Title: Critical Behavior of Non-Hermitian Kondo effect in Pseudogap System

Abstract: The combination of non-Hermitian physics and strong correlation can yield numerous novel and intriguing effects. A previous study on the non-Hermitian Kondo model in ultra-cold atoms reports the reversion of the renormalization group flow. In this work, We investigate the non-Hermitian Kondo effect in system with special form of density of sates $\rho (\omega) \sim |\omega|^{r}(r>0)$, which is called pseudogap system. We find that when $r<1/2$, our conclusion from perturbative renormalization group theory aligns well with previous studies on the traditional pseudogap Kondo problem. In the case of $r$ being equal to $1/2$, a fixed point with reverse property appears in the renormalization group flow. When $r$ is lager than $1/2$, an unstable fixed point appears on the complex plane of the parameter space. Additionally, we validate the conclusions around renormalization group for the $r<1/2$ interval using the large $N$ expansion method.