Holographic s+p superconductors with axion induced translation symmetry breaking

Abstract

We construct a holographic model for an s+p superconductor with axion-induced translation symmetry breaking within the framework of gauge/gravity duality, working in the probe limit. The equations of motion are solved numerically to investigate the influence of the parameter $k/T$ on the competition and coexistence between the s-wave and p-wave orders. We find that increasing $k/T$ suppresses the thermodynamic stability of both the single condensate s-wave and p-wave solutions. With the $k-μ$ phase diagram and the condensate curves, we see that the region dominated by the single condensate p-wave phase gradually decreases with the increasing of $k/T$, finally leaving only the single condensate s-wave phase in the large $k/T$ region, which is explained by the grand potential curves showing a slower decreasing of the thermodynamic stability for the s-wave solution than that for the p-wave solution. Furthermore, a larger minimum ratio of the charges $q_p/q_s$ is required to stabilize the s+p coexistent phase as $k/T$ increases, and we determine the precise dependence of this critical ratio on $k/T$. Finally, our study of the optical conductivity reveals that the gap frequency increases with $k/T$. A characteristic kink, associated with the s+p coexistent phase, is identified in the dependence of gap frenquency on $k/T$, which could serve as a potential experimental signature for detecting multi-condensate superconductivity.

Figures (6)

• Figure 1: The condensate (Left) and grand potential (Right) curves of the s-wave (Solid Curves) and p-wave solutions (Dashed Curves) with $q_s=q_p=1$ and three different values of the translational symmetry breaking strength ($k/T$=0 (red), 2.1855 (green), and 5.0265 (blue)).
• Figure 2: The $k-\mu$ phase diagram with $q_p=0.7225$ and $q_s=1$. The white region is dominated by the normal phase. The cyan region is dominated by the single condensate s-wave phase and the magenta region is dominated by the single condensate p-wave phase. The blue region is dominated by the multi condensate s+p phase. The symbols o, *, + mark the three special points and their values of $k/T$ are given by $(k/T)^o=2.0461$, $(k/T)^*=3.1311$ and $(k/T )^+=3.6118$, respectively.
• Figure 3: The condensate curves for various fixed values of the translational symmetry breaking strength $k/T =1.2758, 3.0862, 3.1929$, and $3.7512$. The red and black lines indicate the condensate values of the s-wave and p-wave orders, respectively. Solid lines denote the condensate values of the most stable solutions, while dashed lines denote the condensate values in the unstable sections of the single condensate solutions.
• Figure 4: The grand potential curves for $k/T=0$ (blue), $1.2758$ (red), $3.0862$ (cyan), $3.1929$ (magenta), and $3.7512$ (green), respectively. The relative value of the grand potential for the single condensate p-wave solution with respective to the single condensate s-wave solution (or normal solution below the critical point of the s-wave solution) is presented.
• Figure 5: The $q_p-\mu$ phase diagrams with $k/T=1.2758$ (Left) and $k/T = 2.1855$ (Middle), and the dependence of $q_{min}$ on $k/T$ (Right). In the left and middle panels, the white region represents the normal phase, the cyan region represents the s-wave phase, the magenta region represents the p-wave phase and the blue region represents the s+p phase. The symbol "+" signifies the minimum charge of the p-wave order $q_p=q_p/q_s=q_{Pmin}$ required for the existence of the multi condensate s+p solutions, the dependence of which on $k/T$ is presented in the right panel.