Abstract We numerically study the complexity of a strip-shaped subregion by the subregion CV conjecture in backreacting holographic p-wave superconductors in low-dimensional Einstein–Maxwell-complex vector field theory. Based on our results with different mass and charge of the vector field, holographic complexity serves as a good probe to the superconductor phase transitions and exhibits characteristic qualitative behaviors across different types of transition. At the second order phase transition point, the complexity is continuous but non-differentiable, while at the first order and the zeroth order transition points, the complexity has a discontinuous change. The distinction between the first order and zeroth order phase transitions lies in the fact that in the former, the complexity for the normal phase and the superconducting phase can be connected by a metastable condensed phase, whereas in the latter such a metastable phase does not exist. A thermodynamically sub-dominant condensed phase known as “retrograde condensation” is also imprinted on the complexity. For this phase, the complexity only appears in the high temperature regime. Furthermore, the holographic complexity has many different behaviors from the holographic entanglement entropy. Particularly, we find that the complexity for the normal phase shows inconsistent temperature dependence in the case of d=2 d = 2 and d=3 d = 3, while such a discrepancy between different dimensions is not observed in the entanglement entropy.
Lai et al. (Thu,) studied this question.