Roberto Anglani, Roberto Casalbuoni, Marco Ciminale, Raoul Gatto, Nicola Ippolito, Massimo Mannarelli, Marco Ruggieri
备注：
Added 3 figures, added section II F, added section with conclusions. Several references added. Improved the quality of the presentation and removed various typos. Almost matches the version accepted for publication of Reviews of Modern Physics
Inhomogeneous superconductors and inhomogeneous superfluids appear in a variety of contexts including quark matter at extreme densities, fermionic systems of cold atoms, type-II cuprates, and organic superconductors. In the present review the focus is on properties of quark matter at high baryonic density, which may exist in the interior of compact stars. The conditions realized in these stellar objects tend to disfavor standard symmetric BCS pairing and may favor an inhomogeneous color superconducting phase. The properties of inhomogeneous color superconductors are discussed in detail and in particular of crystalline color superconductors. The possible astrophysical signatures associated with the presence of crystalline color superconducting phases within the core of compact stars are also reviewed.
This paper is a brief journey into the amazing realm of crystalline color superconductors. Starting from a qualitative description of superfluids, superconductors and supersolids, we show how inhomogeneous phases may arise when the system is under stress. These basic concepts are then extended to quark matter, in which a richer variety of phases can be realized. Then, the most interesting properties of the crystalline color superconductors are presented. This brief journey ends with a discussion of crystalline color superconductors in compact stars and related astrophysical observables. We aim…
The Ginzburg-Landau (GL) free energy of crystalline color superconductors is important for understanding the nature of the phase transition to the normal quark matter and predicting the preferred crystal structure. So far the GL free energy at zero temperature has only been evaluated up to the sixth order in the condensate. To give quantitative reliable predictions we need to evaluate the higher-order terms. In this work, we present a new derivation of the GL free energy by using the discrete Bloch representation of the fermion field. This derivation introduces a simple matrix formalism withou…
Inhomogeneous phases may appear when a stress is applied to a system and the system can minimize the free energy breaking the rotational invariance. Various examples are known in Nature of this sort, as the paramagnetic to ferromagnetic phase transition, or the fluid/solid phase transition. If the rotational symmetry is broken down to a discrete symmetry, the system is typically named a crystal. We breifly review crystalline color superconductors, which arise in cold quark matter with mismatched Fermi spheres.
It is generally believed that the inhomogeneous Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase appears in a color superconductor when the pairing between different quark flavors is under the circumstances of mismatched Fermi surfaces. However, the real crystal structure of the LOFF phase is still unclear because an exact treatment of 3D crystal structures is rather difficult. In this work we present a solid-state-like calculation of the ground-state energy of the body-centered cubic (BCC) structure for two-flavor pairing by diagonalizing the Hamiltonian matrix in the Bloch space without assumin…
Deconfined quark matter may exist in a crystalline color-superconducting phase in the interiors of compact stars. In this paper, we study the torsional oscillations of compact stars featuring a crystalline color-superconducting quark-matter core in general relativity. Depending on the size of the crystalline core and the value of the gap parameter $Δ$, we find that the frequencies of the torsional oscillation modes can range from a few hundred hertz to a few kilohertz for our canonical $1.4 M_\odot$ compact star models. We have also studied the prospect for detecting the gravitational-wave sig…
We investigate the effects of pairing fluctuations in fermionic superfluids/superconductors where pairing occurs among three species (colors) of fermions. Such color superfluids/superconductors can be realized in three-component atomic Fermi gases and in dense quark matter. The superfluidity/superconductivity is characterized by a three-component order parameter which denotes the pairing among the three colors of fermions. Due to the SU$(3)$ symmetry of the Hamiltonian, one color does not participate pairing. This branch of fermionic excitation is gapless in the naive BCS mean-field descriptio…
We present the first calculation of the shear viscosity for two-flavor plane wave (FF) color superconducting quark matter. This is a member of the family of crystalline color superconducting phases of dense quark matter that may be present in the cores of neutron stars. The paired quarks in the FF phase feature gapless excitations on surfaces of crescent shaped blocking regions in momentum space and participate in transport. We calculate their contribution to the shear viscosity. We note that they also lead to dynamic screening of transverse $t^1$, $t^2$, $t^3$ gluons which are undamped in the…
We consider the phenomenon of the Andreev reflection of "hadrons" at the interface between hadronic and color superconducting phases, which are expected to appear in the neutron star interior. Here, hadrons are defined as a superposition of constituent quarks, each of which is Andreev-reflected. We study what kind of reflections are possible to come out of incident mesons and baryons in the hadronic phase, attached to different color superconducting phases. Then, some peculiar patterns of the reflections are obtained.
Following closely the logic of the London phenomenological macroscopic theory of the Meissner effect in superconductors we describe the origin of the short-range behavior of the chromo-electric field, the necessary ingredient for color confinement in QCD. The genuinely non-Abelian model is specified by the strong-coupling colored-gluon current. Its first term, as the superconductivity current, is proportional to the gauge potential. The new term is simply related to the chromo-magnetic pseudo-vector current of the non-Abelian Bianchi identity. We suggest that this London dual color superconduc…
Spin-one color superconductor is a viable candidate phase of dense matter in the interiors of compact stars. Its low-energy excitations will influence the transport properties of such matter and thus have impact on late-stage evolution of neutron stars. It also provides a good example of spontaneous symmetry breaking with rich breaking patterns. In this contribution, we reanalyze the phase diagram of a spin-one color superconductor and point out that a part of it is occupied by noninert states, which have been neglected in literature so far. We classify the collective Nambu--Goldstone modes, w…
The interiors of high mass compact (neutron) stars may contain deconfined quark matter in a crystalline color superconducting (CCS) state. On a basis of microscopic nuclear and quark matter equations of states we explore the internal structure of such stars in general relativity. We find that their stable sequence harbors CCS quark cores with masses M_core \le (0.78-0.82)M_{sun} and radii R_core \le 7 km. The CCS quark matter can support nonaxisymmetric deformations, because of its finite shear modulus, and can generate gravitational radiation at twice the rotation frequency of the star. Assum…
Near the flavor SU(3) limit, we propose an analytical description for color-flavor-locked-type Bardeen-Cooper-Schrieffer (BCS) phase in the Nambu Jona-Lasinio (NJL) model. The diquark behaviors in light-flavor and strange-flavor-involved channels and Bose-Einstein condensation (BEC) of bound diquark states are studied. When the attractive interaction between quarks is strong enough, a BCS-BEC crossover is predicted in the environment with color-flavor-locked pairing pattern. The resulting Bose-Einstein condensed phase is found to be an intergrade phase before the emergence of the previous-pred…
