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arXiv:2506.20710 · Physical review. B./Physical review. B

Beyond one-loop calculation: Higher-order effects on Gross-Neveu-Yukawa tensorial criticality

SangEun Han, Igor F. Herbut

Published Oct 7, 2025• 2 citations
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Abstract

We study the Gross-Neveu-Yukawa field theory for the $\mathrm{SO}(N$) symmetric traceless rank-2 tensor order parameter coupled to Majorana fermions using the $\ensuremath{\epsilon}$-expansion around upper critical dimensions of $3+1$ to two loops. Previously we established in the one-loop calculation that the theory does not exhibit a critical fixed point for $N\ensuremath{\ge}4$, but that nevertheless the stable fixed point inevitably emerges at a large number ${N}_{f}$ of fermion flavors. For ${N}_{f}<{N}_{f,c1}\ensuremath{\approx}N/2$, no critical fixed point exists; for ${N}_{f,c1}<{N}_{f}<{N}_{f,c2}$, a real critical fixed point emerges from the complex plane but fails to satisfy the additional stability conditions necessary for a continuous phase transition; and finally only for ${N}_{f}>{N}_{f,c2}\ensuremath{\approx}N$ does the fixed point satisfy the stability conditions as well. In the present work we compute the $O(\ensuremath{\epsilon})$ (two-loop) corrections to the critical flavor numbers ${N}_{f,c1}$ and ${N}_{f,c2}$. Most importantly, we observe a sharp decrease in ${N}_{f,c2}$ from its one-loop value, which brings it closer to the point ${N}_{f}=1$ relevant to the standard Gross-Neveu model. Some three-loop results are also presented and discussed.

DOI
10.1103/n4vy-dl5v

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Evidence for relativistic quantum criticality of antiferromagnetism and superconductivity in two-dimensional Dirac fermion systems has been found in large-scale quantum Monte Carlo simulations. However, the corresponding (2+12+12+1)-dimensional Gross--Neveu--Yukawa field theory with Nf=2{N}_{f}=2Nf​=2 four-component Dirac fermions coupled to two triplets of order parameters does not exhibit a renormalization group fixed point at one-loop order. Instead, the theory only features a critical point for a large or very small fractional number of fermion flavors Nf{N}_{f}Nf​, which disappears for a broad range of flavor numbers around the physical case, Nf=2{N}_{f}=2Nf​=2, due to fixed-point annihilation. This raises the question on how to explain the observed scaling collapse in the quantum Monte Carlo data. Here, we extend previous renormalization-group analyses by studying a generalized model at two-loop order in $4\ensuremath{-}\ensuremath{\epsilon}$ spacetime dimensions. We determine the $\ensuremath{\epsilon}$ correction to the upper and lower critical flavor numbers for the fixed-point annihilation and find that they both go towards the physical case Nf=2{N}_{f}=2Nf​=2. However, this only happens very slowly, such that an extrapolation to $\ensuremath{\epsilon}=1$ still suggests the absence of criticality in 2+12+12+1 dimensions. Thereby, we consolidate the finding that the continuum field theory does not feature a stable renormalization-group fixed point and no true quantum criticality would be expected for the considered system. We briefly discuss a possible reconciliation in terms of a complex conformal field theory. Further, we also explore the fixed-point structure in an enlarged theory space and identify a candidate stable fixed-point solution.