ar X iv : h ep - p h / 02 09 16 8 v 1 1 6 Se p 20 02 1 The crystallography of color superconductivity ∗

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 mu,d = 0 and ms 6= 0 features a nonzero electron density (μe ≈ m2s/4μ) and three disparate quark Fermi momenta: pdF ≈ puF +m2s/4μ, psF ≈ puF −ms/4μ. (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 ≈ m2s/4∆0, where ∆0 is the CFL gap. Its value and that of ms 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