling large enough to justify the instanton liquid picture of Rapp et al. (1998); Alford et al. (1998) also yields a large gap when substituted in Bailin and Love’s results. After all, instantons are suppressed by an exponential factor . Bailin and Love merely suffered from the good taste not to extrapolate their results to large values of ! ”
Color superconductivity offers a clue to the behavior of strong interactions at very high baryonic densities, an issue of paramount relevance for the understanding of the physics of compact stars and of heavy ion collisions. In the asymptotic regime it is possible to understand the structure of the quark condensate from basic considerations. Consider the matrix element
(1) 
where , represent the quark fields, and , , are color, spin and flavor indices, respectively. For sufficiently high quark chemical potential, , color, spin and flavor structure can be completely fixed by the following arguments:

Antisymmetry in color indices in order to have attraction.

Antisymmetry in spin indices in order to have a spin zero condensate. The isotropic structure of the spin zero condensate is favored with respect to the spin (or higher spin) condensate since a larger portion of the phase space around the Fermi surface is available for pairing.

Given the structure in color and spin, Fermi statistics requires antisymmetry in flavor indices.
Since the quark spin and momenta in the pair are opposite, it follows that the left(right)handed quarks can pair only with left(right)handed quarks. Considering threeflavor quark matter at large baryonic density, the socalled color–flavor locked (CFL) phase turns out to be thermodynamically favored, with condensate
(2) 
where is the pairing gap, we have suppressed spinorial indices and neglected pairing in the color sextet channel. Pairing in the color sextet channel is automatically induced by the quark color structure, but the condensate in this channel is much smaller than in the color antitriplet channel and in most cases can be neglected. The CFL condensate in Eq. (2) was introduced in Alford et al. (1999c) and the reason for its name is that only simultaneous transformations in color and in flavor spaces leave the condensate invariant. The symmetry breaking pattern turns out to be
(3) 
where is the diagonal global subgroup of the three groups and the group means that the quark fields can still be multiplied by 1. According with the symmetry breaking pattern, the generators of chiral symmetry, color symmetry and symmetry are spontaneously broken. The 8 broken generators of the color gauge group correspond to the 8 longitudinal degrees of freedom of the gluons and according with the HiggsAnderson mechanism these gauge bosons acquire a Meissner mass. The diquark condensation induces a Majoranalike mass term in the fermionic sector which is not diagonal in color and flavor indices. Thus, the fermionic excitations consist of gapped modes with mass proportional to ^{1}^{1}1This is a feature of all homogenous superconducting phases: the fermionic excitations which are charged with respect to the condensate acquire a Majoranalike mass term proportional to the pairing gap. The lowenergy spectrum consists of 9 NambuGoldstone bosons (NGB) organized in an octet, associated with the breaking of the flavor group, and in a singlet, associated with the breaking of the baryonic number. For nonvanishing quark masses the octet of NGBs becomes massive, but the singlet NGB is protected by the symmetry; it remains massless and determines the superfluid properties of the CFL phase. The effective theory describing the NGBs for the CFL phase has been studied in Casalbuoni and Gatto (1999); Son and Stephanov (2000a); Son (2002). The CFL condensate also breaks the axial symmetry; given that at very high densities the explicit axial symmetry breaking is weak, one has to include the corresponding pseudoNGB in the lowenergy spectrum.
After the first attempts with instantoninduced interaction many authors tried various approaches in order to calculate the magnitude of gap parameters in the CSC phases (for references see the review by Rajagopal and Wilczek Rajagopal and Wilczek (2000)). Dealing with QCD the ideal situation would be if these kind of calculations could fall within the scope of lattice gauge theories. Unfortunately, lattice methods rely on Monte Carlo sampling techniques that are unfeasible at finite density because the fermion determinant becomes complex. Although various approximation schemes have been developed, for instance, Taylor expansion in the chemical potential Allton et al. (2003), reweighting techniques Fodor and Katz (2002), analytical continuation of calculation employing imaginary baryonic chemical potential Roberge and Weiss (1986); Alford et al. (1999b) or heavy Wilson quarks Fromm et al. (2012), no definite results have been obtained so far for large values of the baryonic chemical potential and physical quark masses.
In the absence of suitable lattice methods, quantitative analyses of color superconductivity have followed two distinct paths. The first path is semiphenomenological, and based on simplified models. The main feature of these models is that they should incorporate the most important physical effects but being at the same time tractable within present mathematical techniques. All these models have free parameters that are adjusted in such a way to give rise to a reasonable vacuum physics.
Examples of these kind of techniques include NambuJona Lasinio (NJL) models in which the interaction between quarks is replaced by a fourfermion interaction originating from instanton exchange Alford et al. (1998); Rapp et al. (1998); Berges and Rajagopal (1999) or where the fourfermi interaction is modeled by that induced by singlegluon exchange Alford et al. (1998, 1999a). Random matrix models have been studied in Vanderheyden and Jackson (2000) and instanton liquid models have been investigated in Carter and Diakonov (1999); Rapp et al. (2000, 2001), while renormalization group methods have been used in Evans et al. (1999); Schafer and Wilczek (1999b). Although none of these methods has a firm theoretical basis, all of them yield results all in fairly qualitative agreement. This is probably due to the fact that what really matters is the existence of an attractive interaction between quarks and that the parameters of the various models are chosen in such a way to reproduce the chirally broken ground state. The gap parameter evaluated within these models varies between tens of MeV up to MeV. The critical temperature is typically the same found in normal superconductivity, that is about one half of the gap.
The second path starts from first principles and relies on the property of asymptotic freedom of QCD. Various results have been obtained starting from the QCD action, employing renormalization group techniques and through the SchwingerDyson equation Son (1999); Schafer and Wilczek (1999c); Pisarski and Rischke (2000a); Hong et al. (2000); Brown et al. (2000); Evans et al. (2000). In particular, Son Son (1999), using the renormalization group near the Fermi surface has obtained the asymptotic form of the gap. However, it has been argued in Rajagopal and Shuster (2000) that the weakcoupling calculations employed in this case are valid only for extremely large densities corresponding to chemical potentials larger than MeV. Moreover, also in this second path, it is not possible to control the approximation, because we do not know how to evaluate higher order corrections. Interestingly enough, extrapolating the results from asymptotic densities to values of quark chemical potential of order MeV, one obtains a magnitude of the gap in agreement with the results of more phenomenological approaches.
The result of the analyses of the abovementioned methods is that the CFL phase is the thermodynamically favored state of matter at asymptotic densities. Qualitatively one can understand this result considering that in the CFL phase quarks of all threeflavors participate coherently in pairing. Since superconductivity is a cooperative phenomenon, the larger the number of fermions that participate in pairing, more energetically favored is the superconducting phase.
