Crystalline color superconductors

Inhomogeneous superconductors and inhomogeneous superfluids appear in a variety of contexts including quark matter at extreme densities, fermionic systems of cold atoms, type-II cuprates, and organic superconductors. In the present review the focus is on properties of quark matter at high baryonic density, which may exist in the interior of compact stars. The conditions realized in these stellar objects tend to disfavor standard symmetric BCS pairing and may favor an inhomogeneous color superconducting phase. The properties of inhomogeneous color superconductors are discussed in detail and in particular of crystalline color superconductors. The possible astrophysical signatures associated with the presence of crystalline color superconducting phases within the core of compact stars are also reviewed.

Figure 1

Schematic representation of the hierarchy of effective Lagrangians characteristic of high-density QCD.

Figure 2

Schematic phase diagram of strongly interacting matter as a function of the baryonic chemical potential and temperature. At low temperatures and low densities matter consists of confined hadrons. At high temperatures quark and gluons degrees of freedom are liberated forming the quark-gluon plasma. At low temperatures and very high densities, the CFL phase is favored. At densities and temperatures relevant for compact stellar objects, the CFL phase may be superseded by some different color superconducting phase or by some other phase of matter. The thick segment represents the possible range of baryonic chemical potential reachable in compact stars.

Figure 3

Pictorial description of the behavior of the Fermi spheres of two different populations of fermions, with up ($\mathbf{u}$) and down ($\mathbf{d}$) spins, with increasing $\delta \mu$. Left panel: the dashed black lines correspond to the Fermi spheres of the two populations in the noninteracting case. In the presence of a weak attractive interaction the BCS pairing produces a smearing of the Fermi spheres corresponding to the gray region. Right panel: for $\delta \mu >\delta {\mu }_1={\mathrm{\Delta }}_0/\sqrt{2}$ the Fermi spheres of the two populations (solid black lines) are widely separated and the BCS homogeneous phase is no more energetically favored.

Figure 4

Exclusion plot in the ($\delta \mu /{\mathrm{\Delta }}_0$, $\mu /{\mathrm{\Delta }}_0$) plane according to conditions (1), (2), and (3) (see text). Condition (1) excludes the region above the dot-dot-dashed line; condition (2) excludes the region above the dot-dashed line; and condition (3) excludes the region above the dotted line. The region above the dashed line corresponds to ${\mathrm{\Omega }}_s>{\mathrm{\Omega }}_n$ and is therefore metastable or unstable. On the top of the figure the Chandrasekhar-Clogston limit $\delta \mu /{\mathrm{\Delta }}_0=1/\sqrt{2}\simeq 0.707$ is indicated. Regions featuring zero (0), one (I), or two (II) gapless surfaces in momentum space are separated by solid lines. These lines are determined by finding the zeros of $E_b$, given in Eq. (10). Adapted from [128].

Figure 5

Pictorial description of the LOFF pairing in the weak coupling approximation. When $\delta \mu >{\mathrm{\Delta }}_0/\sqrt{2}$ the BCS homogeneous pairing is not energetically allowed, but pairing between fermions with total nonvanishing momentum can be realized. Left panel: In the FF phase pairing takes place in two ribbons on the top the Fermi spheres of up and down fermions, such that ${\mathit{P}}_u+{\mathit{P}}_d=2\mathit{q}$, having opening angle $2{\vartheta }_q\simeq 2\arccos (\delta \mu /q)\simeq {67}^{\circ }$, thickness $\mathrm{\Delta }$, and angular width $\mathrm{\Delta }/q$; see Sec. 2e1. Right panel: Structure obtained with two plane waves corresponding to two vectors $2{\mathit{q}}^1$ (light gray) and $2{\mathit{q}}^2$ (dark gray) with relative angle $\varphi$. The size and the opening angle of each ribbon is as in the FF phase. The structure with $\varphi ={180}^{\circ }$ is called the “strip.” For illustrative purposes, we exaggerated the splitting between the Fermi surfaces, relative to the values used in the calculations reported in Sec. 2c.

