Quantum multicriticality and emergent symmetry in Dirac systems with two order parameters at three-loop order

TL;DR

This paper investigates quantum multicritical behavior in Dirac systems with two order parameters using three-loop renormalization group analysis, revealing stability conditions, critical exponents, and the challenges of extrapolation to physical dimensions.

Contribution

It provides a detailed three-loop RG analysis of multicritical points in Dirac systems with two order parameters, including stability criteria and critical exponents, and discusses the implications for physical systems.

Findings

Stability of multicritical points depends on the number of Dirac fermion flavors.

Critical exponents for $SO(4)$ and $SO(5)$ models are computed up to third order in $$.

A critical flavor number $N_c^> \approx; 16.83$ is estimated for the $SO(4) \imes; SO(3)$ model.

Abstract

Two-dimensional materials with interacting Dirac excitations can host quantum multicritical behavior near the phase boundaries of the semimetallic and two-ordered phases. We study such behavior in Gross--Neveu--Yukawa field theories where $N_f$ flavors of Dirac fermions are coupled to two order-parameter fields with $SO(N_A)$ and $SO(N_B)$ symmetry, respectively. To that end, we employ the perturbative renormalization group up to three-loop order in $4-\epsilon$ spacetime dimensions. We distinguish two key scenarios: (i) The two orders are compatible as characterized by anticommuting mass terms, and (ii) the orders are incompatible. For the first case, we explore the stability of a quantum multicritical point with emergent $SO(N_A\!+\!N_B)$ symmetry. We find that the stability is controlled by increasing the number of Dirac fermion flavors. Moreover, we extract the series expansion of the leading critical exponents for the chiral $SO(4)$ and $SO(5)$ models up to third order in $\epsilon$. Notably, we find a tendency towards rapidly growing expansion coefficients at higher orders, rendering an extrapolation to $\epsilon=1$ difficult. For the second scenario, we study a model with $SO(4) \simeq SO(3) \times SO(3)$ symmetry, which was recently suggested to describe criticality of antiferromagnetism and superconductivity in Dirac systems. However, it was also argued that a physically admissible renormalization-group fixed point only exists for $N_f$ above a critical number $N_{c}^>$. We determine the corresponding series expansion at three-loop order as $N_{c}^>\approx 16.83-7.14\epsilon-7.12\epsilon^2$. This suggests that the physical choice of $N_f=2$ may be a borderline case, where true criticality and pseudocriticality, as induced by fixed-point annihilation, are extremely challenging to distinguish.