We build a holographic $s$-wave conductor/superconductor model and an insulator/superconductor model in the four-dimensional conformal anomaly corrected~(CAC) AdS gravity. The effects of CAC parameter $alpha$ are studied using both numerical and anal
ytical methods in the probe approximation. Concretely, when the CAC parameter increases, the critical temperature increases for the conductor/superconductor phase transition, while the critical chemical potential decreases for the insulator/superconductor case, which suggests that the increasing CAC parameter enhances both superconducting phase transitions. Meanwhile, below the critical temperature or beyond the critical chemical potential, the scalar hair begins to condense, and the condensed phases are found to be thermodynamically stable. The critical behaviors obtained from numerics are confirmed by our analytical analysis. For the parameters we are considering, the energy gap in the conductor/superconductor model decreases monotonically by increasing the CAC parameter, while for the insulator/superconductor model the energy of quasiparticle excitations decreases with the CAC parameter.
Based on the dynamics of single scalar field slow-roll inflation and the theory of reheating, we investigate the generalized natural inflationary (GNI) model. Concretely, we give constraints on the scalar spectral index $n_{s}$ and tensor-to scalar r
atio $r$ for $Lambda$CDM $+r$ model according to the latest data from Plack 2018 TT,TE,EE+lowE+lensing (P18) and BICEP2/Keck 2015 season (BK15), i.e., $n_{s}=0.9659pm0.0044$ at $68%$ confidence level (CL) and $r<0.0623$ at $95%$CL. We find that the GNI model is favored by P18 plus BK15 in the ranges of $log_{10}(f/M_{p})=0.62^{+0.17}_{-0.18}$ and $m=0.35^{+0.13}_{-0.23}$ at $68%$CL. In addition, the corresponding predictions of the general and two-phase reheating are respectively discussed. It follows that the parameter $m$ has the significant effect on the model behaviors.
In the probe limit, we numerically build a holographic $p$-wave superfluid model in the four-dimensional Lifshitz black hole coupled to a Maxwell-complex vector field. We observe the rich phase structure and find that the Lifshitz dynamical exponent
$z$ contributes evidently to the effective mass of the matter field and dimension of the gravitational background. Concretely, we obtain the Cave of Winds appeared only in the five-dimensional anti-de Sitter~(AdS) spacetime, and the increasing $z$ hinders not only the condensate but also the appearance of the first-order phase transition. Furthermore, our results agree with the Ginzburg-Landau results near the critical temperature. In addition, the previous AdS superfluid model is generalized to the Lifshitz spacetime.
In the probe limit, we numerically construct a holographic p-wave superfluid model in the 4D and 5D AdS black holes coupled to a Maxwell-complex vector field. We find that, for the condensate with the fixed superfluid velocity, the results are simi
lar to the s-wave cases in both 4D and 5D spacetimes. In particular, The Cave of Winds and the phase transition always being the second order take place in the 5D case. Moreover, we find the second-first order translating point $frac{S_y}{mu}$ increases with the mass squared. Furthermore, for the supercurrent with the fixed temperature, the results agree with the GL prediction near the critical temperature. In addition, this complex vector superfluid model is still a generalization of the SU(2) superfluid model, and also provides a holographic realization of the $He_3$ superfluid system.