Figure 6

Velocity of the fermionic quasiparticles as a function of $z=\cos \vartheta$ for $2\mathrm{\Delta }/qv=0.1$ (solid line), $2\mathrm{\Delta }/qv=0.2$ (dashed line), and $2\mathrm{\Delta }/qv=0.3$ (dotted line) for the strip structure given in Eq. (61).

Figure 7

Velocity of the fermionic quasiparticles in the BCC crystalline structure (left panel) and in the FCC crystalline structure (right panel) as a function of the polar angles $\vartheta$ and $\phi$, for $2\mathrm{\Delta }/(\mathit{qv})=0.3$.

Figure 8

Free energy as a function of $\delta \mu /{\mathrm{\Delta }}_{2\mathrm{SC}}$, for the octahedron (dashed line) and the fcc (solid line). Free energies are measured respect to the normal phase. The octahedron is the favored structure up to $\delta \mu \simeq 0.95{\mathrm{\Delta }}_{2\mathrm{SC}}$. In the range $0.95{\mathrm{\Delta }}_{2\mathrm{SC}}\lesssim \delta \mu \lesssim 1.32{\mathrm{\Delta }}_{2\mathrm{SC}}$ the fcc is favored. For larger values of $\delta \mu$ the normal phase is energetically favored. In this plot the values of $\delta \mu$, $z_q$, and $\mathrm{\Delta }$ are those that minimize the free energy. Adapted from [71].

Figure 9

Left panel: Profile for the gap function, at several values of $q$, obtained from the self-consistent solution of the gap equation, at $\delta \mu =0.7{\mathrm{\Delta }}_{\mathrm{BCS}}$. Right panel: Physical wave vector magnitude as a function of $\delta \mu$. In both plots ${\mathrm{\Delta }}_{\mathrm{BCS}}\equiv {\mathrm{\Delta }}_{2\mathrm{SC}}$. From [196].

Figure 10

Free energy as a function of $\delta \mu$, for the BCS phase (solid line), FF phase (dotted line), and solitonic phase (dashed line). Free energies are measured with respect to the normal phase. From [196].

Figure 11

Fermi momenta differences $P_d^F-P_u^F$ (black solid line) and $P_u^F-P_s^F$ (dashed line) as a function of $M_s$, for $\mu =500\mathrm{MeV}$.

Figure 12

Left panel: Gap parameters ${\mathrm{\Delta }}_3$, ${\mathrm{\Delta }}_2$, and ${\mathrm{\Delta }}_1$ as a function of $M_s^2/\mu$ for $\mu =500\mathrm{MeV}$, in a model where ${\mathrm{\Delta }}_{\text{CFL}}=25\mathrm{MeV}$ (see text). Right panel: Free energy of the CFL-gCFL phase, relative to that of neutral noninteracting quark matter and that of the 2SC-g2SC and 2SCus phases. At $M_s^2/\mu \approx 47.1\mathrm{MeV}$ (vertical dotted line) there is a continuous phase transition between the CFL and gCFL phases. Above $M_s^2/\mu \approx 130\mathrm{MeV}$ (vertical dash-dotted line) unpaired quark matter has a lower free energy than the gCFL phase. From [18].

Figure 13

Plot of $\mathrm{\Delta }/{\mathrm{\Delta }}_0$ (left panel), with ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_{\mathrm{CFL}}$, and the free energy relative to neutral noninteracting quark matter (right panel) as a function of $M_s^2/\mu$ for four values of the angle $\varphi$ between ${\mathit{q}}_2$ and ${\mathit{q}}_3$. The various lines correspond to the calculations done in the GL approximation whereas dots correspond to the NJL calculation, done without making a GL approximation. The solid lines and circles correspond to $\varphi =0$, the dash-dotted lines, and diamonds correspond to $\varphi =2\pi /3$, the dashed lines and squares correspond to $\varphi =7\pi /8$, the dotted lines and triangles correspond to $\varphi =31\pi /32$. From [187].

