Phenomenological Models of Holographic Superconductors and Hall currents

TL;DR

This work develops a phenomenological holographic framework for superconductors described by a neutral scalar $\eta$ and bulk couplings $G(\eta)$, $U(\eta)$, $J(\eta)$, plus a theta term $\Theta(\eta)$. The authors derive analytic expressions for critical exponents and demonstrate Rushbrooke scaling, classifying transitions into mean-field (Model I) and non-analytic (Model II) regimes with tunable exponents, including higher-order transitions. They construct a generalized Landau-Ginzburg free energy near $T_c$, compute the condensate's temperature dependence and the specific heat, and analyze conductivity, uncovering resonance peaks that intensify at low $T$ without BF-bound tuning. They also show that a parity-violating theta term yields Hall conductivity without an external magnetic field, with DC Hall response vanishing above $T_c$ and a nonzero, temperature-dependent $\sigma_{xy}$ in the condensed phase, highlighting connections to quantum criticality and potential experimental contexts.

Abstract

We study general models of holographic superconductivity parametrized by four arbitrary functions of a neutral scalar field of the bulk theory. The models can accommodate several features of real superconductors, like arbitrary critical temperatures and critical exponents in a certain range, and perhaps impurities, boundary or thickness effects. We find analytical expressions for the critical exponents of the general model and show that they satisfy the Rushbrooke identity. An important subclass of models exhibits second order phase transitions. A study of the specific heat shows that general models can also describe holographic superconductors undergoing first, second and third (or higher) order phase transitions. We discuss how small deformations of the HHH model lead to the appearance of resonance peaks in the conductivity, which become narrower as the temperature is gradually decreased, without the need for tuning mass of the scalar to be close to the Breitenlohner-Freedman bound. Finally, we investigate the inclusion of a generalized "theta term" producing Hall effect without magnetic field.

Figures (8)

• Figure 1: $T_c^2$ vs. $\kappa$ in second (or higher) order phase transitions in Model I or Model II ($q=\mu=1$).
• Figure 2: $\Delta c_v$ as a function of temperature close to the phase transition for $J=\eta^2 - \eta^c + \eta^4$, with $c=3+k/5$, $k=0,\ldots,4$.
• Figure 3: (a) Real part of the conductivity as a function of frequency for $J=\eta^2 +j_0 \eta^4$ with $j_0=0.6$ in the $\langle O_1\rangle = 0$ scheme. The curves correspond to different values of $T/T_c$ equal to $0.24,\ 0.29,\ 0.50,\ 0.61,\ 0.81$ (the curves with lower temperatures are those that go to zero more rapidly as $\omega \to 0$). (b) Real and imaginary part of the conductivity for the same model at $T=0.20$.
• Figure 4: Conductivity as a function of frequency for HHH (dashed-dotted line) and for the different deformations given by the models (\ref{['J_conductivity']}), (\ref{['JKKK']}) and (\ref{['JLLL']}) in the $\langle O_1\rangle = 0$ scheme at $T=0.415\ T_c$.
• Figure 5: (a) Potential for the model (\ref{['J_conductivity']}) with $j_0=0$ (dashed line), corresponding to the HHH model, and for $j_0=0.05$ (solid line) in the scheme where $\langle O_2\rangle \neq 0$. The temperatures are $0.78,\ 0.24$, from bottom to top. (b) Maximum of the potential as a function of the temperature. The dashed line is the model (\ref{['JLLL']}) with $G=1/(1+0.1\eta^4)$; the solid lines correspond to (from bottom to top) HHH model, $J=\eta^2+0.05\eta^4$ and $J=\eta^2+0.2\eta^4$.