Condensation for non-relativistic matter in Ho\v{r}ava-Lifshitz gravity

We study condensation for non-relativistic matter in a Ho\v{r}ava-Lifshitz black hole without the condition of the detailed balance. We show that, for the fixed non-relativistic parameter ${\alpha }_2$ (or the detailed balance parameter $ϵ$), it is easier for the scalar hair to form as the parameter $ϵ$ (or ${\alpha }_2$) becomes larger, but the condensation is not affected by the non-relativistic parameter ${\beta }_2$. We also find that the ratio of the gap frequency in conductivity to the critical temperature decreases with the increase of $ϵ$ and ${\alpha }_2$, but increases with the increase of ${\beta }_2$. The ratio can reduce to the Horowitz-Roberts relation ${\omega }_g\mathrm{/}{T}_c\approx 8$ obtained in the Einstein gravity and Cai’s result ${\omega }_g\mathrm{/}{T}_c\approx 13$ found in a Ho\v{r}ava-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 ${\omega }_g\mathrm{/}{T}_c\approx 3.5$ by taking proper values of $ϵ$, ${\alpha }_2$, ${\beta }_2$ and $m$.