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Condensation for non-relativistic matter in Hořava-Lifshitz gravity

We study condensation for non-relativistic matter in a Hořava-Lifshitz black hole without the condition of the detailed balance. We show that, for the fixed non-relativistic parameter $α_2$ (or the detailed balance parameter $ε$), it is easier for the scalar hair to form as the parameter $ε$ (or $α_2$) becomes larger, but the condensation is not affected by the non-relativistic parameter $β_2$. We also find that the ratio of the gap frequency in conductivity to the critical temperature decreases with the increase of $ε$ and $α_2$, but increases with the increase of $β_2$. The ratio can reduce to the Horowitz-Roberts relation $ω_g/T_c\approx 8$ obtained in the Einstein gravity and Cai's result $ω_g/T_c\approx 13$ found in a Hořava-Lifshitz gravity with the condition of the detailed balance for the relativistic matter. Especially, we note that the ratio can arrive at the value of the BCS theory $ω_g/T_c\approx 3.5$ by taking proper values of $ε$, $α_2$, $β_2$ and $m$.