Color Superconductivity in Dense, but not Asymptotically Dense, Quark Matter

At ultra-high density, matter is expected to form a degenerate Fermi gas of quarks in which there is a condensate of Cooper pairs of quarks near the Fermi surface: color superconductivity. In this chapter we review some of the underlying physics, and discuss outstanding questions about the phase structure of ultra-dense quark matter. We then focus on describing recent results on the crystalline color superconducting phase that may be the preferred form of cold, dense but not asymptotically dense, three-flavor quark matter. The gap parameter and free energy for this phase have recently been evaluated within a Ginzburg-Landau approximation for many candidate crystal structures. We describe the two that are most favorable. The robustness of these phases results in their being favored over wide ranges of density. However, it also implies that the Ginzburg-Landau approximation is not quantitatively reliable. We describe qualitative insights into what makes a crystal structure favorable which can be used to winnow the possibilities. We close with a look ahead at the calculations that remain to be done in order to make quantitative contact with observations of compact stars.Comment: 37 pages, 7 figures. To appear as a Chapter in "Pairing in Fermionic Systems: Basic Concepts and Modern Applications", published by World Scientifi

Shift of the critical temperature in superconductors: a self-consistent approach

Within the Ginzburg-Landau functional framework for the superconducting transition, we analyze the fluctuation-driven shift of the critical temperature. In addition to the order parameter fluctuations, we also take into account the fluctuations of the vector potential above its vacuum. We detail the approximation scheme to include the fluctuating fields contribution, based on the Hartree-Fock-Bogoliubov-Popov framework. We give explicit results for $d=2$ and $d=3$ spatial dimensions, in terms of easily accessible experimental parameters such as the Ginzburg-Levanyuk number $\text{Gi}_{(d)}$, which is related to the width of the critical region where fluctuations cannot be neglected, and the Ginzburg-Landau parameter $\kappa$, defined as the ratio between the magnetic penetration length and the coherence one.Comment: 12 pages, 2 figures. Layout issue with Fig. 1 fixed. Editorially accepted for publication in Scientific Report

Testing the Ginzburg-Landau approximation for three-flavor crystalline color superconductivity

It is an open challenge to analyze the crystalline color superconducting phases that may arise in cold dense, but not asymptotically dense, three-flavor quark matter. At present the only approximation within which it seems possible to compare the free energies of the myriad possible crystal structures is the Ginzburg-Landau approximation. Here, we test this approximation on a particularly simple "crystal" structure in which there are only two condensates $\sim \Delta \exp(i {\bf q_2}\cdot {\bf r})$ and $\sim \Delta \exp(i {\bf q_3}\cdot {\bf r})$ whose position-space dependence is that of two plane waves with wave vectors ${\bf q_2}$ and ${\bf q_3}$ at arbitrary angles. For this case, we are able to solve the mean-field gap equation without making a Ginzburg-Landau approximation. We find that the Ginzburg-Landau approximation works in the $\Delta\to 0$ limit as expected, find that it correctly predicts that $\Delta$ decreases with increasing angle between ${\bf q_2}$ and ${\bf q_3}$ meaning that the phase with ${\bf q_2}\parallel {\bf q_3}$ has the lowest free energy, and find that the Ginzburg-Landau approximation is conservative in the sense that it underestimates $\Delta$ at all values of the angle between ${\bf q_2}$ and ${\bf q_3}$.Comment: 16 pages, 6 figures. Small changes only. Version to appear in Phys. Rev.

The Crystallography of Color Superconductivity

We develop the Ginzburg-Landau approach to comparing different possible crystal structures for the crystalline color superconducting phase of QCD, the QCD incarnation of the Larkin-Ovchinnikov-Fulde-Ferrell phase. In this phase, quarks of different flavor with differing Fermi momenta form Cooper pairs with nonzero total momentum, yielding a condensate that varies in space like a sum of plane waves. We work at zero temperature, as is relevant for compact star physics. The Ginzburg-Landau approach predicts a strong first-order phase transition (as a function of the chemical potential difference between quarks) and for this reason is not under quantitative control. Nevertheless, by organizing the comparison between different possible arrangements of plane waves (i.e. different crystal structures) it provides considerable qualitative insight into what makes a crystal structure favorable. Together, the qualitative insights and the quantitative, but not controlled, calculations make a compelling case that the favored pairing pattern yields a condensate which is a sum of eight plane waves forming a face-centered cubic structure. They also predict that the phase is quite robust, with gaps comparable in magnitude to the BCS gap that would form if the Fermi momenta were degenerate. These predictions may be tested in ultracold gases made of fermionic atoms. In a QCD context, our results lay the foundation for a calculation of vortex pinning in a crystalline color superconductor, and thus for the analysis of pulsar glitches that may originate within the core of a compact star.Comment: 41 pages, 13 figures, 1 tabl

The bifurcation diagrams for the Ginzburg-Landau system for superconductivity

In this paper, we provide the different types of bifurcation diagrams for a superconducting cylinder placed in a magnetic field along the direction of the axis of the cylinder. The computation is based on the numerical solutions of the Ginzburg-Landau model by the finite element method. The response of the material depends on the values of the exterior field, the Ginzburg-Landau parameter and the size of the domain. The solution branches in the different regions of the bifurcation diagrams are analyzed and open mathematical problems are mentioned.Comment: 16 page

Supeconductivity in the Pseudogap State in "Hot - Spots" Model: Ginzburg - Landau Expansion

We analyze properties of superconducting state (for both s-wave and d-wave pairing), appearing on the "background" of the pseudogap state, induced by fluctuations of "dielectric" (AFM(SDW) or CDW) short -- range order in the model of the Fermi surface with "hot spots". We present microscopic derivation of Ginzburg - Landau expansion, taking into account all Feynman diagrams of perturbation theory over electron interaction with this short - range order fluctuations, leading to strong electronic scattering in the vicinity of "hot spots". We determine the dependence of superconducting critical temperature on the effective width of the pseudogap and on correlation length of short - range order fluctuations. We also find similar dependences of the main characteristics of such superconductor close to transition temperature. It is shown particularly, that specific heat discontinuity at the transition temperature is significantly decreased in the pseudogap region of the phase diagram.Comment: 35 pages, 12 figures, RevTeX 3.0, minor additions to text and improved figure

Thermal fluctuations of gauge fields and first order phase transitions in color superconductivity

We study the effects of thermal fluctuations of gluons and the diquark pairing field on the superconducting-to-normal state phase transition in a three-flavor color superconductor, using the Ginzburg-Landau free energy. At high baryon densities, where the system is a type I superconductor, gluonic fluctuations, which dominate over diquark fluctuations, induce a cubic term in the Ginzburg-Landau free energy, as well as large corrections to quadratic and quartic terms of the order parameter. The cubic term leads to a relatively strong first order transition, in contrast with the very weak first order transitions in metallic type I superconductors. The strength of the first order transition decreases with increasing baryon density. In addition gluonic fluctuations lower the critical temperature of the first order transition. We derive explicit formulas for the critical temperature and the discontinuity of the order parameter at the critical point. The validity of the first order transition obtained in the one-loop approximation is also examined by estimating the size of the critical region.Comment: 12 pages, 4 figures, final version published in Phys. Rev.