Abstract
We study the phase structure of a backreacted Einstein–Maxwell holographic superconductor with a gauge-invariant quartic scalar self-interaction V(ψ)=m2|ψ|2+λ|ψ|4 in (3+1)-dimensional asymptotically AdS spacetime. The reduced equations of motion are derived from the action by symbolic tensor calculus and validated against the exact Reissner–Nordström–AdS4 solution to better than one part in 106. At κ2=1, λ˜=0.1, standard quantization, a scan over seventeen charges q ∈ [1.0, 4.0] shows second-order-like behavior for q ≲ 1.8 and q ≳ 3.3, and a first-order window in between, confirmed by exact thermodynamic integration of the free energy and by a positive latent heat at two representative points (ℓ/μ3 ≃ 0.106 and 0.277), bounded by two tricritical points located at qtrilower=1.79±0.01 and qtriupper=3.28±0.01 (bisection resolution). This two-tricritical-point structure differs from the single-TCP phase diagram previously reported for this model and persists at a second coupling, λ˜=0.05. A numerical fit of the order-parameter critical exponent on an ordinary second-order line gives β=0.25–0.40 depending on the fitting window, not yet resolving between the mean-field value 1/2 and the tricritical value 1/4 at the precision and proximity to Tc accessible with the present solver. We derive the coupled gauge-field/metric-perturbation equations governing the optical conductivity; a first-order constraint from the linearized Einstein zx equation reduces the associated shooting problem to a single complex parameter, which we solve numerically and validate against the known normal-phase asymptotic conductivity, |σ(ω → ∞)| → 1 to four decimal places. A representative superconducting-phase point does not show a resolved spectral gap within the frequency range explored, and the zero-frequency spectral weight is comparable in magnitude to its normal-phase, momentum-conservation counterpart, so it cannot be attributed to superfluid density without a momentum-relaxation deformation not implemented here. A holographic entanglement-entropy calculation for a boundary strip is implemented, validated against the expected universal UV divergence, and restricted to the thermodynamically stable branch. We do not find, and do not claim, topological superconductivity in the sense of a bulk topological invariant, protected edge state, or Majorana mode.