Figure 14

$I(\varphi )$, defined in Eq. (159), vs the angle $\varphi$ between the wave vectors ${\mathit{q}}_2$ and ${\mathit{q}}_3$. This function is proportional to the free energy cost for having overlap between ribbons associated with different wave vectors. For ${\mathit{q}}_2$ parallel to ${\mathit{q}}_3$, corresponding to $\varphi =0$, there is no overlap between ribbons and no free energy cost has to be paid. The free energy cost diverges for perfectly overlapping ribbons, corresponding to $\varphi =\pi$.

Figure 15

Sketch showing where on the Fermi surfaces pairing occurs for condensates in which ${\mathit{q}}_2$ and ${\mathit{q}}_3$ are at an angle $\varphi$. The light gray ribbons on the $d$ and $u$ Fermi surfaces indicate those quarks that contribute the most to the $⟨ud⟩$ condensate with gap parameter ${\mathrm{\Delta }}_3$ and wave vector ${\mathit{q}}_3$, which points upward in both panels. The dark gray ribbons on the $u$ and $s$ Fermi surfaces indicate those quarks that contribute most to the $⟨us⟩$ condensate with gap parameter ${\mathrm{\Delta }}_2$ and wave vector ${\mathit{q}}_2$. For descriptive reasons, in the left panel the case $\varphi ={70}^{\circ }$ is shown, and in the right panel the antipodal case, with the two vectors pointing in opposite directions. We exaggerated the splitting between the Fermi surfaces, relative to the values used in the calculations reported in Sec. 3b.

Figure 16

(a) Tadpolelike diagram and (b) charmlike diagram contributing to the gluon masses in the HDET. The curly lines correspond to the gluon fields; the solid lines correspond to the positive energy quark fields; see Eq. (136). The interaction vertices can be obtained from the expansion of the effective Lagrangian in Eq. (160), including terms of order $1/\mu$.

Figure 17

Squared Meissner masses of the gluons ${\tilde{A}}_3$ (solid line), $A_6$ (dotted line), ${\tilde{A}}_8$ (dashed black line), and $A_1$ (dot-dashed line), in units of the CFL squared Meissner mass, vs $M_s^2/\mu$. Left panel: longitudinal masses. Right panel: transverse masses. Adapted from [88].

Figure 18

Left panel: Quark chemical potential difference as a function of $M_s^2/\mu$. The dashed line corresponds to $|\delta {\mu }_2|$ and the dash-dotted line corresponds to $|\delta {\mu }_3|$; their common value $M_s^2/(8\mu )$, obtained neglecting $\mathcal{O}(1/\mu )$ corrections, corresponds to the solid black line. Right panel: Gap parameters as a function of $M_s^2/\mu$. The solid black line represents the solution $\mathrm{\Delta }={\mathrm{\Delta }}_2={\mathrm{\Delta }}_3$ obtained neglecting $\mathcal{O}(1/\mu )$ corrections; the dashed and dash-dotted lines represent, respectively, ${\mathrm{\Delta }}_2$ and ${\mathrm{\Delta }}_3$ and are obtained including $\mathcal{O}(1/\mu )$ corrections. All gap parameters are normalized to the CFL gap ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_{\text{CFL}}=25\mathrm{MeV}$. Adapted from [70].

