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 $V(ϕ) = λϕ^4$, 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 $ϕ=0.40\pm0.03$, deviating significantly from mean-field predictions ($ϕ=2/3$) due to finite-size effects and holographic geometric corrections. The order parameter critical exponent $β=0.50\pm0.02$ 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 $ω_{g}/T_{c}=3.18\pm0.05$, 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.

💡 Research Summary

In this work the authors study a holographic model of a three‑dimensional topological superconductor by embedding a charged scalar field with a quartic self‑interaction into four‑dimensional Einstein‑Maxwell gravity with a negative cosmological constant. The key novelty is that the gravitational backreaction is fully taken into account (κ²=1) and the scalar potential includes V(ϕ)=−m²ϕ²+λϕ⁴ with λ>0. The scalar mass is chosen as m²L²=−2 so that the dual operator has dimension Δ=2. By scaling all quantities with the chemical potential μ, the authors work with dimensionless charge q and temperature T.

The coupled nonlinear ordinary differential equations for the metric functions f(z), χ(z), the gauge potential A_t(z) and the scalar profile ϕ(z) are solved numerically using an adaptive fourth‑order Runge‑Kutta scheme together with a Newton‑Raphson shooting method. Near the horizon regularity conditions are imposed, and at the AdS boundary holographic renormalization is performed by adding the Gibbons‑Hawking term and appropriate counterterms. This yields a finite grand potential Ω for both the normal (Reissner‑Nordström‑AdS) and superconducting phases, allowing a reliable comparison of free energies.

Scanning the (q,T) plane the authors map out two distinct transition lines. For large charge q the scalar condenses continuously at a critical temperature T_c, signalling a second‑order phase transition (blue solid line). For smaller q the free‑energy landscape exhibits the characteristic swallow‑tail structure, indicating a first‑order transition (red dashed line). The two lines intersect at a tricritical point (TCP) located at (q_tri,T_tri)=(2.00±0.02, 0.1521±0.0003). By varying the self‑interaction coupling λ (0.05, 0.10, 0.15, 0.20) the authors demonstrate that λ controls the position of the TCP: larger λ shifts the TCP to higher q and higher T, confirming that the λϕ⁴ term is not merely a technical requirement but an active tuning parameter of the phase diagram.

Critical scaling near the TCP is analyzed by fitting the order‑parameter condensate ⟨O₂⟩ versus the reduced temperature ΔT. The tricritical exponent ϕ, defined by ⟨O₂⟩∝(ΔT)^ϕ, is found to be ϕ=0.40±0.03, significantly lower than the mean‑field value 2/3, indicating non‑mean‑field behavior likely due to finite‑size effects and holographic geometric corrections. In contrast, the standard critical exponent β governing the second‑order line remains β=0.50±0.02, consistent with mean‑field theory because large‑N suppression damps quantum fluctuations.

The optical conductivity σ(ω) is computed by perturbing the bulk gauge field. The frequency‑dependent real and imaginary parts display a clear superconducting gap with ω_g/T_c=3.18±0.05, about 10 % lower than the BCS prediction (≈3.5), reflecting the strong‑coupling nature of the holographic superconductor.

Finally, holographic entanglement entropy S_EE is evaluated for strip regions on the boundary. Across the first‑order line S_EE jumps discontinuously, mirroring the swallow‑tail in the free energy, while across the second‑order line S_EE varies smoothly but with a kink in its derivative, providing an independent quantum‑information diagnostic of the transition order.

Overall, the paper establishes that incorporating full gravitational backreaction together with a quartic scalar self‑interaction is essential for generating a tricritical point in holographic topological superconductors. Backreaction raises the critical temperature by a factor of ≈1.22 relative to the probe limit, while the λϕ⁴ term determines both the existence and the location of the tricritical point. The work clarifies the universal (mean‑field) and non‑universal (tricritical) scaling exponents in the large‑N holographic setting and enriches the phenomenology of strongly coupled topological superconductors with concrete predictions for conductivity and entanglement measures.