In the description of color superconductivity one has to deal with various scales, the chemical potential , the gap parameter, which we shall generically indicate with , the strange quark mass and the screening/damping scale , where is the QCD coupling. One typically has that , whereas the strange quark mass can be considered as a free parameter, although in some models it can be computed selfconsistently.
QCD at high density is conveniently studied through a hierarchy of effective field theories, schematically depicted in Fig. 1.
The starting point is the fundamental QCD Lagrangian, then one can obtain the low energy effective Lagrangian through different methods. One way is to integrate out highenergy degrees of freedom as shown in Polchinski (1992). The physics is particularly simple for energies close to the Fermi energy where all the interactions are irrelevant except for a fourfermi interaction coupling pair of fermions with opposite momenta. This is nothing but the interaction giving rise to BCS condensation, which can be described using the High Density Effective Theory (HDET) Nardulli (2002); Hong (2000a, b); Beane et al. (2000); Casalbuoni et al. (2001b); Schafer (2003a), see Sec. III.2.1 for a brief description of the method. The HDET is based on the fact that at vanishing temperature and large chemical potentials antiparticle fields decouple and the only relevant fermionic degrees of freedom are quasiparticles and quasiholes close to the Fermi surface. In the HDET Lagrangian the great advantage is that the effective fermionic fields have no spin structure and therefore the theory is particularly simple to handle.
This description is supposed to hold up to a cutoff , with smaller than the Fermi momentum, , but bigger than the gap parameter, i.e. . Considering momenta much smaller than all the gapped particles decouple and one is left with the low energy modes as NGBs, ungapped fermions and holes and massless gauged fields according with the symmetry breaking scheme. In the case of CFL and other CSC phases, such effective Lagrangians have been derived in Casalbuoni and Gatto (1999); Casalbuoni et al. (2000); Rischke et al. (2001). The parameters of the effective Lagrangian can be evaluated at each step of the hierarchy by matching the Green’s functions with the ones evaluated at the upper level. For the CFL phase the effective Lagrangian of the superfluid mode associated with the breaking of may also be determined by symmetry arguments alone as in Son (2002).
In the high density limit one neglects the quark masses, but in realistic situations the quark chemical potential may be of the same order of magnitude of the strange quark mass. The typical effect of quark masses is to produce a mismatch between Fermi surfaces. Neglecting light quark masses and assuming (for simplicity) that quarks have all the same chemical potentials, the Fermi spheres have now different radii
(4) 
where is the constituent strange quark mass. Thus, increasing for a fixed value of , increases the mismatch between the Fermi surface of strange quarks and the Fermi surfaces of up and down quarks (which in this simplistic case are equal).
The standard BCS mechanism assumes that the Fermi momenta of the fermionic species that form Cooper pairs are equal. When there is a mismatch it is not guaranteed that BCS pairing takes place, because the condensation of fermions with different Fermi momenta has a freeenergy cost. For small mismatches there is still condensation Chandrasekhar (1962); Clogston (1962), and in the case at hand it means that the CFL phase is favored. However, for large values of the strange quark mass the assumptions leading to prove that the favored phase is CFL should be reconsidered. According with Eq. (4) if the strange quark mass is about the quark chemical potential, then strange quarks decouple, and the corresponding favored condensate should consist of only up and down quarks. With only two flavors of quarks, and due to the antisymmetry in color, the condensate must necessarily choose a direction in color space and one possible pairing pattern is
(5) 
This phase of matter is known as twoflavor color superconductor (2SC) and is the corresponding gap parameter. This phase is characterized by the presence of 2 ungapped quarks, and 4 gapped quasiparticles given by the combinations and of the quark fields, where the color indices of the fundamental representation have been identified with (red, green and blue). In case massive strange quarks are present the corresponding phase is named 2SC+s and eventually strange quarks may by themselves form a spin1 condensate Pisarski and Rischke (2000b).
In the 2SC phase the symmetry breaking pattern is completely different from the threeflavor case and it turns out to be
(6) 
The chiral group remains unbroken, meaning that there are no NGBs. The original color symmetry group is broken to and since three color generators are unbroken, only five gluons acquire a Meissner mass. Even though is spontaneously broken there is an unbroken global symmetry, where is given by a combination of and of the eighth color generator, , playing the same role of the original baryonic number symmetry. In particular, this means that unlike CFL matter, 2SC matter is not superfluid. One can construct an effective theory to describe the emergence of the unbroken subgroup and the low energy excitations, much in the same way as one builds up chiral effective Lagrangian with effective fields at zero density. This development can be found in Casalbuoni et al. (2000); Rischke et al. (2001).
The problem of condensation for imbalanced Fermi momenta is quite general and can be described assuming that different populations have different chemical potentials, see e.g. Alford and Rajagopal (2002). We shall present a discussion of a simple twolevel system in Sec. II.1, which allows to clarify the main effects of a chemical potential difference, .
Regarding quark matter a similar behavior can be found when one considers realistic condition, i.e. conditions that can be realized in a compact stellar object (CSO). This is a real possibility since the central densities for these stars could reach g/cm, whereas the temperature is of the order of tens of keV, much less than the critical temperature for color superconductivity. Matter inside a CSO should be electrically neutral and in a color singlet state. Also conditions for equilibrium should be fulfilled. As far as color is concerned, it is possible to impose a simpler condition, that is color neutrality, since in Amore et al. (2002) it has been shown (in the twoflavor case) that there is a small freeenergy cost in projecting color singlet states out of color neutral ones. If electrons are present (as generally required by electrical neutrality) the equilibrium condition forces the chemical potentials of quarks with different electric charges to be different, thus Eqs. (4) are no more valid and more complicated relations hold Alford and Rajagopal (2002). The effect of the mass of the strange quark, equilibrium and color and electric neutrality, is to pull apart the Fermi spheres of up, down and strange quarks and many different phases may be realized, depending on the parameters of the system. Besides the abovementioned standard 2SC and 2SC+s phases, the twoflavor superconducting phase 2SCus, with pairing between up and strange quarks can be favored; see e.g. Iida et al. (2004); Ruester et al. (2006a) for different pairing patterns. For very large mismatches among the three flavors of quarks only the interspecies singleflavor spin1 pairing may take place Bailin and Love (1979); Alford et al. (2003, 1998); Buballa et al. (2003); Schafer (2000); Schmitt (2005); Schmitt et al. (2002), see e.g. Alford et al. (2008) for an extended discussion on these topics.
In the 2SC phase, the above conditions tend to separate the Fermi spheres of up and down quarks and, as discussed in Sec. II.2, for gapless modes appear. The corresponding phase has been named g2SC Shovkovy and Huang (2003); Huang and Shovkovy (2003), with “g” standing for gapless. The g2SC phase with pairing between up and down quarks has the same condensate of the 2SC phase reported in Eq. (5), and therefore the ground states of the 2SC and of the g2SC phases share the same symmetry. However, these two phases have a different low energy spectrum, due to the fact that in the g2SC phase only two fermionic modes are gapped. The g2SC phase is energetically favored with respect to the 2SC phase and unpaired quark matter in a certain range of values of the fourfermi interaction strength when one considers equilibrium, color and electrical neutrality Shovkovy and Huang (2003).
