Critical behavior of Dirac fermions from perturbative renormalization

TL;DR

This paper analyzes the critical behavior of Dirac fermions in condensed matter systems using advanced perturbative renormalization techniques, providing refined estimates of critical exponents and comparing them with other methods.

Contribution

It offers a comprehensive four-loop perturbative analysis of the Gross-Neveu universality class, combining multiple dimensional expansions and resummation techniques to improve critical exponent predictions.

Findings

Excellent agreement with conformal bootstrap for large N and N=1.

Deviations from Monte Carlo results at intermediate N are critically discussed.

Provides refined critical exponent estimates for Dirac fermions in 2+1 dimensions.

Abstract

Gapless Dirac fermions appear as quasiparticle excitations in various condensed-matter systems. They feature quantum critical points with critical behavior in the 2+1 dimensional Gross-Neveu universality class. The precise determination of their critical exponents defines a prime benchmark for complementary theoretical approaches, such as lattice simulations, the renormalization group and the conformal bootstrap. Despite promising recent developments in each of these methods, however, no satisfactory consensus on the fermionic critical exponents has been achieved, so far. Here, we perform a comprehensive analysis of the Ising Gross-Neveu universality classes based on the recently achieved four-loop perturbative calculations. We combine the perturbative series in $4-\epsilon$ spacetime dimensions with the one for the purely fermionic Gross-Neveu model in $2+\epsilon$ dimensions by employing polynomial interpolation as well as two-sided Pad\'e approximants. Further, we provide predictions for the critical exponents exploring various resummation techniques following the strategies developed for the three-dimensional scalar $O(n)$ universality classes. We give an exhaustive appraisal of the current situation of Gross-Neveu universality by comparison to other methods. For large enough number of spinor components $N\geq 8$ as well as for the case of emergent supersymmetry $N=1$, we find our renormalization group estimates to be in excellent agreement with the conformal bootstrap, building a strong case for the validity of these values. For intermediate $N$ as well as in comparison with recent Monte Carlo results, deviations are found and critically discussed.