Crystalline Color Superconductivity

TL;DR

The paper demonstrates that crystalline color superconductivity (the LOFF phase) can arise in dense quark matter when quark species have mismatched Fermi momenta, using an NJL-type four-fermion interaction to derive a self-consistent LOFF gap equation that yields spatially modulated condensates. It shows that a LOFF window exists in weak coupling with ${oldsymbol{ extdelta} extmu}_1 o oldsymbol{ riangle}_0/\sqrt{2}$ and ${oldsymbol{ extdelta} extmu}_2 o 0.754oldsymbol{ riangle}_0$, with a nonzero induced $J=1$ condensate $oldsymbol{ riangle}_B$ and a preferred momentum $|oldsymbol{q}| oughly 1.2{oldsymbol{ extdelta} extmu}$; stronger coupling narrows the window and altering gluon couplings ($G_E$, $G_M$) can shift the window. These crystalline phases could reside in neutron star cores and may pin rotational vortices, offering a potential mechanism for pulsar glitches and affecting stellar dynamics, while motivating extensions to three-flavor quark matter and crystal-structure determinations to quantify astrophysical impact.

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 (LOFF). 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 star) 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.

Figures (7)

• Figure 1: The momenta ${\mathbf k}_u$ and ${\mathbf k}_d$ of the two members of a LOFF-state Cooper pair. We choose the vector ${\mathbf q}$, common to all Cooper pairs, to coincide with the $z$-axis. The angles $\alpha_u({\mathbf p})$ and $\alpha_d({\mathbf p})$ indicate the polar angles of ${\mathbf k}_u$ and ${\mathbf k}_d$, respectively.
• Figure 2: The LOFF phase space, as a function of ${\mathbf p}$ (Eq. (\ref{['LOFF:mom']})). We show the $p_y=0$ plane. (a) The phase space in the limit of arbitrarily weak interactions. In the shaded blocking regions ${\cal B}_u$ and ${\cal B}_d$, no pairing is possible. In the inner unshaded region, an interaction can induce hole-hole pairs. In the outer unshaded region, an interaction can induce particle-particle pairs. The region ${\cal P}$ (Eq. (\ref{['LOFF:ansatz']})) is the whole unshaded area. (b) When the effects of interactions and the formation of the LOFF state are taken into account, the blocking regions shrink. The BCS singularity occurs on the dashed ellipse, defined by $\epsilon_u+\epsilon_d=\mu_u+\mu_d$, where making a Cooper pair costs no free energy in the free case.
• Figure 3: (a) The zero-gap curves for the LOFF state. To the right of a solid curve, there is no solution to the LOFF gap equation, to the left of the curve there is a solution, and on the curve the gap parameter is zero. The three curves are (from strongest to weakest coupling): $\Delta_0=0.1, 0.04, 0.01~{\rm GeV}$. The region $q<{\delta\mu}$ is complicated to describe FF, and solutions found in this region never give the lowest free energy state at a given ${\delta\mu}$. (b) Here, we choose $\Delta_0=0.04~{\rm GeV}$ and focus on the region near ${\delta\mu_2}$, the maximum value of ${\delta\mu}$ at which the LOFF state exists. The dashed curve shows the value of $|{\mathbf q}|$ which minimizes the free energy of the LOFF state at a given ${\delta\mu}$. ${\delta\mu_1}$, discussed below, is also indicated.
• Figure 4: LOFF and BCS gaps and free energies as a function of ${\delta\mu}$, with coupling chosen so that $\Delta_0=40{\rm ~MeV}$ and with ${\bar{\mu}}=0.4~{\rm GeV}, \Lambda=1~{\rm GeV}$. Free energies are measured relative to the normal state. At each ${\delta\mu}$ we have varied $q$ to find the best LOFF state. The vertical dashed line marks ${\delta\mu}={\delta\mu}_1$, the value of ${\delta\mu}$ above which the LOFF state has lower free energy than BCS. The expanded inset (wherein $s=10^{-7}~{\rm GeV}^4$) focuses on the region ${\delta\mu}_1<{\delta\mu}<{\delta\mu}_2$ where the LOFF state has the lowest free energy.
• Figure 5: The two LOFF condensates $\Gamma_A$$(J=0)$ and $\Gamma_B$$(J=1)$ for the same choice of parameters as in Figure \ref{['fig:F_plot']}. We focus on the region ${\delta\mu_1}<{\delta\mu}<{\delta\mu_2}$. For reference, in the BCS phase $\Gamma_A=\Delta_0/G=0.00583~{\rm GeV}^3$ and $\Gamma_B=0$.