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Asymmetric RG flow to lower-dimensional effective theories

Chanyong Park

Abstract

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.

Asymmetric RG flow to lower-dimensional effective theories

Abstract

We investigate the emergence of locality in infrared (IR) physics, which indicates an asymmetric renormalization group (RG) flow from a -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 -dimensional AdS charged black hole. At zero temperature, in particular, it has a -dimensional anti-de Sitter (AdS) in the asymptotic region and AdS near the horizon, whose dual field theory is mapped to IR quantum field theory (QFT) defined in . We show that the IR QFT can be reduced to effective conformal quantum mechanics due to the rapid suppression of the correlation in . 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.

Paper Structure

This paper contains 4 sections, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two-point functions in the IR limit
  3. Holographic localization to two-dimensional CFT
  4. Discussion

Figures (2)

  • Figure 1: We plot (a) the profiles of $V(r)$ and $W(r)$ and (b) their derivatives in the $r$-directions. We take $R=1$, $B=10$, and $c_2 =84.948$.
  • Figure 2: We depict (a) the suppression power of the transverse two-point function and (b) the effective conformal dimension of the longitudinal two-point function depending on the RG scale.