⚛️ high-energy theory

Spontaneous symmetry breaking of $\mathrm{SO}(2N)$ in Gross--Neveu theory from $2+\epsilon$ expansion

This paper employs a $2+\epsilon$ expansion to construct a Fierz-complete renormalizable Lagrangian for the Gross--Neveu model, demonstrating that the symmetric-tensor and adjoint-nematic fixed points remain critical for $N_f > N_{f,c}^{\mathrm{ST}}(N)$ and clarifying the nature of the spontaneous symmetry breaking of $\mathrm{SO}(2N)$ down to $\mathrm{SO}(N)\times\mathrm{SO}(N)$ .

Original authors: Bilal Hawashin, Max Uetrecht

Published 2026-03-17

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CC BY 4.0

Original authors: Bilal Hawashin, Max Uetrecht

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Dance of Electrons

Imagine a crowded dance floor where electrons (the dancers) are moving around. In most materials, these dancers move freely. But in special materials like graphene or twisted bilayer graphene (two sheets of graphene stacked at a tiny angle), the dancers move in a very specific, relativistic way, almost like light particles.

Physicists want to know: What happens if these dancers start interacting strongly? Do they keep dancing freely, or do they suddenly pair up, stop moving, and form a rigid, ordered structure (like a crystal or an insulator)?

This paper investigates a specific type of interaction called the Gross–Neveu model. It's a mathematical playground used to predict how these electrons behave when they get too close to each other.

The Mystery: Hidden Superpowers

For a long time, physicists thought the symmetry (the rules of the dance) of these electrons was limited. They thought the electrons could only follow a specific set of rules, like a standard dance routine.

However, the authors of this paper discovered something surprising: If you tune the interactions just right, the electrons reveal a hidden "superpower."

The Analogy: Imagine a group of dancers who usually only know how to dance in pairs. But if you remove a few specific distractions (suppress certain interactions), they suddenly realize they can dance in a massive, coordinated group formation involving everyone on the floor.
The Science: The paper shows that under these conditions, the symmetry of the system upgrades from a smaller group to a much larger, more complex group called SO(2N). This is a "hidden symmetry" that was there all along but was hard to see until they changed the mathematical perspective (switching from "Dirac fermions" to "Majorana fermions," which is like switching from looking at the dancers' faces to looking at their footwork).

The Three Possible Endings (Fixed Points)

When the electrons interact, the system tries to settle into a stable state. The paper identifies three different ways the system can settle down, which they call "fixed points." Think of these as three different endings to the story:

The "Quantum Anomalous Hall" Ending (The Ising Transition):
- What happens: The dancers suddenly decide to all face the same direction and stop moving freely. They form a specific, ordered pattern.
- The Result: The material becomes an insulator with a special magnetic property. This is a "second-order" transition, meaning it happens smoothly, like water slowly freezing into ice.
The "Symmetric Tensor" Ending:
- What happens: The dancers split into two large groups that dance in harmony with each other but differently from the first group.
- The Result: This was the most controversial part. Previous studies suggested this ending might be possible, but only if there were many copies of the electrons (a high "flavor number"). The authors found that if there are too few copies (like in real-world graphene), this smooth transition breaks down. Instead of a smooth change, the system snaps abruptly.
- The Analogy: Imagine trying to organize a smooth transition from a chaotic mosh pit to a synchronized line dance. If you don't have enough people, the transition fails, and the crowd just violently shoves into the new formation. This is a first-order transition (a sudden jump, not a smooth slide).
The "Adjoint-Nematic" Ending:
- What happens: The dancers break the rules of the dance floor itself, changing the geometry of the space they occupy.
- The Result: Interestingly, the authors found that for the specific case of real-world materials (where there is only one "copy" of the electron system), this ending actually merges with the first ending. It's not a unique path; it's just a different way of describing the same outcome.

The Main Discovery: The "Snap" vs. The "Slide"

The most important finding of this paper is about how the transition happens.

The Old View: Some previous theories suggested that the electrons could smoothly transition into a new, ordered state (a "symmetric tensor" phase) even in standard materials.
The New View: The authors used a mathematical technique called the 2 + $\epsilon$ expansion. Imagine you are trying to understand a 3D object by looking at it in 2D, then slowly adding a tiny bit of 3D depth.
- They found that for the specific case of real materials (like twisted bilayer graphene), the "Symmetric Tensor" transition does not exist as a smooth slide.
- Instead, the system is unstable. It cannot smoothly become that specific ordered state. It has to snap into a different state.
- The Verdict: The transition is likely first-order (abrupt and discontinuous), not second-order (smooth).

