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

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