Pinning down the correct ground state of neutral quark matter in equilibrium is not simple because another difficulty emerges. This problem, which is already present in the simple twolevel system discussed in Sec. II.1, has a rather general character Alford and Wang (2005), and is due to an instability connected to the Meissner mass. In particular, when the system becomes magnetically unstable, meaning that the Meissner mass becomes imaginary. In the 2SC phase the color group is broken to and 5 out of 8 gluons acquire a mass. Four of these masses turn out to be imaginary in the 2SC phase for , thus in this range of the 2SC phase is chromomagnetically unstable Huang and Shovkovy (2004b, a). Increasing the chemical potential difference the instability gets worse, because at the phase transition from the 2SC phase to the g2SC phase all the five gluon masses become pure imaginary.
An analogous phenomenon arises in threeflavor quark matter because the gapless CFL (gCFL) phase Alford et al. (2004, 2005c, 2005b); Fukushima et al. (2005) turns out to be chromomagnetically unstable Casalbuoni et al. (2005b); Fukushima (2005). The gCFL phase has been proposed as the favored ground state for sufficiently large mismatch between up, down and strange quarks and occurs in color and electrically neutral quark matter in equilibrium for . Some properties of the gCFL phase and a brief discussion of the corresponding instability will be presented in Sec. III.1.
There is a variety of solutions that have been proposed for the chromomagnetic instability and that can be realized depending on the particular conditions considered. As we have already discussed, the chromomagnetic instability is a serious problem not only for the gapless phases (g2SC and gCFL) but also for the 2SC phase. In the latter case, it has been shown in Gorbar et al. (2006a) that vector condensates of gluons with a value of about MeV can cure the instability. The corresponding phase has been named gluonic phase and is characterized by the nonvanishing values of the chromoelectric condensates and which spontaneously break the SO(3) rotational symmetry. It is not clear whether the same method can be extended to the gapless phases. The chromomagnetic instability of the gapped 2SC phase can also be removed by the formation of an inhomogeneous condensate of charged gluons Ferrer and de la Incera (2007).
For the cases in which the chromomagnetic instability is related to the presence of gapless modes, in Hong (2005) the possibility that a secondary gap opens at the Fermi surface is studied. The solution of the instability is due to a mechanism that stabilizes the system preventing the appearance of gapless modes. However, the secondary gap turns out to be extremely small and at temperatures typical of CSOs it is not able to fix the chromomagnetic instability Alford and Wang (2006).
The imaginary value of the Meissner mass can be understood as a tendency of the system toward a nonhomogeneous phase Iida and Fukushima (2006); Gubankova et al. (2010); Hong (2005). This can be easily seen in the toymodel system discussed in Sec. II.1 for the case of a symmetry, where one can show that the coefficient of the gradient term of the low energy fluctuations around the ground state of the effective action is proportional to the Meissner mass squared Gubankova et al. (2010).
For threeflavor quark matter two nonhomogenous superconducting phases have been proposed. If kaon condensation takes place in the CFL phase Bedaque and Schafer (2002); Kaplan and Reddy (2002), the chromomagnetic instability might drive the system toward a nonhomogeneous state where a kaon condensate current is generated, balanced by a counterpropagating current in the opposite direction carried by gapless quark quasiparticles. This phase of matter, named curCFL, has been studied in Kryjevski (2008) and turns out to be chromomagnetically stable.
The second possibility is the crystalline color superconducting (CCSC) phase Alford et al. (2001); Bowers et al. (2001); Kundu and Rajagopal (2002); Leibovich et al. (2001); Bowers and Rajagopal (2002); Casalbuoni et al. (2001a, 2002c, 2002b, 2003, 2004, 2005a); Mannarelli et al. (2006b), which is the QCD analogue of a form of nonBCS pairing first proposed by Larkin, Ovchinnikov, Fulde and Ferrell (LOFF) Larkin and Ovchinnikov (1964); Fulde and Ferrell (1964). This phase is chromomagnetically stable as we shall discuss in Sec. II.6, for twoflavor quark matter, and in Sec. III.2.4, for threeflavor quark matter. The condensate characteristic of this phase is given by
(7) 
which is similar to the condensate reported in Eq. (2) but now there are three gap parameters, each having a periodic modulation in space. The modulation of the ’th condensate is defined by the vectors , where is the index which identifies the elements of the set . In position space, this corresponds to condensates that vary like , meaning that the ’s are the reciprocal vectors which define the crystal structure of the condensate.
In Sec. II we shall discuss various properties of the the twoflavor CCSC phase. This phase, first proposed in Alford et al. (2001), corresponds to the case where only one gap parameter is nonvanishing. The chromomagnetic stability of a simple twoflavor periodic structure with a gap parameter modulated by a single plane wave (hereafter we shall refer to this phase as FuldeFerrell (FF) structure Fulde and Ferrell (1964)) has been considered in Giannakis and Ren (2005a, b); Giannakis et al. (2005) where it has been shown that

The presence of the chromomagnetic instability in g2SC is exactly what one needs in order that the FF phase is energetically favored Giannakis and Ren (2005a).
The stability of the FF phase in the strong coupling case has been studied in Gorbar et al. (2006b), where it is shown that for large values of the gap parameter the FF phase cannot cure the chromomagnetic instability. In Nickel and Buballa (2009) it has been questioned whether among the possible onedimensional modulations, the periodic LOFF solution is the favored one. As we shall discuss in Sec. II.7, it is found that for twoflavor quark matter, a solitoniclike ground state is favored with respect to FF in the range of values . However, at least in weak coupling, the FF phase is not the crystalline structure one should compare to. The FF phase is slightly energetically favored with respect to unpaired quark matter and 2SC quark matter for , but more complicated crystalline structures have larger condensation energies in a larger range of values of Bowers and Rajagopal (2002).
The stability analysis of the threeflavor CCSC phase is discussed in Sec. III.2.4, where we report on the results obtained for a simple structure made of two plane waves by a GinzburgLandau (GL) expansion Ciminale et al. (2006). This particular threeflavor CCSC phase turns out to be chromomagnetically stable, but the stability of more complicated crystalline structures has not been studied, although by general arguments they are expected to be stable, at least in the weak coupling limit.
