# Quantum Criticality of Semi-Dirac Fermions in 2+1 Dimensions

## TL;DR

This paper studies the quantum critical behavior of semi-Dirac fermions in 2+1 dimensions, analyzing instabilities towards various ordered phases using RG techniques and large-Nf expansion.

## Contribution

It provides a systematic RG analysis of semi-Dirac fermions' instabilities and calculates critical exponents including 1/Nf corrections, revealing small corrections and anisotropic correlations.

## Key findings

- Critical exponents are computed at one-loop order with small 1/Nf corrections.
- Order-parameter correlations exhibit anisotropic divergence along different spatial directions.
- Proximity to criticality may lead to stabilization of modulated order phases.

## Abstract

Two-dimensional semi-Dirac fermions are quasiparticles that disperse linearly in one direction and quadratically in the other. We investigate instabilities of semi-Dirac fermions towards charge, spin-density wave and superconducting orders, driven by short-range interactions. We analyze the critical behavior of the Yukawa theories for the different order parameters using Wilson momentum shell RG. We generalize to a large number $N_f$ of fermion flavors to achieve analytic control in 2+1 dimensions and calculate critical exponents at one-loop order, systematically including $1/N_f$ corrections. The latter depend on the specific form of the bosonic infrared propagator in 2+1 dimensions, which needs to be included to regularize divergencies. The $1/N_f$ corrections are surprisingly small, suggesting that the expansion is well controlled in the physical dimension. The order-parameter correlations inherit the electronic anisotropy of the semi-Dirac fermions, leading to correlation lengths that diverge along the spatial directions with distinct exponents, even at the mean-field level. We conjecture that the proximity to the critical point may stabilize novel modulated order phases.