Novel five-dimensional rotating Lifshitz black holes with electric and axionic charges

TL;DR

The paper constructs exact five-dimensional rotating Lifshitz black holes with electric and axionic charges in a dilaton–gauge theory supplemented by two generalized Chern-Simons terms, valid for the dynamical exponent $z\in(2,3]$. It derives the metric and matter fields, analyzes horizon structure, and computes thermodynamic quantities, proving the first-law $dM=TdS+\Phi_e dQ_e+\Omega dJ+\sum_i\Psi_a^{(i)} d\omega_a^{(i)}$ and the Smarr relation $M=\frac{1}{z+3}(3TS+4\Omega J+4\Psi_a^{(1)}\omega_a^{(1)}+4\Psi_a^{(2)}\omega_a^{(2)})$. The authors then place a holographic s-wave superconductor in the probe limit on these backgrounds, establishing how rotation ($J$) and anisotropy ($z$) affect condensation and AC conductivity, with rotation suppressing and higher $z$ enhancing superconductivity. This work provides a controlled setting to study non-relativistic holography with rotation and Lifshitz scaling, and opens avenues for stability analysis, backreacted matter configurations, and holographic hydrodynamics in rotating Lifshitz spacetimes.

Abstract

In the present paper, we construct a new family of exact charged and rotating asymptotically Lifshitz black hole solutions in five dimensions. The spacetime solves Einstein equations coupled to a dilaton, two Abelian gauge fields, and axionic scalars supplemented by two generalized Chern-Simons terms. This configuration is characterized by a range of the free dynamical exponent $z$ and possesses nontrivial thermodynamical parameters, where we verify the first law of black hole thermodynamics and derive the corresponding Smarr relation. Motivated by applications to gauge/gravity duality, we then investigate a holographic superconductor in the rotating Lifshitz background. We study the condensation of the scalar operator and the AC conductivity of the dual system. These results show that increasing the rotation parameter suppresses the condensate and weakens the superconducting phase, while increasing the dynamical critical exponent enhances the superconducting order. To the best of our knowledge, the solutions presented here are the first to demonstrate five-dimensional rotating Lifshitz black holes supported by both electric and axionic charges. This opens up a new avenue to investigate non-relativistic holography beyond static backgrounds.

Figures (4)

• Figure 1: Graphical representation of the function ${\ell^2 f(r)}/{r^{2}}$ with $\ell=1$ and $z=5/2$ for our calculations. Here, the red curve indicates the presence of a naked singularity ($f(r_{\tiny{\hbox{ext}}})>0$). The black curve corresponds to the extremal case ($f(r_{\tiny{\hbox{ext}}})=0$). Finally, the blue curve reveals the situation $f(r_{\tiny{\hbox{ext}}})<0$, where the existence of an inner and outer horizon is ensured.
• Figure 2: The condensation profiles for the operator $\mathcal{O}_1$ as function of temperature, considering different values of the rotation parameter $J$ (left panel) and the dynamical exponent $z$ (right panel).
• Figure 3: Real (left) and imaginary (right) parts of the conductivity as functions of the frequency for different values of the rotation parameter. These plots were generated at $T\approx 0.036\,T_c$, considering $\Delta =1$, $z=5/2$ and $\ell =1$.
• Figure 4: Real (left) and imaginary (right) parts of the conductivity as functions of the frequency for different values of the dynamical critical exponent. These curves were generated at $T\approx 0.036\,T_c$, considering $\Delta =1$, $J=1/4$ and $\ell =1$.