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

Abstract We study the holographic s-wave superfluid model with fourth- and sixth-power self-interaction terms $$\lambda |\psi |^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>|</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:math> and $$\tau |\psi |^6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>|</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mn>6</mml:mn> </mml:msup> </mml:math> , 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 $$\lambda _s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> of the parameter $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> 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.