Emergent Lorentz symmetry near fermionic quantum critical points in two and three dimensions

TL;DR

This paper investigates how Lorentz symmetry emerges at quantum critical points involving fermions in two and three dimensions, showing that velocities tend to unify, leading to an effective relativistic invariance in the infrared limit.

Contribution

It demonstrates that Lorentz symmetry emerges universally in fermionic quantum critical theories in 2D and 3D, with velocities flowing to a common terminal value regardless of initial conditions.

Findings

Velocities flow to the speed of light in 2D.

Velocities unify to a non-universal terminal velocity in 3D.

Lorentz symmetry emerges in the infrared regime.

Abstract

We study the renormalization group flow of the velocities in the field theory describing the coupling of the massless quasi-relativistic fermions to the bosons through the Yukawa coupling, as well as with both bosons and fermions coupled to a fluctuating $U(1)$ gauge field in two and three spatial dimensions. Different versions of this theory describe quantum critical behavior of interacting Dirac fermions in various condensed-matter systems. We perform an analysis using one-loop $\epsilon-$expansion about three spatial dimensions, which is the upper critical dimension in the problem. In two dimensions, we find that velocities of both charged fermions and bosons ultimately flow to the velocity of light, independently of the initial conditions, the number of fermionic and bosonic flavors, and the value of the couplings at the critical point. In three dimensions, due to the analyticity of the gauge field propagator, both the $U(1)$ charge and the velocity of light flow, which leads to a richer behavior than in two dimensions. We show that all three velocities ultimately flow to a common terminal velocity, which is non-universal and different from the original velocity of light. Therefore, emergence of the Lorentz symmetry in the ultimate infrared regime seems to be a rather universal feature of this class of theories in both two and three dimensions.