Whether or not the crystalline color superconducting phase is the correct ground state for quark systems with mismatched Fermi surfaces has not yet been proven. In any case it represents an appealing candidate because in this phase quark pairing has no energy cost proportional to . The reason is that pairing occurs between quarks living on their own Fermi surfaces. However, this kind of pairing can take place only if Cooper pairs have nonzero total momentum and therefore it has an energy cost corresponding to the kinetic energy needed for the creation of quark currents. Moreover, pairing can take place only in restricted phase space regions, meaning that the condensation energy is smaller than in the homogeneous phase. The vector has a magnitude proportional to the chemical potential splitting between Fermi surfaces, whereas its direction is spontaneously chosen by the system. In case one considers structures composed by a set of vectors , one has to find the arrangement that minimizes the freeenergy of the system Bowers and Rajagopal (2002); Rajagopal and Sharma (2006). This is a rather complicated task which is achieved by analyzing some ansatz structures and comparing the corresponding freeenergies. We shall report on this subject in Sec. II.3 and II.5 for the twoflavor case and in Sec. III.3 for the threeflavor case.
Summarizing, we can say that the state of matter at asymptotic densities is well defined and should correspond to the CFL condensate. At intermediate and more realistic densities it is not clear which is the ground state of matter. Our knowledge of the phases of matter can be represented in the socalled QCD phase diagram, which is schematically depicted in Fig. 2. At lowdensity and low temperature quarks are confined in hadrons but increasing the energy scale quarks and gluons degrees of freedom are liberated. At high temperature this leads to the formation of a plasma of quarks and gluons (QGP), while at large densities matter should be in a color superconducting phase. Apart from the phases we have discussed other possibilities may be realized in the density regime relevant for CSO. Here we only mention that recently it has been proposed one more candidate phase, the socalled quarkyonic phase McLerran and Pisarski (2007), which is characterized by a nonvanishing baryon number density and found to be a candidate phase at least for a large number of colors. Another possibility is that the constituent value of the strange quark mass is so small that the CFL phase is the dominant one down to the phase transition to the hadronic phase. In this case, a rather interesting possibility is that there is no phase transition between the CFL phase and the hadronic phase (hypernuclear matter), in the socalled quarkhadron continuity scenario Schafer and Wilczek (1999a).
In Section IV we shall discuss whether the presence of a nonhomogenous color superconducting phase within the core of a compact star may lead to observable effects. The different possible signatures are associated with

Gravitational wave emission.

Anomalies in the rotation frequency (known as glitches).

Cooling processes.

Massradius relation.
Point 1 is discussed in Sec. IV.1, and relies on the observation that pulsars can be continuous sources of gravitational waves if their mass distribution is not axissymmetric. The large shear modulus characteristic of the CCSC phase allows the presence of big deformations of the star, usually called “mountains”.
Regarding point 2, the threeflavor CCSC phase is characterized by an extremely large rigidity Mannarelli et al. (2007), with a shear modulus which is larger than that of a conventional neutron star crust by a factor of 20 to 1000. This fact makes the crystalline phases of quark matter unique among all forms of matter proposed as candidates for explaining stellar glitches. This topic is detailed in Sec. IV.2.
Point 3 is discussed in Sec. IV.3; we report the results of a first step toward calculating the cooling rate for neutron stars with a CCSC core taken in Anglani et al. (2006), where a simple two plane waves structure was considered. Because the crystalline phases leave some quarks at their respective Fermi surfaces unpaired, their neutrino emissivity and heat capacity are only quantitatively smaller than those of unpaired quark matter Iwamoto (1980, 1981), not parametrically suppressed. This suggests that neutron stars with crystalline quark matter cores will cool down by the direct Urca reactions, i.e. more rapidly than in standard cooling scenarios Page et al. (2004).
Point 4 is discussed in Sec. IV.4, where it is shown that recent observations of very massive compact stars do not exclude the possibility that CSO have a CCSC core.
Finally, studying the damping mechanisms of the radial and nonradial star oscillations (starseismology) might be useful to infer the properties of the CCSC core, and to have information about the width of the various internal layers of the star. We are not aware of any paper discussing this topic, which however might be relevant for restricting the parameter space of the CCSC phase.
Ii The twoflavor nonhomogenous phases
The nonhomogeneous twoflavor crystalline color superconducting phase is an extension to QCD of the phase proposed in condensed matter systems by Fulde and Ferrell Fulde and Ferrell (1964) and by Larkin and Ovchinnikov Larkin and Ovchinnikov (1964) (LOFF). Some aspects of this state have been previously discussed in the review Casalbuoni and Nardulli (2004). Therefore, we will focus here on recent results, and in particular we discuss one of the main properties of these phase, namely its chromomagnetic stability. This important property is not shared with other homogeneous gapless color superconducting phases, and therefore it strongly motivates its study.
ii.1 Mismatched Fermi spheres
Before discussing the case of twoflavor quark matter, in order to show how gapless superconductivity may arise, it is useful to consider the simpler case of a nonrelativistic fermionic gas avoiding the formal complications due to flavor and color degrees of freedom. We consider a system consisting of two unbalanced populations of different species and , with opposite spin, at vanishing temperature, having the hamiltonian density
(8) 
where is the four fermion coupling constant. The chemical potentials of the two species can be written as and , so that is the average of the two chemical potentials and their difference. The effect of the attractive interaction between fermions is to induce the difermion condensate
(9) 
which spontaneously breaks the global symmetry corresponding to particle number conservation. As a result the fermionic excitation spectrum consists of two Bogolyubov modes with dispersion laws
(10) 
with and is the homogenous mean field solution. Without loss of generality we take , then from Eq. (10) we infer that tuning the chemical potential difference to values , the mode becomes gapless. This phase corresponds to a superconductor with one gapped and one gapless fermionic mode and is named gapless homogeneous superfluid.
In the above naive discussion we did not take into account that increasing the difference between the freeenergy of the superfluid phase, , and of the normal phase, decreases; eventually the normal phase becomes energetically favored for sufficiently large . In weak coupling it is possible to show that the two freeenergies become equal at (corresponding to the socalled ChandrasekharClogston limit Chandrasekhar (1962); Clogston (1962)), that is before the fermionic excitation spectrum becomes gapless. At this critical value of a first order transition to the normal phase takes place and the superfluid phase becomes metastable. The reason for this behavior can be qualitatively understood as follows. Pairing results in an energy gain of the order of , however BCS pairing takes place between fermions with equal and opposite momenta. When a mismatch between the Fermi sphere is present it tends to disfavor the BCS pairing, because in order to have equal momenta, fermions must pay an energy cost of the order of . Therefore, when , where is some number, pairing cannot take place. In the weak coupling limit, one finds that . This behavior is pictorially depicted in Fig. 3.
