Asymmetric RG flow to lower-dimensional effective theories

We investigate an asymmetric RG flow from a $d$-dimensional UV CFT to a lower-dimensional IR CFT. The asymptotic geometry of a $(d+1)$-dimensional AdS charged black hole is given by a $(d+1)$-dimensional AdS space while its near horizon geometry is reduced to AdS$_2 \times \Sigma_{d-1}$. According to the AdS/CFT correspondence, the dual field theory describes an RG flow from a $d$-dimensional CFT to IR QFT defined in $R_t \times \Sigma_{d-1}$. We show that one-dimensional conformal quantum mechanics can effectively describe the IR QFT due to the rapid suppression of the correlation in the $\Sigma_{d-1}$ sector. In this case, the IR conformal dimension crucially relies on the blackening factor, which on the dual QFT side is associated with the matter's density. We further study an asymmetric RG flow from a four-dimensional UV CFT to a two-dimensional IR CFT by turning on an external magnetic field, which leads to a localization in the directions perpendicular to the magnetic field.