Phase transitions in a holographic superfluid model with non-linear terms beyond the probe limit

TL;DR

This work extends the holographic s-wave superfluid model by including full back-reaction and nonlinear scalar self-interactions with terms $\lambda|\psi|^4$ and $\tau|\psi|^6$ in a 3+1 dimensional bulk. By solving the coupled Einstein–matter system and computing the grand potential, it reveals that back-reaction acts as an effective coupling analogous to the sextic term, enabling zeroth-, first-, second-order, and COW phase transitions, while shifting the critical temperature $T_c$. A key finding is the non-monotonic dependence of the special value $\lambda_s(b)$ on the back-reaction strength, marking how back-reaction modulates the condensate growth at criticality and the order of transitions. Overall, the results demonstrate the persistence of universal control by $\lambda$ and $\tau$ beyond the probe limit and outline future avenues for analytic back-reaction expansions and exploration of entanglement and complexity in these holographic phase transitions.

Abstract

We study the holographic s-wave superfluid model with 4th and 6th power self-interaction terms $λ|ψ|^4$ and $τ|ψ|^6$ with considering the full back-reaction of the matter fields on the metric in the 3+1 dimensional bulk. The self-interaction terms are good at controlling the condensate to realize various phase transitions, such as the zeroth-order, first-order, and second-order phase transitions within the single condensate s-wave superfluid model. Therefore, in this work, we are able to investigate the influence of the back-reaction strength on the various phase transitions, including the zeroth and first order phase transitions. In addition, we confirm that the influence of the 4th and 6th power terms on the superfluid phase transition in the case of finite back-reaction are qualitative the same as in the probe limit, thus present universality. We also plot the special value $λ_s$ of the parameter $λ$ at different back-reaction strength, below which the condensate grows to an opposite direction and is important in controlling the order of the superfluid phase transitions. Comparing the influence of the back-reaction parameter and that of the higher-order nonlinear coefficients, we see that the back-reaction strength brings in both the effective couplings similar to the 4th power and 6th power terms.

Figures (10)

• Figure 1: The $b-T$ phase diagram of temperature with $\lambda=0$ and $\tau=0$. The blue solid line represents the critical points of the second-order superfluid phase transitions.
• Figure 2: The dependence of the condensates as well as grand potential curves on the back-reaction strength $b$ with $\lambda=-0.2$ and $\tau=0$ . The left panel depicts the condensates, while the right panel represents the corresponding grand potential curves. The red, blue, green, magenta, and gray lines represent solutions with $b=0.010$, $0.200$, $0.310$, $0.330$, and $0.400$, respectively.
• Figure 3: The dependence of the condensates as well as grand potential curves on the back-reaction strength $b$ with parameters $\lambda=-0.78$ and $\tau=0.4$. The left panel depicts the condensates, while the right panel represents the corresponding grand potential curves. The red, blue, green, magenta, and gray lines represent solutions with $b=0.010$, $0.270$, $0.500$, $0.800$, and $1.100$, respectively.
• Figure 4: The dependence of the condensates as well as grand potential curves on the back-reaction strength $b$ with $\lambda=-0.75$ and $\tau=0.5$. The left panel depicts the condensates, while the right panel represents the corresponding grand potential curves. The red, blue, green, magenta, and gray lines represent solutions with $b=0.010$, $0.380$, $0.580$, $0.880$, and $1.025$, respectively.
• Figure 5: The phase diagram of temperature $T/\mu$ and back-reaction parameter $b$ with $\lambda=-0.75$ and $\tau=0.5$, the right panel is an enlarged view of a section of the left panel. The blue solid line represents the phase transition points of second-order phase transition. The blue dashed line represents the quasi-critical points of first-order phase transition at which the superfluid solution first appears. The red solid line represents the phase transition points of first-order phase transition. The red dashed line represents the turning points of first-order phase transition. The blue region corresponds to the superfluid phase.