Considering homogeneous phases, a metastable superconducting phase exists for , but still the system cannot develop fermionic massless modes, because at various instabilities appear Pao et al. (2006); Sheehy and Radzihovsky (2006); Mannarelli et al. (2006a); Gubankova et al. (2006, 2010); Wu and Yip (2003). In order to explain what happens, let us consider the low energy spectrum of the system, which can be described considering the fluctuations of around the mean field solution . The oscillations in the magnitude of the condensate are described by the Higgs mode, , while the phase fluctuations are described by the NambuGoldstone (or AnderssonBogolyubov) mode . Integrating out the fermionic degrees of freedom results in the Lagrangian densities Gubankova et al. (2010)
(11)  
(12) 
The stability of the system is guaranteed when all the coefficients , , , , are positive. and turn out to be always positive, then we define the three stability conditions

The Higgs has a positive squared mass:

The space derivative of the Higgs must be positive:

The space derivative of the NGB must be positive:
For the three conditions above are not simultaneously satisfied. Condition 1 is not satisfied because we are expanding around a maximum of the freeenergy, and indeed a freeenergy analysis shows that at the homogenous superfluid phase becomes unstable. The fact that the condition 2 is not satisfied signals that the system is unstable toward space fluctuations of the absolute value of the condensate, while condition 3 is not fulfilled when the system is unstable toward space fluctuations of the phase of the condensate. Clearly the conditions 2 and 3 are related and tell us that when a large mismatch between the Fermi sphere is present, the system prefers to move to a nonhomogenous phase.
In the present toymodel the three conditions above are simultaneously violated in weak coupling at , but they are violated at different values of this ratio in the strong coupling regime. Moreover, with increasing coupling strength it is possible to force the system into a homogeneous gapless phase, but this happens when , deep in the BoseEinstein condensate (BEC) limit, see e.g. Gubankova et al. (2010).
For , the homogenous BCS might also be energetically favored if there is a way of reducing , and this is indeed what happens in the CFL phase where multiple interaction channels are available and the color and electrical neutrality conditions may disfavor the normal phase. This corresponds to the fact that and thus the condition 1 above is satisfied. However, the conditions 2 and 3 must be satisfied as well, but decreasing does not per se guarantee that those constraints are satisfied. Indeed, it is possible to show that in the weak coupling regime the gapless homogeneous phase is in general not accessible, because when , the solution with is unstable with respect to the conditions 2 and 3.
Gauging the global symmetry, it is possible to show that the condition is equivalent to the condition that the Meissner mass squared of the gauge field becomes negative, which corresponds to a magnetic instability. Therefore the magnetic instability is related to the fact that we are expanding the freeenergy around a local maximum. This statement is rather general and indeed in Sec. II.6 we shall see that an analogous conclusion can be drawn for the 2SC phase. Notice that increasing the temperature of the system does not help to recover from this instability Alford and Wang (2005). Indeed, the effect of the temperature is to produce a smoothing of the dispersion law, which first has the effect of increasing the instability region to values .
Summarizing, we have seen that for the simplest case of a weakly interacting twoflavor system, for the superfluid homogenous phase is metastable, while for , it does not have a local minimum in and it is unstable toward fluctuations of the condensate. In general, the three conditions above should be simultaneously satisfied in order to have a stable (or metastable) vacuum. The gapless phase is only accessible for homogenous superfluids deep in the strong coupling regime, for negative values of the chemical potential.
A different possibility is that gapless modes arise at weak coupling in a nonhomogenous superfluid. As we shall see in the following sections, the nonhomogenous LOFF phase is energetically favored in a certain range of values of larger than the ChandrasekharClogston limit, see Sec. II.3, it is (chromo)magnetically stable, see Sec. II.6, and it has gapless fermionic excitations, see Sec. II.4. It is important to remark that the presence of a gapless fermionic spectrum is not in contrast with the existence of superconductivity de Gennes (1966), e.g. type II superconductors have gapless fermionic excitations for sufficiently large magnetic fields de Gennes (1966); SaintJames et al. (1969).
ii.2 Gapless 2SC phase of QCD
The gapless 2SC phase (g2SC in the following) of QCD was proposed in Shovkovy and Huang (2003) (see also Huang and Shovkovy (2003)) as a color superconducting phase which may sustain large Fermi surface mismatches. However, it was soon realized by the same authors that this phase is chromomagnetic unstable Huang and Shovkovy (2004b, a), meaning that the masses of some gauge fields become imaginary. In the following we briefly discuss the properties of the g2SC phase at vanishing temperature, and then we deal with the problem of chromomagnetic instability.
We consider neutral twoflavor quark matter at finite chemical potential. The system is described by the following Lagrangian density:
(13) 
where , corresponds to a quark spinor of flavor and color . The current quark mass is denoted by (we take the isospin symmetric limit ), and is an interaction Lagrangian that will be specified later.
In Eq. (13), is the quark chemical potential matrix with color and flavor indices, given by
(14) 
is the quark electric charge matrix and is the color generator along the direction eight in the adjoint color space; , denote respectively the electron and the color chemical potential. Since is diagonal in color and flavor spaces, we can indicate its element with , e.g. is the chemical potential of up blue quarks. A chemical potential along the third direction of color, , can be introduced besides , but, for all the cases that we discuss in this section, we require the ground state to be invariant under the color subgroup; this makes the introduction of unnecessary.
As interaction Lagrangian density we consider the NJLlike model
(15) 
where denotes the chargeconjugate spinor, with the charge conjugation matrix. The matrices and denote the antisymmetric tensors in flavor and color space, respectively; we used in the second term on the right hand side of Eq. (15) the shorthand notation
(16) 
and an analogous expression for the other bilinear. In Eq. (15), two coupling constants are introduced in the scalarpseudoscalar quarkantiquark channel, denoted by , and in the scalar diquark channel, denoted by . In Huang and Shovkovy (2003) the parameters of the model are chosen to reproduce the pion decay constant in the vacuum, MeV, and the vacuum chiral condensate = = MeV. Moreover, an ultraviolet cutoff is introduced to regularize the divergent momentum integrals. The parameter set of Huang and Shovkovy (2003) is given by
(17) 
The relative strength between the couplings in the quarkantiquark and quarkquark channels could be fixed by a Fierz rearranging of the quarkantiquark interaction, see for example Buballa (2005). The Fierz transformation gives . However, non perturbative inmedium effects might change this value. Therefore, in Huang and Shovkovy (2003) the ratio of to is considered as a free parameter.
In Huang and Shovkovy (2003) the authors consider only the case and vanishing chiral condensate. When the chiral condensate in the ground state does not vanish, but its effects are presumably negligible, giving a small shift of the quark Fermi momenta. This shift of Fermi momenta might change the numerical value of the electron chemical potential only of some few percent. Hence, the main results of Huang and Shovkovy (2003) should not change much if a finite value of the current quark mass is considered.
Once the Lagrangian density is specified, the goal is to compute the thermodynamic potential. In the mean field (and one loop) approximation, this can be done easily using standard bosonization techniques. Neglecting the chiral condensate, the mean field Lagrangian density can be written within the NambuGorkov formalism, see Sec. III.2.1, in the compact form
(18) 
where
(19) 
is the NambuGorkov spinor and the the gap parameter, , is included in the inverse propagator
(20) 
as an offdiagonal term in the “NambuGorkov space”.
