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Classifying GNY-like models

Matthew S. Mitchell, David Poland

TL;DR

This work provides a systematic group-theory–driven classification of (2+1)d GNY-like fermion-scalar models with N_D Dirac fermions and an O(M) order-parameter symmetry, allowing a controlled analysis in the $4-\epsilon$ expansion. It develops a detailed framework to determine the full symmetry group G from Yukawa matrices and their commutants, and enumerates six universality classes, including two new families: quarter GNY and orthogonal Heisenberg. By computing two-loop beta functions, the authors identify perturbative fixed points for various M and N_D, uncovering a novel M=3 fixed point—the orthogonal Heisenberg CFT—and providing explicit scaling dimensions. The results offer concrete targets for conformal bootstrap investigations and lay a pathway to explore richer symmetry-breaking patterns, subgroups, and Majorana extensions in 2+1 dimensions with potential condensed-matter realizations.

Abstract

We perform a systematic classification of (2+1)d Gross--Neveu--Yukawa-like models built out of one or more 4-component Dirac fermions and $M$ scalar fields, which preserve an O($M$) symmetry rotating the scalars. We then identify the perturbative fixed points of these models in the $4-ε$ expansion. Our classification highlights several targets for the conformal bootstrap and reveals a new fixed point with $M=3$, which we call the "orthogonal Heisenberg" CFT.

Classifying GNY-like models

TL;DR

This work provides a systematic group-theory–driven classification of (2+1)d GNY-like fermion-scalar models with N_D Dirac fermions and an O(M) order-parameter symmetry, allowing a controlled analysis in the expansion. It develops a detailed framework to determine the full symmetry group G from Yukawa matrices and their commutants, and enumerates six universality classes, including two new families: quarter GNY and orthogonal Heisenberg. By computing two-loop beta functions, the authors identify perturbative fixed points for various M and N_D, uncovering a novel M=3 fixed point—the orthogonal Heisenberg CFT—and providing explicit scaling dimensions. The results offer concrete targets for conformal bootstrap investigations and lay a pathway to explore richer symmetry-breaking patterns, subgroups, and Majorana extensions in 2+1 dimensions with potential condensed-matter realizations.

Abstract

We perform a systematic classification of (2+1)d Gross--Neveu--Yukawa-like models built out of one or more 4-component Dirac fermions and scalar fields, which preserve an O() symmetry rotating the scalars. We then identify the perturbative fixed points of these models in the expansion. Our classification highlights several targets for the conformal bootstrap and reveals a new fixed point with , which we call the "orthogonal Heisenberg" CFT.

Paper Structure

This paper contains 14 sections, 9 theorems, 57 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction and motivation
  2. Representation theory
  3. Review of spinors in (2+1)d
  4. Global symmetry
  5. Orthogonal subgroups
  6. Commutants
  7. List of GNY-like models
  8. Perturbative fixed points
  9. GNY, quarter GNY, and chiral Ising
  10. Chiral XY
  11. Chiral Heisenberg and a new $M=3$ CFT
  12. Spontaneous symmetry breaking and CPT
  13. Discussion and future directions
  14. Real representation theory of $\mathrm{Spin}(M)$

Key Result

Theorem 1

$G$ is the subgroup of $\mathrm{O}(4N_D) \times \mathrm{O}(M)$ that preserves $1_{N_D \times N_D} \otimes R_m$.

Figures (1)

  • Figure 1: $\beta$-functions for the chiral Heisenberg and orthogonal Heisenberg universality classes at $N_D=1$ and $\epsilon=0.1$. Perturbative fixed points with $g \ne 0$ are indicated in red (note that the free theory and Wilson--Fisher fixed points are also visible).

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • proof
  • ...and 6 more