Crystalline color superconductivity

Abstract

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...

Inhomogeneous chiral condensates

“…This ansatz is formally analogous to the Fulde-Ferrell solutions introduced in the context of superconductivity [84] and color superconductivity [25]. According to Eqs.…”

“…Charge and spin density waves are commonly found in solid-state physics (for a review see, e.g., [22]), and crystalline phases have also been discussed long time ago for superconductors by Fulde and Ferrell as well as by Larkin and Ovchinikov [23,24] and more recently for color superconductors [25] (see also [26,27] for reviews). Inhomogeneous phases have also been shown to appear in imbalanced cold atom Fermi gases [28,29].…”

Phonons and gluons in the crystalline color superconducting phase of QCD

“…Closeness to their Fermi surfaces for both quarks in the pair will correspond to a lower free energy. Such a possibility for the quark pairing leads to the suggestion of a crystalline color superconductivity [5,6,7,8,9]. We assume the existence of this phase and we will limit our analysis, as in the existing literature, to two flavors, with δµ = 0 (δµ the difference in the quark chemical potentials).…”

“…Each quark momentum will lie near its Fermi surface (Fermi spheres will now be of different radii) [5,6,7,8,9]. Closeness to their Fermi surfaces for both quarks in the pair will correspond to a lower free energy.…”

Competing order parameters and a tricritical point with a difference

“…In past ten years, these exotic superfluid phases have been extensively studied, both theoretically [1,2,3,4,5] and experimentally [6,7,8,9,10,11] in ultra-cold atomic gases and in quark-gluon plasma which resides in the core of neutron stars [12,13]. Similar imbalanced fermionic systems were studied in electron superconductors in a magnetic field as early as nineteen-sixties [14,15], and showed the possibility of tricritical points where the second-order and the first-order lines meet along with the line separating stable and unstable superconducting phases.…”