Why Does This Matter?

This is crucial for understanding Moiré materials (like twisted bilayer graphene). These materials are the "hot new thing" in physics because they can turn from conductors to insulators easily.

If the transition is smooth, we can tune the material gently to get new properties.
If the transition is abrupt (a "snap"), the material behaves differently, potentially jumping straight into a new state without passing through a critical "tipping point."

The authors also compared their results with other methods (like computer simulations and different mathematical expansions). They found that their results align with recent computer simulations suggesting that the "smooth" transition is actually a "snap."

Summary in a Nutshell

Hidden Symmetry: Electrons in these materials have a hidden, larger symmetry that only shows up under specific conditions.
Three Paths: The system could theoretically settle into three different ordered states.
The Reality Check: For real-world materials (with one set of electrons), two of those paths merge or disappear.
The "Snap": The transition to the ordered state isn't a smooth slide; it's a sudden jump. The "Symmetric Tensor" phase, which some thought was possible, is actually unstable in these materials.

The Takeaway: Nature is often more dramatic than we expect. Instead of a gentle evolution into a new state, the electrons in these special materials seem to prefer a sudden, dramatic change of state.

1. Problem Statement and Motivation

The paper investigates the phase transitions and critical behavior of the Gross–Neveu (GN) model with $N$ copies of two-dimensional Dirac fermions. Recent studies have established that under specific conditions (suppressing certain interactions), this model exhibits an enhanced global symmetry from $SU(N) \times U(1)$ to $SO(2N)$.

The central problem addressed is the nature of the spontaneous symmetry breaking (SSB) of this $SO(2N)$ symmetry in $d=2+1$ dimensions. Previous mean-field and Wilsonian Renormalization Group (RG) analyses in $d=3$ suggested the existence of three distinct critical points:

Gross–Neveu–Ising (GNI): Corresponds to the Quantum Anomalous Hall (QAH) state.
Symmetric-Tensor (ST): Separates the $SO(2N)$ symmetric phase from a phase where $SO(2N) \to SO(N) \times SO(N)$ .
Adjoint-Nematic (N): Involves the breaking of both $SO(2N)$ and Lorentz symmetry.

However, a discrepancy exists in the literature regarding the Symmetric-Tensor (ST) and Adjoint-Nematic (N) fixed points. While Wilsonian RG in $d=3$ suggests these fixed points disappear (implying a first-order transition) for small flavor numbers ( $N_f$ ), $4-\epsilon$ expansion studies suggest a different mechanism involving fixed-point collisions. The authors aim to resolve this by approaching the problem from the lower critical dimension ( $d=2+\epsilon$ ) using a Fierz-complete analysis.

2. Methodology

The authors employ a perturbative Renormalization Group (RG) approach near the lower critical dimension of two spacetime dimensions ( $d = 2 + \epsilon$ ).

Symmetry Manifestation: They reformulate the theory using Majorana fermions. By choosing a specific representation of the Clifford algebra where $\gamma_0^* = -\gamma_0$ and $\gamma_i^* = \gamma_i$ , they decompose the complex Dirac spinor into real Majorana components. This makes the $SO(2N)$ symmetry manifest, treating the fermions as vectors under $SO(2N)$.
Fierz-Complete Lagrangian: To ensure no relevant operators are missed, they construct a Fierz-complete basis of symmetry-allowed four-fermion interactions. For $N_f > 1$ flavors, this results in four linearly independent couplings ( $g_1, g_2, g_3, g_4$ ) corresponding to different tensor structures (scalar, symmetric tensor, antisymmetric tensor, etc.).
Special Case ( $N_f=1$ ): They demonstrate that for a single flavor ( $N_f=1$ ), Fierz identities reduce the four couplings to a single independent interaction, recovering the canonical Gross–Neveu model.
Calculations:
- Computed one-loop $\beta$ -functions for the four couplings.
- Computed the fermion anomalous dimension ( $\eta_\chi$ ) at two-loop order.
- Computed the order parameter anomalous dimensions ( $\eta_\phi$ ) via a susceptibility analysis.
- Used the MaRTIn toolkit for the calculation of Feynman diagrams and extraction of UV divergences.

