Phase structure of a holographic topological superconductor beyond the probe limit

TL;DR

This work analyzes tricritical phase transitions in a holographic superconductor by including gravitational backreaction and a quartic scalar self-interaction $V(\phi)=\lambda\phi^4$ in Einstein–Maxwell theory on AdS$_4$. The authors map the phase diagram in the $(q,T)$ plane, locating a tricritical point at $$(q_{\mathrm{tri}},T_{\mathrm{tri}})=(2.00\pm0.02,0.1521\pm0.0003)$, where second-order and first-order transitions meet. They report a nonmean-field tricritical scaling with exponent $\phi=0.40\pm0.03$ (vs $\phi_{\mathrm MF}=2/3$) and a mean-field-like order parameter exponent $\beta=0.50\pm0.02$, along with a superconducting gap ratio $\omega_{g}/T_{c}=3.18\pm0.05$ and entanglement-entropy signatures that distinguish transition types. Gravitational backreaction enhances $T_c$ by up to a factor of $1.22$ and is essential for tricritical behavior, while the quartic term provides thermodynamic instability sufficient for first-order tendencies. The results imply cooperative strong-coupling effects in holographic duals of topological superconductors and offer concrete experimental signatures for multicritical phenomena in strongly coupled quantum systems.

Abstract

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.

Figures (7)

• Figure 1: Phase diagram in $(q,T)$ space obtained from numerical solutions of the coupled Einstein-Maxwell-scalar system with gravitational backreaction. The blue curve delineates second-order transitions for $q > q_{\mathrm{tri}} = 2.0$, while the red curve shows first-order transitions for $q < q_{\mathrm{tri}}$. The yellow point marks the tricritical point at $(q_{\mathrm{tri}},T_{\mathrm{tri}}) = (2.0, 0.152)$. The shaded regions indicate the normal phase (light blue) and superconducting phase (light red).
• Figure 2: Tricritical scaling analysis in log-log coordinates showing the approach to the tricritical point. Red circles with solid line represent numerical data yielding $\phi = 0.40 \pm 0.03$, while the green dashed line shows the mean-field prediction $\phi = 2/3 \approx 0.667$. The significant deviation reflects finite-size effects, holographic geometric corrections, and non-trivial renormalization group flow under backreaction.
• Figure 3: Order parameter critical exponent analysis in log-log coordinates. Red circles represent numerical data points, the blue solid line shows the fitted scaling $\beta = 0.50 \pm 0.02$, and the green dashed line indicates the mean-field reference $\beta = 1/2$. The excellent agreement with mean-field theory reflects the large-$N$ suppression of quantum fluctuations in the holographic framework.
• Figure 4: Free energy difference $\Delta G = G_{\mathrm{SC}} - G_{\mathrm{normal}}$ versus temperature for a first-order transition at $q < q_{\mathrm{tri}}$. The blue curve shows the characteristic swallowtail behavior with metastable regions (orange shaded area). The red dashed line marks the Maxwell construction determining the true transition temperature where free energies are equal. The swallowtail structure provides definitive evidence for first-order behavior and enables calculation of latent heat.
• Figure 5: Order parameter evolution: (a) Second-order transition showing continuous onset with critical scaling $\langle\mathcal{O}\rangle \propto (T_c - T)^{1/2}$ for $q > q_{\mathrm{tri}}$. (b) First-order transition exhibiting discontinuous jump with finite order parameter below $T_c$ for $q < q_{\mathrm{tri}}$. The red dashed lines mark the respective critical temperatures. This comparison clearly demonstrates the qualitative difference between transition types.