In any context in which color superconductivity arises in nature, it is likely to involve pairing between species of quarks with differing chemical potentials. For suitable values of the differences between chemical potentials, Cooper pairs with nonzero total momentum are favored, as was first realized by Larkin, Ovchinnikov, Fulde, and Ferrell ͑LOFF͒. Condensates of this sort spontaneously break translational and rotational invariance, leading to gaps which vary periodically in a crystalline pattern. Unlike the original LOFF state, these crystalline quark matter condensates include both spin-zero and spin-one Cooper pairs. We explore the range of parameters for which crystalline color superconductivity arises in the QCD phase diagram. If in some shell within the quark matter core of a neutron star ͑or within a strange quark star͒ the quark number densities are such that crystalline color superconductivity arises, rotational vortices may be pinned in this shell, making it a locus for glitch phenomena.The attraction between two quarks which are antisymmetric in color renders cold dense quark matter unstable to the formation of quark Cooper pairs in a color superconducting state ͓1-7͔. If two ͑or more͒ different quark flavors are involved, and their Fermi momenta are the same, they pair as in the standard BCS state. The pairing is guaranteed because in the absence of an interaction each pair costs no free energy-each quark can be created at its Fermi surface-and the interaction then makes the system unstable against formation of a condensate of pairs.In this paper we study the situation, generic in the real world, where the Fermi momenta of the two species are different. If the Fermi momenta are far apart, no pairing between the species is possible. The transition between the BCS and unpaired states as the splitting between Fermi momenta increases has been studied in electron ͓8͔ and QCD ͓9-11͔ superconductors, assuming that no other state intervenes. However, there is good reason to think that another state can occur. This is the ''LOFF'' state, first explored by Larkin and Ovchinnikov ͓12͔ and Fulde and Ferrell ͓13͔ in the context of electron superconductivity in the presence of magnetic impurities. They found that near the unpairing transition it is favorable to form a crystalline state in which the Cooper pairs have nonzero momentum. This is favored because it gives rise to a region of phase space where each of the two quarks in a pair can be close to its Fermi surface, and such pairs can be created at low cost in free energy.We study the pairing between two species whose chemical potentials differ by 2␦ and find that for a large class of interactions there is a window of ␦ within which states of the LOFF type are preferred over the BCS and unpaired states. This has important ramifications for compact star phenomenology, since it means that there may be a layer of crystalline quark matter inside the star. This could pin rotational vortices, and lead to the kind of glitch phenomena that have up to now been thoug...

We describe the crystalline phase of color superconducting quark matter. This phase may occur in quark matter at densities relevant for compact star physics, with possible implications for glitch phenomena in pulsars [ 1]. We use a Ginzburg-Landau approach to determine that the crystal has a face-centered-cubic (FCC) structure [ 2]. Moreover, our results indicate that the phase is robust, with gaps, critical temperature, and free energy comparable to those of the color-flavor-locked (CFL) phase [ 2]. Our calculations also predict "crystalline superfluidity" in ultracold gases of fermionic atoms [ 3].Cold dense quark matter is a color superconductor [ 4]. At asymptotically high densities, the ground state of QCD with quarks of three flavors (u, d, and s) is the color-flavorlocked (CFL) phase [ 5]. This phase features a BCS condensate of Cooper pairs of quarks that includes ud, us, and ds pairs. At intermediate densities, however, the CFL phase can be disrupted by any flavor asymmetry (such as a chemical potential difference or a mass difference) that would, in the absence of pairing, separate the Fermi surfaces. In the absence of pairing, electrically neutral bulk quark matter with m u,d = 0 and m s = 0 features a nonzero electron density (µ e ≈ m 2 s /4µ) and three disparate quark Fermi momenta:(Note that decreasing µ enhances the flavor disparity.) Accounting for pairing effects modifies this picture: starting in the CFL phase at large µ, as we decrease µ the CFL phase remains "rigid" [ 6], with coincident quark Fermi surfaces and no electrons, until either hadronization or a firstorder unlocking transition, whichever comes first. Unlocking occurs at µ ≈ m 2 s /4∆ 0 , where ∆ 0 is the CFL gap. Its value and that of m s are density dependent and sufficiently uncertain that we do not know whether unlocking occurs before hadronization. Here, we pursue the consequences of assuming that unlocking occurs first.In quark matter below the unlocking transition, pairing can still occur. One option is single-flavor pairing (uu, dd, ss), but these J = 1 condensates have very small gaps [ 8]. Crystalline color superconductivity is more robust [ 1,2]. We propose that unlocked quark matter is in the crystalline phase, which therefore occupies the window in the QCD phase diagram between the CFL and hadronic phases (Fig. 1). The crystalline phase has the unique virtue of allowing pairing between quarks with unequal Fermi surfaces. It was originally described by Larkin, Ovchinnikov, Fulde, and Ferrell (LOFF) [ 9] as a novel pairing mechanism for an electron superconductor with a Zeeman splitting between † Speaker. * We thank the Kavli Institute for Theoretical Physics in Santa Barbara for support and hospitality.

After a brief review of the phenomena expected in cold dense quark matter, colour superconductivity and colour-flavour locking, we sketch some implications of recent developments in our understanding of cold dense quark matter for the physics of compact stars. We give a more detailed summary of our recent work on crystalline colour superconductivity and the consequent realization that (some) pulsar glitches may originate in quark matter.

We survey the non-locked color-flavor-spin channels for quark-quark (color superconducting) condensates in QCD, using an NJL model. We also study isotropic quark-antiquark (mesonic) condensates. We make mean-field estimates of the strength and sign of the self-interaction of each condensate, using four-fermion interaction vertices based on known QCD interactions. For the attractive quark pairing channels, we solve the mean-field gap equations to obtain the size of the gap as a function of quark density. We also calculate the dispersion relations for the quasiquarks, in order to see how fully gapped the spectrum of fermionic excitations will be. We use our results to specify the likely pairing patterns in neutral quark matter, and comment on possible phenomenological consequences.

Abstract. After a brief review of the phenomena expected in cold dense quark matter, color superconductivity and color-flavor locking, we sketch some implications of recent developments in our understanding of cold dense quark matter for the physics of compact stars. We give a more detailed summary of our recent work on crystalline color superconductivity and the consequent realization that (some) pulsar glitches may originate in quark matter. Color Superconductivity and Color-Flavor LockingBecause QCD is asymptotically free, its high temperature and high baryon density phases are more simply and more appropriately described in terms of quarks and gluons as degrees of freedom, rather than hadrons. The chiral symmetry breaking condensate which characterizes the vacuum melts away. At high temperatures, in the resulting quark-gluon plasma phase all of the symmetries of the QCD Lagrangian are unbroken and the excitations have the quantum numbers of quarks and gluons. At high densities, on the other hand, quarks form Cooper pairs and new condensates develop. The formation of such superconducting phases [1,2,3,4,5,6] requires only weak attractive interactions; these phases may nevertheless break chiral symmetry [5] and have excitations with the same quantum numbers as those in a confined phase [5,7,8,9]. These cold dense quark matter phases may arise in the cores of neutron stars; understanding this region KR thanks the organizers all three meetings named above for providing stimulating environments, thanks the organizers of SEWM2K for an excuse to visit Provence for the first time, and thanks the organizers of the Trento workshop for providing the venue within which many of the helpful discussions acknowledged above took place.