We shall focus on the zero temperature regime (for a discussion of the rather uncommon temperature behavior of the g2SC phase see Huang and Shovkovy (2003)), which is relevant for astrophysical applications. The one loop expression of the thermodynamic potential can be determined from the inverse propagator in Eq.(20); for vanishing temperature it is given by
(21) 
for a derivation see e.g. Buballa (2005). In the above equation is an irrelevant constant which is chosen in order to have a vanishing pressure in the vacuum. The second addendum corresponds to the electron freeenergy (electron masses have been neglected). The last addendum is the contribution due to the quark determinant. The sum runs over the twelve fermion propagator poles, six of them corresponding to quarks and the other six corresponding to antiquarks quasiparticles:
(22)  
(23)  
(24)  
(25) 
and , , , . Here we have introduced the shorthand notation
(26) 
Using the explicit form of the dispersion laws, the previous equation can be written as
(27)  
where .
The value of is determined by the solution of the equation
(28) 
with the neutrality constraints,
(29) 
which fix the values of and .
The numerical analysis of Huang and Shovkovy (2003) shows that is much smaller than and , both for and for . As a consequence, it is possible to simplify the equations for the gap parameter and the electron chemical potential, (28) and (29) respectively, by putting . Therefore, the properties of the system depend only on the values and and on the couplings and . The result of Huang and Shovkovy (2003) can be summarized as follows:

For , strong coupling, the 2SC phase is the only stable phase

For , intermediate coupling, the g2SC phase is allowed for

For , weak coupling, only unpaired quark matter is favored.
In the g2SC phase the quasiparticle fermionic spectrum consists of four gapless modes and two gapped modes, whereas in the 2SC phase there are two gapless fermionic modes and four gapped fermionic modes. In the latter case the only gapless modes correspond to the up and down blue quarks that do not participate in pairing.
ii.2.1 Meissner masses of gluons in the g2SC phase
The diquark condensate of the 2SC phase induces the symmetry breaking pattern reported in Eq. (6); in particular the group is broken down to , where is the gauge group corresponding to the rotated massless photon associated to the unbroken generator
(30) 
where and denote the strong and the electromagnetic couplings respectively, is the generator of the electromagnetic gauge transformations and is the eighth generator of the color transformations. The mixing coefficients have been determined in Alford et al. (2000b) (see also Gorbar (2000)) and are given by
(31) 
The linear combination
(32) 
is orthogonal to and gives the broken generator; the corresponding gauge field, which we shall refer to as the mode, acquires a Meissner mass. Actually, the NJLlike Lagrangian in Eqs.(13) and (15), has only global symmetries, but gauging the group and the subgroup of , one has that the spontaneous symmetry breaking leads to the generation of Meissner masses for the five gluons associated to the broken generators. In order to compute these masses, we define the gauge boson polarization tensor, see e.g. Le Bellac (2000),
(33) 
where is the quark propagator in momentum space, which can be obtained from Eq. (20), and
(34)  
(35) 
are the interaction vertex matrices. The trace in Eq. (33) is taken over Dirac, NambuGorkov, color and flavor indices; indicate the adjoint color and we use the convention that the component with corresponds to the photon.
The screening masses of the gauge bosons are defined in terms of the eigenvalues of the polarization tensor and in the basis in which is diagonal the Debye masses and the Meissner masses are respectively defined as
(36)  
(37) 
Both masses have been evaluated in the 2SC phase in Rischke (2000b); Rischke and Shovkovy (2002); Schmitt et al. (2004). The Debye masses of all gluons are related to the chromoelectric screening and are always real, therefore do not affect the stability of the 2SC and g2SC phases. The Meissner masses of the gluons with adjoint color are always zero, because they are associated to the unbroken color subgroup .
For nonvanishing values of the Meissner screening masses have been evaluated in Huang and Shovkovy (2004b, a). Gluons with adjoint color , are degenerate and in the limit their Meissner masses are given by
(38) 
The squared Meissner mass turn out to be negative not only in the gapless phase, , but also in the gapped phase, when . This result seems in contrast with the result of the previous section, where an imaginary Meissner mass was related to the existence of a local maximum of the freeenergy arising at . However, from the analysis of the 2SC freeenergy of the system, one can see that when the state with corresponds to a saddle point in the plane. The neutrality condition transforms this saddle point into a local minimum. However, as explained in the previous section, the gauge fields can be related to the fluctuations of the gap parameter. These fluctuations can probe all the directions in the plane around the stationary point and would result in a low energy Lagrangian with dispersion laws akin to the those discussed in Eqs.(11) and (12) with the coefficients and negative.
Finally let us consider the mode, associated to the broken generator defined in Eq. (32). The corresponding Meissner mass can be obtained diagonalizing the polarization tensor in Eq. (33) in the subspace , or more directly, by substituting in the vertex factors of the polarization tensor. The squared Meissner mass of the mode turns out to be
(39) 
and becomes negative for . As shown in Gatto and Ruggieri (2007), the instability in this sector is transmitted to a gradient instability of the pseudoGoldstone boson related to the symmetry which is broken by the diquark condensate. Although in the vacuum the symmetry is explicitly broken by instantons, at finite chemical potential instantons are Debye screened and can be considered as an approximate symmetry, which is then spontaneously broken by the diquark condensate.
ii.3 The twoflavor crystalline color superconducting phase
Since the homogeneous g2SC phase is chromomagnetically unstable, the question arises of the possible existence of a different superconducting phase for large mismatch between the Fermi spheres. There are several candidate phases, which include the gluonic phase Gorbar et al. (2006a), the solitonic phase Nickel and Buballa (2009) and the crystalline color superconducting phases.
In this section we review some of the main results about the twoflavor crystalline phase. Firstly, we describe the one plane wave ansatz in the framework of the symple model discussed in Sec. II.1; then, we turn to the crystalline color superconducting phase and report on the GinzburgLandau (GL) analysis of various crystalline structures. We relax the constraint of electrical and color neutrality, and treat the difference of chemical potentials between and quarks, 2, as a free parameter.
ii.3.1 The one plane wave ansatz
In Sec. II.1 we have shown that in weak coupling the ChandrasekharClogston limit Chandrasekhar (1962); Clogston (1962) signals that the standard BCS phase becomes metastable, but does not forbid the existence of different forms of superconductivity. In particular, it does not forbid the existence of Cooper pairs with nonvanishing total momentum. It was shown by Larkin and Ovchinnikov Larkin and Ovchinnikov (1964) and Fulde and Ferrel Fulde and Ferrell (1964), in the context of electromagnetic superconductivity, that in a certain range of values of it might be energetically favorable to have Cooper pairs with nonzero total momentum. For the simple twolevel system discussed in Sec. II.1, Cooper pairs with momentum can be described by considering the difermion condensate in Eq. (9) given by
(40) 
and we shall call this state of matter the FF phase.
