Asymmetric RG flow to lower-dimensional effective theories

We investigate the emergence of locality in infrared (IR) physics, which indicates an asymmetric renormalization group (RG) flow from a $d$-dimensional ultraviolet (UV) conformal field theory (CFT) to a lower-dimensional IR effective theory. In the holographic setup, this phenomenon can be described by a $(d+1)$-dimensional AdS charged black hole. At zero temperature, in particular, it has a $(d+1)$-dimensional anti-de Sitter (AdS) in the asymptotic region and AdS$_2 \times R^{d-1}$ near the horizon, whose dual field theory is mapped to IR quantum field theory (QFT) defined in $R_t \times R^{d-1}$. We show that the IR QFT can be reduced to effective conformal quantum mechanics due to the rapid suppression of the correlation in $R^{d-1}$. In this case, the IR conformal dimension of an operator crucially relies on the details of the UV theory. We further investigate another 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 localization in the directions perpendicular to the magnetic field.