Renormalization group analysis of Dirac fermions with random mass

TL;DR

This paper uses two-loop and four-loop renormalization group analyses to study the quantum multicritical behavior of 2D Dirac fermions with random mass, revealing fixed points and critical exponents relevant to disordered superconductors.

Contribution

It provides a detailed RG analysis of the tricritical point in 2D disordered superconductors with Dirac fermions, including higher-loop calculations and mappings to the Gross-Neveu model.

Findings

Identification of an IR unstable fixed point at finite disorder strength.

Evaluation of critical and dynamical exponents at the tricritical point.

Insights into the control of phase transitions by saddle-point fixed points.

Abstract

Two-dimensional (2D) disordered superconductor (SC) in class D exhibits a disorder-induced quantum multicritical phenomenon among diffusive thermal metal (DTM), topological superconductor (TS), and conventional localized (AI) phases. To characterize the quantum tricritical point where these three phases meet, we carry out a two-loop renormalization group (RG) analysis for 2D Dirac fermion with random mass in terms of the $\epsilon$-expansion in the spatial dimension $d=2-\epsilon$. In 2D ($\epsilon=0$), the random mass is marginally irrelevant around a clean-limit fixed point of the gapless Dirac fermion, while there exists an IR unstable fixed point at finite disorder strength that corresponds to the tricritical point. The critical exponent, dynamical exponent, and scaling dimension of the (uniform) mass term are evaluated around the tricritical point by the two-loop RG analysis. Using a mapping between an effective theory for the 2D random-mass Dirac fermion and the (1+1)-dimensional Gross-Neveu model, we further deduce the four-loop evaluation of the critical exponent, and the scaling dimension of the uniform mass around the tricritical point. Both the two-loop and four-loop results suggest that criticalities of a AI-DTM transition line as well as TS-DTM transition line are controlled by other saddle-point fixed point(s) at finite uniform mass.