Notice that this ansatz breaks rotational symmetry because there is a privileged direction corresponding to . In the left panel of Fig. 4 the two Fermi spheres of fermions are pictorially shown and the red ribbons correspond to the regions in momentum space where pairing occurs. Pairing between fermions of different spin can only take place in a restricted region of momentum space and this implies that . The reason why the FF phase is energetically favorable with respect to the normal phase, is that no energy cost proportional to has to be payed for allowing the formation of Cooper pairs. The only energetic price to pay is due to the kinetic energy associated to Cooper pairs. Actually, there is a spontaneous generation of a supercurrent in the direction of , which is balanced by a current of normal fermions in the opposite direction Fulde and Ferrell (1964). The gap parameter in Eq. (40) can be determined solving a gap equation under the constraint that the modulus of the Cooper momentum, , minimizes the freeenergy. The result is that at there is a first order phase transition from the homogeneous BCS phase to the FF phase. Increasing further results in a smooth decreasing of the gap function of the FF phase, until at a critical value a second order phase transition to the normal phase takes place. In the weak coupling limit ; the range is called the LOFF window. In the LOFF window, the optimal value of turns out to be approximately constant, .
In Alford et al. (2001) the twoflavor QCD analog of the FF phase was presented. In this case the condensate has the same color, spin and flavor structure of the 2SC condensate, but with the plane wave space dependence characteristic of the FF phase, that is
(41) 
As noticed in Alford et al. (2001), the FF condensate induces a spin1 condensate as well; however, its effect is found to be numerically small, and it will be neglected here. As in the simple twolevel system, in the twoflavor FF phase it is possible to determine the freeenergy employing a NJLlike model with gap parameter in (41), which can be determined solving the corresponding gap equation under the constraint that the value of in Eq. (41) minimizes the freeenergy. The results are the same obtained in the twolevel system; in particular the LOFF window and have the same expressions reported above (but now is the the difference of chemical potentials between and quarks).
ii.3.2 GinzburgLandau analysis
From Fig. 4, it seems clear than an immediate generalization of the FF phase, can be obtained adding more ribbons on the top of the Fermi spheres, corresponding to different vectors , with , where is some set of vectors to be determined by minimizing the freeenergy of the system and is a label that identifies the vectors of the set. This in turn, corresponds to consider inhomogeneous CSC phases with a more general ansatz than in Eq. (41), where the single plane wave is replaced by a superposition of plane waves, that is
(42) 
Assuming that the set of vectors identifies the vertices of a crystalline structure it follows that at each set corresponds a particular crystalline phase. We shall assume that the vectors have equal length and then we can write , therefore the crystalline structure is determined by the set of unit vectors . We shall also consider the simplified case that the coefficients do not depend on and we shall indicate their common value with . In other words, we shall consider condensates with
(43) 
where is the number of vectors . The simplest example is clearly the FF condensate, depicted in the left panel of Fig. 4, characterized by a single plane wave, thus corresponding to . The case with is reported in the right panel of Fig.4, in this case the “crystalline” structure is completely determined by , the relative angle between and ; more complicated structures can be pictorially represented in a similar way.
It is important to stress that the crystalline structure is determined by the modulation of the condensate, but the underlying fermions are not arranged in an ordered pattern, indeed fermions are superconducting, that is they form a superfluid of charged carriers.
The computation of the freeenergy of a system with a general crystalline condensate cannot be obtained analytically and also the numerical evaluation is quite involved. As a consequence, the use of some approximation is necessary. A viable method is the GL expansion of the freeenergy, which is obtained expanding in powers of Bowers and Rajagopal (2002):
(44) 
where the coefficients , and are computed, in the oneloop approximation using, as microscopic model, a NJL model. The GL expansion is well suited for studying second order phase transitions but might give reasonable results for soft first order phase transitions as well. In the present case the expansion is under control for and if the coefficient is positive, meaning that the freeenergy is bounded from below.
For a given crystalline structure, the coefficients in Eq. (44) depend on and on the magnitude of ; the latter is fixed, in the calculation of Bowers and Rajagopal (2002), to the weak coupling value . For any value of the thermodynamic potential of a given structure is computed by minimization with respect to and then the optimal crystalline structure is identified with that with the lowest freeenergy. It is possible to compute analytically the GL coefficients only for few structures; in general, they have to be computed numerically. We refer to the appendix of Bowers and Rajagopal (2002) for details.
In Bowers and Rajagopal (2002), twentythree crystalline structures have been studied and among them, those with more than nine rings turn out to be energetically disfavored. This has been nicely explained in the weak coupling regime: in this case, as shown in the left panel of Fig. 4 for the FF phase, the pairing regions of the inhomogeneous superconductor can be approximated as rings on the top of the Fermi surfaces; one ring per wave vector in the set . The computation of the lowest order GL coefficients shows that the intersection of two rings is energetically disfavored Bowers and Rajagopal (2002). As a consequence, it is natural to expect that in the most favored structure no intersecting rings appear. Since each ring has an opening angle of approximately degrees, a maximum of nine rings can be accommodated on a spherical surface.
This result can be quantitatively understood as follows. For the case of the two plane waves structure (right panel of Fig. 4), in the weak coupling approximation there is one pairing ring for each of the two wave vectors.
The quartic coefficient depends on the angle between the two wave vectors and it diverges at
(45) 
which corresponds to the angle at which the two pairing rings are contiguous, meaning that for the two rings overlap. The latter case is energetically disfavored because, being large and positive, the freeenergy would be smaller.
The divergence of the coefficient at is due to the two limits that have been taken to compute the freeenergy, namely the GinzburgLandau and weak coupling limits. A detailed explanation of what happens will be given in Sec. III.2 when discussing a simple crystalline structure in the threeflavor case. In any case it is clear that the divergency of a GL coefficient means that the expansion is not under control, or more precisely, that the radius of convergence of the series (44) tends to zero.
Among the structures with no intersecting rings, seven are good candidates to be the most favored structure. Within these seven structures, the octahedron, which corresponds to a crystal with and whose wave vectors point into the direction of a bodycenteredcube (BCC), is the only one with effective potential bounded from below (that is, with ). The remaining six structures, with and , are characterized by a potential which is unbounded from below, at least at the order . Even if in this case the freeenergy cannot be computed, qualitative arguments given in Bowers and Rajagopal (2002) suggest that the facecenteredcube (FCC), with and with wave vectors pointing towards the vertices of a facecenteredcube, is the favored structure.
Of course, as the authors of Bowers and Rajagopal (2002) admit, their study cannot be trusted quantitatively, because of the several limitations of the GL analysis. First of all, the GL expansion formally corresponds to an expansion in powers of and therefore it is well suited for the study of second order phase transitions, but the condition that is not satisfied by all crystalline structures considered in Bowers and Rajagopal (2002). In some cases the GL analysis predicts a strong firstorder phase transition to the normal state, with a large value of the gap at the transition point. Moreover, it may happen that the local minimum for small values of is not a global minimum of the system, as discussed in Sec. II.7. In this case the GL expansion in Eq. (44) underestimates the freeenergy of the system and is not able to reproduce the correct order of the phase transition. For a more reliable determination of the ground state one should consider terms of higher power in , which are difficult to evaluate. Finally, the claimed favored crystalline structure, namely the facecenteredcube, has and a global minimum cannot be found unless the coefficient (or of higher order) is computed and found to be positive.
