Solving QED$_3$ with Conformal Bootstrap

Zhijin Li

Published 2018-12-21Version 1

We study the conformal bootstrap for $3D$ Quantum Electrodynamic (QED$_3$) coupled to $N_f$ flavors of two-component Dirac fermions $\Psi_i$. We bootstrap four point correlator of fermion bilinear operator, which forms an adjoint representation of the flavor symmetry $SU(N_f)$. We obtain rigorous upper bounds on the scaling dimensions of the parity even $SU(N_f)$ singlets, i.e., the fermion quadrilinear operators. We find strong evidence that the IR fixed points of standard QED$_3$ and QED$_3$-Gross-Neveu model saturate the bound with large $N_f$. The two kinks merge near $N_f=3$ and disappear for $N_f\leq2$. The $SU(N_f)$ singlets related to these kinks are irrelevant. Our results support the "merger and annihilation of fixed points" scenario. Besides, it provides a solution to the long-standing problem of the critical flavor symmetry of QED$_3$: $N_{\mathrm{crit}}=2$. Our results shed lights on several interesting problems, including high-temperature superconducting, N\'eel-Valence Bond Solid quantum phase transition and the duality web of $3D$ theories.