[Paper Review] Holographic Superconductors in $z=3$ Hořava-Lifshitz gravity without condition of detailed balance

Jiliang Jing, Liancheng Wang+1 more|arXiv (Cornell University)|Jan 10, 2010

Black Holes and Theoretical Physics24 references3 citations

TL;DR

This paper investigates holographic superconductors in z=3 Hořava-Lifshitz gravity without the detailed balance condition, using numerical and analytical methods to study scalar condensation and electrical conductivity. It finds that the ratio ωg/Tc decreases almost linearly with increasing balance parameter ε, bridging Cai’s result (ωg/Tc ≈ 13 at ε=0) and the Horowitz-Roberts relation (ωg/Tc ≈ 8 as ε→1).

ABSTRACT

We study holographic superconductors in a Hořava-Lifshitz black hole without the condition of the detailed balance. We show that it is easier for the scalar hair to form as the parameter of the detailed balance becomes larger, but harder when the mass of the scalar field larger. We also find that the ratio of the gap frequency in conductivity to the critical temperature, $ω_{g}/T_c$, almost linear decreases with the increase of the balance constant. For $ε= 0$ the ratio reduces to Cai's result $ω_g/T_c\approx 13$ found in the Hořava-Lifshitz black hole with the condition of the detailed balance, while as $ε ightarrow 1$ it tends to Horowitz-Roberts relation $ω_g/T_c\approx 8$ obtained in the AdS Schwarzschild black hole. Our result provides a bridge between the results for the Hǒrava-Lifshitz theory with the condition of the detailed balance and Einstein's gravity.

Motivation & Objective

• To investigate the formation of scalar hair in holographic superconductors within z=3 Hořava-Lifshitz gravity without the detailed balance condition.
• To analyze the impact of the detailed balance parameter ε on scalar condensation and critical temperature Tc.
• To compute the electrical conductivity and determine the ratio ωg/Tc, where ωg is the gap frequency in conductivity.
• To establish a continuous connection between results in Hořava-Lifshitz gravity (with detailed balance) and Einstein gravity (AdS Schwarzschild black hole).

Proposed method

• Constructing a planar black hole solution in z=3 Hořava-Lifshitz gravity with a general action that includes the detailed balance parameter ε.
• Solving the coupled Einstein-Maxwell-scalar equations numerically to study scalar condensation and critical temperature Tc.
• Applying analytical matching techniques at an intermediate radius to validate numerical results for the condensate gap.
• Computing the frequency-dependent conductivity σ(ω) by solving the Maxwell equation for the perturbed gauge field.
• Extracting the gap frequency ωg from the conductivity spectrum and calculating the ratio ωg/Tc for varying ε and scalar mass m².
• Using the parameter ε ∈ [0,1] to interpolate between Hořava-Lifshitz gravity (ε=0) and Einstein gravity (ε→1).

Experimental results

Research questions

• RQ1 How does the detailed balance parameter ε affect the formation of scalar hair in holographic superconductors in z=3 Hořava-Lifshitz gravity?
• RQ2 What is the behavior of the critical temperature Tc and condensate gap as ε increases from 0 to 1?
• RQ3 How does the ratio ωg/Tc, where ωg is the gap frequency in conductivity, vary with ε?
• RQ4 Does the model interpolate smoothly between the Cai-Zhang result (ωg/Tc ≈ 13 at ε=0) and the Horowitz-Roberts relation (ωg/Tc ≈ 8 at ε→1)?
• RQ5 How does the scalar field mass m² influence the condensation and the ωg/Tc ratio in the absence of detailed balance?

Key findings

• The scalar hair forms more easily as the detailed balance parameter ε increases, indicating a lower critical temperature Tc for larger ε.
• For a fixed ε, increasing the scalar field mass m² (making it less negative) makes scalar hair formation harder, increasing the critical temperature Tc.
• The ratio ωg/Tc decreases almost linearly with increasing ε, transitioning from approximately 14.6 at ε=0 (m²L²=0) to 8.6 at ε=0.99.
• At ε=0, the ratio ωg/Tc ≈ 13 matches Cai et al.'s result in Hořava-Lifshitz gravity with detailed balance.
• As ε→1, the ratio ωg/Tc approaches 8.1–8.6, consistent with the Horowitz-Roberts relation in Einstein gravity (AdS Schwarzschild black hole).
• The results provide a continuous bridge between holographic superconductivity in Hořava-Lifshitz gravity and Einstein gravity via the parameter ε.