Because of these reasons, the quantitative predictions of the GL analysis should be taken with a grain of salt. One should not trust the order of the phase transition obtained by the GL expansion and also the comparison among various crystalline structures may be partially incorrect, because it is not guaranteed that one is comparing the energies of the true ground states.
On the other hand, the qualitative picture that we can draw from it, namely the existence of crystalline structures with lower freeenergy than the single plane wave, is quite reasonable: crystalline structures benefit of more phase space available for pairing, thus lowering the freeenergy. The symmetry argument is quite solid too, because it is based on the fact that configurations with overlapping pairing regions are disfavored, and as we shall see for one particular configuration in the threeflavor case in Sec. III.2, one can prove that this statement is correct without relying on the GL expansion. Also, we shall show an interesting point, that the GL expansion underestimates the freeenergy of the crystalline structures. And this happens not only in the presence of a global minimum different from the local minimum around which the GL expansion is performed, see Sec. II.7, but also comparing the GL freeenergy with the freeenergy evaluated without the expansion.
ii.4 Dispersion laws and specific heats
The thermal coefficients (specific heat, thermal conductivity etc.) of quark matter at very low temperature are of fundamental importance for the transport properties and cooling mechanisms of compact stars. The largest contribution to thermal coefficients comes from the low energy degrees of freedom and it is therefore of the utmost importance to understand whether fermionic modes are gapped or gapless. Indeed, the absence of a gap in the spectrum of fermions implies that quasiquarks can be excited even at low temperature and therefore the corresponding thermal coefficients are not suppressed by a factor (which is distinctive of homogeneous BCS superconductors).
In this section, we discuss the fermion and phonon dispersion laws in the twoflavor crystalline phases for low values of momenta. Then we employ the obtained dispersion laws for the computation of the specific heats. The results discussed here do not rely on the GL approximation but are obtained by finding the zeros of the full inverse propagator close to the nodes of the dispersion law Larkin and Ovchinnikov (1964); Casalbuoni et al. (2003).
ii.4.1 Fermi quasiparticle dispersion law: general settings
We consider a general difermion condensate , and determine the quasiparticle dispersion laws looking at the zero modes of the inverse propagator of the system. Arranging the fields in the NambuGorkov spinor as follows,
(46) 
the inverse propagator is given by
(47) 
where is the quasiparticle energy and is the Fermi velocity, that for massless quarks satisfies . Then, the quasiparticle spectrum can obtained solving the eigenvalue equation
(48) 
Performing the unitary transformation
(49) 
it is possible to eliminate the dependence on in the eigenvalue problem and this corresponds to measure the energy of each flavor from its Fermi energy. The resulting equations for and are independent of color and flavor indices, and therefore these indices will be omitted below. The eigenvalue problem reduces to solve the coupled differential equations:
(50)  
(51) 
These equations can be used to find the dispersion laws for any nonhomogeneous , and we shall consider here the periodic structures of the form given in Eq (42) in order to determine whether there are gapless fermionic excitations. We shall prove that for any crystalline structure with realvalued periodic functions there exists a gapless mode iff the set does not contain the vector .
The proof is given below. Here we notice that this theorem does not apply to the case in which is not real, and indeed we shall show that the single plane wave structure does have a gapless mode. The theorem implies that any antipodal structure has a gapless mode, in particular the “strip” (corresponding to the structure depicted in the right panel of Fig. 4 for ) and the cube have gapless modes. On the other hand, the set of vectors which identify a triedral prism or a hexahedral prisms have a vector with , and the corresponding dispersion laws are gapped.
Proof:
For a periodic , a general solution of the system in (51) is given by the Bloch functions
(52) 
where and are periodic functions. Notice that if is real, the dispersion law is ungapped, otherwise a gap is present in the excitation spectrum and the eigenfunctions decrease exponentially with .
Comparing this expression with Eq.(52), it is clear that theses solutions corresponds to Bloch functions with . If has a term with , it means that , where is a constant and is a periodic nonconstant function. In this case Eq. (53) has an exponential behavior of the type and therefore the spectrum is gapped. On the other hand, if in the expansion of no term with is present then the imaginary part of vanishes and the spectrum is gapless c.v.d.
The fermion dispersion law for the gapless modes can be determined using degenerate perturbation theory for , i.e. close to the Fermi sphere. At the lowest order in one finds that
(54) 
where is the velocity of the excitations, and
(55) 
where is the volume of a unit cell of the lattice. The energy of the fermionic excitations depends linearly on the residual momentum , but the velocity of the excitations is not isotropic.
Let us now specialize (54) to the case of the strip,
(56) 
which corresponds to the condensate in Eq.(43) with and and strictly speaking does not describe a crystal, but a condensate that is modulated in the direction. The coefficients in the dispersion law are given by
(57) 
where is the modified Bessel function of the zeroth order. Therefore the velocity of the fermionic quasiparticles has the analytic expression
(58) 
which has the important property to vanish when is orthogonal to . The reason is that, in the direction orthogonal to the gap is constant and its effect is equivalent to a potential barrier. Taking , the dispersion law is symmetric with respect to rotations around the axis, for inversions with respect to the plane and depends only on the polar angle , between and the axis. In Fig. 5 we report a plot of the velocity of fermionic quasiparticles as a function of , for three different values of the ratio .
In the ultrarelativistic case, , the relevant case is and we see that the dispersion law of fermionic quasiparticles is not much affected by the condensate for and it is the same of relativistic fermions. On the other hand for small values of , the fermionic velocity is exponentially suppressed and vanishes for , meaning that fermionic quasiparticles cannot propagate in the plane as discussed above.
The fact that the dispersion law is linear in for small values of the momentum does not assure that it is linear for any value of the momentum. Considering , it is possible to solve the Eq. (51) for in (52) along the direction, without restricting to low momenta Larkin and Ovchinnikov (1964); the result is that
(59) 
meaning that the dispersion law is linear in the residual momentum, only for .
For the octahedron, whose six wave vectors point into the direction of a BCC structure, the corresponding gap parameter can be written as
(60) 
It is easy to show that in this case the integral in Eq. (55) factorizes, and the dispersion law is gapless with velocity
(61) 
The corresponding plot is reported in Fig.6, left panel, where the unit vector has been expressed by the polar angles and . The plot has been obtained for and considering and . Note that according with Eq. (61), the velocity of the fermionic quasiparticles vanishes along the planes , and .
For more complicated crystalline structures it s not possible to have an analytic expression of the fermionic velocity, one notable example is the FCC structure which is defined by the condensate