Various phase transitions in a holographic p-wave superfluid model with nonlinear terms

Abstract

This study investigates various phase transitions, including those of 2nd, 1st, and 0th order, in a holographic p-wave superfluid model incorporating 4th- and 6th-order nonlinear terms with coefficients $λ$ and $τ$. We demonstrate that these nonlinear terms provide universal control over the phase transitions of the p-wave model, qualitatively consistent with findings in the holographic s-wave case. By analyzing the condensate and free energy behavior across typical phase transitions, we quantitatively map out the $λ-τ$ parameter space that characterizes different transition types. For a slightly negative $λ$, we further establish a $τ-ρ$ phase diagram featuring a line of first-order phase transition points that terminates at a critical point, beyond which lies a supercritical region. Our results confirm the precise tunability of the p-wave superfluid phase transitions through $λ$ and $τ$. The comprehensive phase diagrams and quantitative transition criteria we provide offer a valuable resource for future studies.

Figures (6)

• Figure 1: The condensate (Left) and free energy (Right) of the second order phase transition with ($\lambda=\tau=0$). The blue solid line represents the (meta-)stable section of the normal solution while the blue dashed line represents the unstable section of the normal phase beyond the (quasi) critical point $\rho_c$. The solid green line represents the (meta-)stable p-wave superfluid phase. The black triangle indicates the location of the critical point with $\rho_c$ = 3.6496, where the branch of the superfluid phase is connected to the branch of the normal phase.
• Figure 2: The condensate (Left) and free energy (Right) of the zeroth- order phase transition with ($\lambda=-0.7,\tau=0$). The notion of the green and blue lines as well as the black triangle are used as the same as that int Figure \ref{['2nd']}. The solid red line represents the unstable section of the p-wave superfluid solutions, which is connected to the (meta-)stable section at turning point marked by the blue dot with $\rho_t=3.78444$.
• Figure 3: The condensate (Left) and free energy (Right) of the first- order phase transition with ($\lambda=-1.4,\tau=0.8$). The notion of the green and blue lines as well as the black triangle are used as the same as that int Figure \ref{['2nd']}. In this plot, the unstable section of the p-wave superfluid solutions marked by the solid red line is connected to the normal solution on the quasi critical point marked by the black triangle, and is connected to the (meta-)stable section marked by the solid green line on the turning point marked by the blue dot with $\rho_t=3.42887$.
• Figure 4: The condensate (Left) and free energy (Right) of the COW phase transition with $\lambda=-0.7$ and $\tau=0.14$. The notion of the green and blue lines as well as the black triangle are used as the same as that int Figure \ref{['2nd']}. The (meta-)stable section of the p-wave superfluid solutions marked by the solid green line is connected to the normal solution on the quasi critical point marked by the black triangle, and is connected to the unstable section marked by the solid red line on the 1st turning point marked by the blue dot with $\rho_{t1}=3.84522$. The other (meta-)stable section of the p-wave superfluid solutions marked by the solid cyan line is connected to the other side of the solid red line on the 2nd turning point marked by the green dot with $\rho_{t2}=3.71325$. The red dot with $\rho_T=3.78758$ mark the position of the first order phase transition between the two sections of superfluid solutions, which is identified by the intersection point of the free energy curve for the two sections of superfluid solutions.
• Figure 5: The $\tau-\rho$ parameter space for the various phase transitions. In this plot, the symbol "COW" stands for "cave of wind" phase transition and the symbol "NSSP" stands for "no stable superfluid phase". "2nd", "1st" and "0th" denote the parameter space of the first-order phase transitions, the second-order phase transitions, and the zeroth-order phase transitions, respectively.