Phase structure of a holographic topological superconductor beyond the probe limit

We investigate tricritical phase transitions in a holographic model of topological superconductivity using Einstein-Maxwell gravity coupled with a charged scalar field in Anti-de Sitter spacetime. By incorporating both gravitational backreaction and quartic self-interaction $V(\phi) = \lambda \phi^4$, we demonstrate that the system exhibits both second-order and first-order phase transitions separated by a tricritical point at $(q_{\mathrm{tri}},T_{\mathrm{tri}})=(2.00\pm0.02,0.1521\pm0.0003)$ in the $(q,T)$ parameter space, where $q$ is the dimensionless charge parameter. The backreacted critical temperature shows enhancement by a factor of 1.22 compared to the probe limit, revealing the importance of strong coupling effects. Tricritical scaling analysis yields an exponent $\phi=0.40\pm0.03$, deviating significantly from mean-field predictions ($\phi=2/3$) due to finite-size effects and holographic geometric corrections. The order parameter critical exponent $\beta=0.50\pm0.02$ remains consistent with mean-field theory due to large-$N$ suppression of quantum fluctuations. The frequency-dependent conductivity exhibits a superconducting gap with energy ratio $\omega_{g}/T_{c}=3.18\pm0.05$, representing a $10\%$ deviation from BCS theory. Holographic entanglement entropy provides quantum information signatures that clearly distinguish transition types. Our results establish that gravitational backreaction, combined with scalar self-interaction, is essential for generating tricritical behavior in holographic superconductors.