Figure 19

Exclusion plots in the $\mathrm{\Delta }$-$\mu$ plane for the Crab (left panel) and the pulsar J2124-3358 (right panel) obtained assuming that these stars have mass $M=1.4M_{\odot }$, radius $R=10\mathrm{km}$, uniform mass distribution, and are maximally strained. The area above each line is excluded by the corresponding model. The rectangular box is the theoretically allowed region of $\mu$ and $\mathrm{\Delta }$ for the CCSC phase; see [188]. Left panel: The dashed lines correspond to the results of [181] which considers the S3/S4 LIGO data of the Crab pulsar, for the case ${\sigma }_{\max }={10}^{-3}$ (upper heavy dashed line) and ${\sigma }_{\max }={10}^{-2}$ (lower dashed line) and assuming that the whole star consists of maximally strained crystalline color superconducting matter. The solid lines have been determined using Eq. (7) of [181], but with the S5 LIGO data ([3]) and considering the case of ${\sigma }_{\max }={10}^{-3}$ (upper heavy solid line) and ${\sigma }_{\max }={10}^{-2}$ (lower solid line). Right panel: Dependence of the exclusion region on the size of the crystalline core for the pulsar J2124-3358 obtained using the S5 LIGO data ([3]) and considering the case of ${\sigma }_{\max }={10}^{-3}$. The three lines correspond to different values of $x=R_c/R$, where $R_c$ is the radius of the crystalline color superconducting core.

Figure 20

Surface temperature, in Kelvin, as a function of time, in years, for various models of compact stars ([35]). Model I is a neutron star of nuclear matter with uniform density $n=0.24\mathrm{fm}{}^{-3}$ and radius $R=12\mathrm{k}\mathrm{m}$; model II corresponds to a star with $R=10\mathrm{km}$, having an outer part of nuclear matter and a core of radius $R_c=5\mathrm{k}\mathrm{m}$ of unpaired quark matter; model III is like model II, but in the core there is quark matter in a crystalline phase, see the text for more details. For all stars $M=1.4M_{\odot }$ and the temperature at $t=1\mathrm{yr}$ was set to $T={10}^9\mathrm{K}$. Parameters for the core are $\mu =500\mathrm{MeV}$ and $M_s^2/\mu =140\mathrm{MeV}$. Adapted from [35].

Figure 21

Surface temperature, in Kelvin, as a function of time, in years, for various models of compact stars comprising a two-flavor LOFF phase between $u$ and $d$ red and green quarks, with gap parameter $\mathrm{\Delta }$, and pairing between $u$ and $d$ blue quarks, with gap parameter ${\mathrm{\Delta }}_b$. Various values of the central density ${\rho }_c$ of $\zeta =\mathrm{\Delta }/\delta \mu$ and ${\mathrm{\Delta }}_b$ are considered. The solid curve refers to a model with no quark matter, central density $5.1{\rho }_{14}$, where ${\rho }_{14}={10}^{14}\mathrm{g}{\mathrm{cm}}^{-3}$, corresponding to a star with $M\simeq 0.54M_{\odot }$. The long dashed curve corresponds to a model with ${\rho }_c=10.8{\rho }_{14}$, $M\simeq 1.91M_{\odot }$, and a quark core with radius $R_c\simeq 0.68\mathrm{\hspace{0.17em}km}$. The short dashed curve corresponds to a model with ${\rho }_c=11.8{\rho }_{14}$, $M\simeq 1.93M_{\odot }$, and $R_c\simeq 3.41\mathrm{\hspace{0.17em}km}$. The dotted curve corresponds to a model with ${\rho }_c=21.0{\rho }_{14}$, $M\simeq 2.05M_{\odot }$, and $R_c\simeq 6.77\mathrm{\hspace{0.17em}km}$. The symbols correspond to the observational data points [see [138] for an explanation]; here we only remark that the $+$ sign on the top left corner corresponds to Cassiopeia A ([137]; [257]). From [138].

Figure 22

Left panel: Pressure vs matter density for the two considered models. The solid black line labeled nuclear refers to the EOS based on the Dirac-Brueckner-Hartree-Fock approach ([269]; [270]). The dashed line corresponds to model A. The EOS of model A has the low-density EOS corresponding to nuclear and the high-density EOS obtained by a NJL model describing CCSC matter. At the deconfinement phase transition there is a jump in the density at constant pressure. Right panel: Mass-radius diagram for nonrotating configurations. The dashed lines correspond to the lower pulsar mass bound from J0737-3039 and the $2M_{\odot }$ upper bound from various astronomical observations.