3. Key Contributions and Results

A. Fixed Point Structure

The analysis identifies three fixed points (FPs) that correspond to the universality classes mentioned in the introduction:

Gross–Neveu–Ising (GNI) FP: Located in a one-loop closed subspace ( $g_2=g_3=g_4=0$ ). It is IR stable for all $N_f > 1$ .
Adjoint-Nematic (N) FP: Located in a subspace where only $g_3 \neq 0$ .
Symmetric-Tensor (ST) FP: Located in the full coupling space but tractable in the large- $N_f$ limit.

B. Behavior at $N_f = 1$

A crucial finding is the behavior of these fixed points as the number of flavors approaches the physical case of $N_f=1$ :

The Adjoint-Nematic (N) fixed point becomes identical to the GNI fixed point at $N_f=1$ .
The Symmetric-Tensor (ST) fixed point merges with the Gaussian (free) fixed point at $N_f=1$ .
This contrasts with some $d=3$ Wilsonian RG results where the Nematic FP was found to merge with the Gaussian point; here, the authors find it merges with the interacting GNI point.

C. Critical Flavor Number ( $N_{f,c}$ ) and Susceptibility

The authors investigate the stability of the Symmetric-Tensor (ST) transition by analyzing the divergence of the susceptibility. A true second-order phase transition requires a diverging susceptibility ( $\eta_\phi < 2$ ).

They find that the ST fixed point implies a diverging susceptibility only if $N_f > N_{f,c}^{ST}(N)$ .
Below this critical flavor number, the susceptibility remains finite, suggesting the transition becomes first-order.
The critical flavor number is estimated as:
$N_{f,c}^{ST}(N) \approx 0.56 + 1.48 N + O(\epsilon)$
- For Graphene ( $N=4$ ): $N_{f,c} \approx 6.3$ .
- For Twisted Bilayer Graphene ( $N=8$ ): $N_{f,c} \approx 12.5$ .

D. Interpolation with $4-\epsilon$ Results

To improve the estimate for $d=2+1$ ( $\epsilon=1$ ), the authors perform a two-sided Padé interpolation between their $2+\epsilon$ result and the three-loop $4-\epsilon$ results from the Gross–Neveu–Yukawa theory (Ref. [26]).

Interpolated Results:
- $N=4$ : $N_{f,c} \approx 8.9$ .
- $N=8$ : $N_{f,c} \approx 16.1$ .
These values are significantly higher than the $N_f=1$ physical case, reinforcing the conclusion that the canonical Gross–Neveu model ( $N_f=1$ ) does not exhibit a second-order Symmetric-Tensor transition.

4. Significance and Implications

Resolution of Mechanism Discrepancy: The paper clarifies that the absence of the ST critical point in $d=2+1$ is consistent across different methods (Wilsonian RG and $\epsilon$ -expansion), though the mechanisms appear different (fixed-point collision vs. loss of criticality/susceptibility divergence).
First-Order Transition Confirmation: The results strongly support the interpretation that the transition to the $SO(N) \times SO(N)$ broken phase in the canonical Gross–Neveu model ( $N_f=1$ ) is first-order, not second-order. This aligns with recent Quantum Monte Carlo simulations (Ref. [46]) which found weak first-order transitions.
Universality Classes: The work confirms that the canonical Gross–Neveu model features a robust second-order transition only in the Gross–Neveu–Ising (QAH) channel.
Material Relevance: For physical systems like twisted bilayer graphene ( $N=8$ ), the critical flavor number required for a second-order ST transition ( $N_f \approx 16$ ) is far beyond the physical $N_f=1$ . Thus, the observed phase transitions in these materials are likely governed by the GNI universality class or first-order dynamics, rather than the symmetric-tensor criticality.

Conclusion

Hawashin and Uetrecht provide a rigorous perturbative analysis from the lower critical dimension, constructing a complete renormalizable Lagrangian. They demonstrate that while the $SO(2N)$ symmetry allows for multiple fixed points, the specific physical case of $N_f=1$ eliminates the Symmetric-Tensor and Adjoint-Nematic criticalities as second-order transitions. The work bridges the gap between lower and upper critical dimension analyses, confirming that the canonical Gross–Neveu model is dominated by the Gross–Neveu–Ising universality class.