content_image/387519.jpg
pairs carry total spin one. Spin-1 color superconductiv- ity was firstly studied in Ref. [18], for more recent and detailed discussions about the spin-1 gap, its critical tem- perature and Meissner effect, see Refs. [69, 70, 94–97]. For a review, see Ref. [98]. 

## Pairing with mismatch: LOFF, CFL-K, g2SC and gCFL 

To form the Cooper pair, the ideal case is when the two pairing quarks have the same Fermi momenta, i.e., p$_{F,i}$ = p$_{F,j}$ with p$_{F,i}$ = √ µ 2 F,i − m 2 $_{i}$, like in the ideal 2SC, CFL, and spin-1 color superconducting phases. However, in reality, the nonzero strange quark mass or the require- ment of charge neutrality induces a mismatch between the Fermi momenta of the two pairing quarks. When the mismatch is very small, it has little effect on the Cooper pairing. While if the mismatch is very large, the Cooper pair will be destroyed. The most interesting situation happens when the mismatch is neither very small nor very large. 

LOFF: In the regime just on the edge of decoupling of the two pairing quarks (due to the nonzero strange quark mass for the qs Cooper pair with q ∈ ( u, d ) or the chemical potential difference for the ud Cooper pair), a “LOFF” (Larkin-Ovchinnikov-Fulde-Ferrell) state may be formed. The LOFF state was firstly investigated in the context of electron superconductivity in the presence of magnetic impurities [99, 100]. It was found that near the unpairing transition, it is favorable to form a state in which the Cooper pairs have nonzero momentum. This is favored because it gives rise to a regime 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. This sort of condensates spontaneously break translational and rotational invariance, leading to gaps which vary periodically in a crystalline pattern. The crystalline color superconductivity has been investigated in a series of papers, e.g., see Refs. [101–108]. 

CFL-K: The strange quark mass m$_{s}$ induces an effec- tive chemical potential µ$_{s}$ = m 2 $_{s}$/ (2 p$_{F}$ ), and the effects of the strange quark mass can be quite dramatic. In the CFL phase, the K + and K 0 modes may be unstable for large values of the strange quark mass to form a kaon con- densation [109–112]. In the framework of effective theory [91, 92, 113–115], the masses of the Goldstone bosons can be determined as 

<img src='content_image/387519.jpg'>

with A = 3Δ $^{2}$/ (4 π $^{2}$) [92, 116]. It was found that the kaon 

masses are substantially affected by the strange quark mass, the masses of K − and ¯ K 0 are pushed up while K + and K 0 are lowered. As a result, the K + and K 0 become massless if m$_{s}$ | crit = 3 . 03 m 1 / 3 d Δ 2 / $^{3}$. For larger values of m$_{s}$ the kaon modes are unstable, signaling the formation of a kaon condensate. Recently, it was found that in the CFL phase, there also may exist η condensate [117]. 

g2SC and gCFL: When the β -equilibrium and the charge neutrality condition are required for the two-flavor quark system, the Fermi surfaces of the pairing u quark and d quark differ by µ$_{e}$ , here µ$_{e}$ is the chemical potential for electrons. It was found that when the gap parameter Δ < µ$_{e}$/ 2, the system will be in a new ground state called the gapless 2SC (g2SC) phase [118]. The g2SC phase has very unusual temperature properties [119] and chromo- magnetic properties [120]. This phase will be introduced in more detail in Sec. III B. 

Similarly, for a charge neutral 3-flavor system with a nonzero strange quark mass m$_{s}$ , with increasing m$_{s}$ , the CFL phase transfers to a new gapless CFL (gCFL) phase when m 2 $_{s}$/µ ≃ 2Δ [121]. The finite temperature property of the charge neutral three-flavor quark matter was in- vestigated in Ref. [122–124]. Recently, it was shown that the kaon condensate shifts the critical strange quark mass to higher values for the appearance of the gCFL phase [125]. 

In this section, I would like to focus on unconventional color superconductor with mismatched pairing by taking charge neutral two-flavor system as an example. 

It is very likely that the color superconducting phase may exist in the core of compact stars, where bulk mat- ter should satisfy the charge neutrality condition. This is because bulk matter inside the neutron star is bound by the gravitation force, which is much weaker than the electromagnetic and the strong color forces. Any elec- tric charges or color charges will forbid the formation of bulk matter. In addition, matter inside neutron star also needs to satisfy the β -equilibrium. 

In the ideal two-flavor color superconducting (2SC) phase, the pairing u and d quarks have the same Fermi momenta. Because u quark carries electric charge 2 / 3, and d quark carries electric charge − 1 / 3, it is easy to check that quark matter in the ideal 2SC phase is pos- itively charged. To satisfy the electric charge neutrality condition, roughly speaking, twice as many d quarks as u quarks are needed. This induces a large difference be- tween the Fermi surfaces of the two pairing quarks, i.e., µ$_{d}$ − µ$_{u}$ = µ$_{e}$ ≈ µ/ 4, where µ, µ$_{e}$ are chemical poten- tials for quarks and electrons, respectively. Naively, one would expect that the requirement of the charge neutral- ity condition will destroy the ud Cooper pairing in the 2SC phase. 

Indeed, the interest in the charge neutral 2SC phase was stirred by the paper Ref. [126]. It was claimed in this 

## B. Gapless color superconductor 

