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arxiv: 2606.00657 · v1 · pith:DHH3S6YCnew · submitted 2026-05-30 · ❄️ cond-mat.str-el · cond-mat.stat-mech· hep-th

Supersymmetric quantum criticality with discrete symmetry

Teng-Yue Wang , Shuai Yin This is my paper

Pith reviewed 2026-06-28 18:13 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mechhep-th
keywords Gross-Neveu-Yukawa modelemergent supersymmetryquantum criticalityfunctional renormalization groupdiscrete symmetryWess-Zumino modeldangerously irrelevant operator
0
0 comments X

The pith

For n>3 the Z_n anisotropy is irrelevant at the fixed point of (2+1)D Gross-Neveu-Yukawa theories, recovering an N=2 Wess-Zumino supersymmetric critical point.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies Gross-Neveu-Yukawa models in two-plus-one dimensions in which Dirac fermions couple to a complex order parameter that breaks a discrete Z_n rotational symmetry. It demonstrates that when n exceeds three the anisotropy operator is irrelevant under the renormalization group flow, so the interacting fixed point coincides with the one of the N=2 Wess-Zumino model. In the symmetry-broken phase the same operator, now dangerously irrelevant, generates an extra length scale ξ' whose scaling is extracted by following mass thresholds along the flow. The resulting exponents satisfy the relation ν'/ν = 1 + |y_n|/2. This construction shows how supersymmetry can appear at low energies even when the microscopic symmetry is only discrete.

Core claim

Using the functional renormalization group, the authors show that anisotropic perturbations with Z_n symmetry for n>3 flow to zero at the interacting fixed point, so the critical theory is the N=2 Wess-Zumino supersymmetric model. In the ordered phase the same operator remains dangerously irrelevant and produces a second length scale ξ' in addition to the usual correlation length ξ. By tracking mass thresholds along symmetry-broken trajectories the authors extract the exponents ν' and ν without prior scaling assumptions and confirm that they obey ν'/ν = 1 + |y_n|/2 for the Z_4, Z_5 and Z_6 cases.

What carries the argument

Functional renormalization group flow of the effective potential for the Gross-Neveu-Yukawa model with explicit Z_n anisotropy, used to classify operator relevance and to locate mass thresholds along broken-symmetry trajectories.

If this is right

  • The critical theory is controlled by the N=2 Wess-Zumino model whenever n>3.
  • A second length scale ξ' appears in the ordered phase due to the dangerously irrelevant anisotropy.
  • The exponent ratio obeys ν'/ν = 1 + |y_n|/2 for Z_4, Z_5 and Z_6.
  • Supersymmetry emerges at the critical point even though the microscopic symmetry is only discrete.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar discrete-symmetry models in other dimensions or with different fermion representations may also admit emergent supersymmetry.
  • Neutron scattering or thermodynamic measurements could detect the second length scale ξ' as an additional crossover in the ordered phase.
  • The result supplies a concrete target for sign-problem-free lattice simulations of discrete-symmetry Gross-Neveu-Yukawa models.

Load-bearing premise

The chosen functional renormalization group truncation correctly determines whether the Z_n anisotropy is relevant or irrelevant and accurately tracks the mass thresholds without missing operators or introducing uncontrolled errors.

What would settle it

A lattice Monte Carlo simulation of the Z_4 Gross-Neveu-Yukawa model that measures both correlation lengths and finds the ratio ν'/ν differing from 1 + |y_n|/2 by more than numerical uncertainty would falsify the scaling relation.

Figures

Figures reproduced from arXiv: 2606.00657 by Shuai Yin, Teng-Yue Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Ordered-phase RG flow and mass flows for the [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Extraction of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Ordered-phase RG flows for the [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Ordered-phase RG flows for the [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Extraction of [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Extraction of [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
read the original abstract

Supersymmetry, originally proposed in high-energy physics, can emerge as a remarkable low-energy structure in condensed matter systems. While emergent supersymmetry at quantum critical points is widely discussed in models with continuous symmetries, real materials are constrained by microscopic discrete symmetries. To address this, we investigate (2+1)-dimensional Gross-Neveu-Yukawa theories coupling Dirac fermions to a complex order parameter with discrete $Z_n$ anisotropy. Using the functional renormalization group, we find that for $n>3$, the anisotropic perturbations are irrelevant at the fixed point, yielding a $\mathcal{N}=2$ Wess-Zumino supersymmetric critical point. In the ordered phase, this dangerously irrelevant anisotropy gives rise to a second characteristic length scale, $\xi'$, alongside the usual correlation length, $\xi$. By tracking mass thresholds along symmetry-broken renormalization group trajectories, we extract the exponents $\nu'$ and $\nu$ without imposing prior scaling assumptions. For the $Z_4$, $Z_5$, and $Z_6$ models, our results support the scaling relation $\nu'/\nu = 1+|y_n|/p$ with $p=2$ in the isotropic framework used here.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript studies (2+1)-dimensional Gross-Neveu-Yukawa theories with Dirac fermions coupled to a complex order parameter subject to discrete Z_n anisotropy. Using functional renormalization group (FRG) methods, it claims that for n>3 the Z_n anisotropy is irrelevant at the fixed point, producing an emergent N=2 Wess-Zumino supersymmetric quantum critical point. In the ordered phase the dangerously irrelevant anisotropy generates a second length scale ξ' in addition to the usual correlation length ξ; by tracking mass thresholds along symmetry-broken trajectories the authors extract ν' and ν without prior scaling assumptions and report that the ratio satisfies ν'/ν = 1 + |y_n|/p with p=2 for the Z_4, Z_5 and Z_6 cases.

Significance. If the FRG results prove robust, the work would be significant because it extends the discussion of emergent supersymmetry from continuous-symmetry models to the discrete symmetries that constrain real materials. The identification of a second length scale arising from a dangerously irrelevant operator and the direct numerical extraction of the associated scaling relation constitute concrete, testable predictions for future lattice or experimental studies.

major comments (3)
  1. [FRG truncation and fixed-point analysis] The central claim that the Z_n anisotropy is irrelevant for n>3 (and therefore that the fixed point is the N=2 Wess-Zumino theory) rests entirely on the sign of the eigenvalue y_n obtained from the linearized FRG flow around the putative supersymmetric fixed point. The manuscript provides no convergence test against an enlarged truncation that includes additional four-fermion or higher-derivative operators that could mix into the anisotropy channel; without such a check the sign of y_n remains truncation-dependent.
  2. [Ordered-phase analysis and exponent extraction] The extraction of ν' and ν by tracking mass thresholds along symmetry-broken RG trajectories, and the subsequent verification of the relation ν'/ν = 1 + |y_n|/p, likewise depends on the same truncation. The paper reports support for the relation but does not tabulate the individual numerical values of y_n, ν and ν' or demonstrate regulator independence, so it is impossible to judge whether cutoff artifacts in the broken-phase flow could alter the reported ratio.
  3. [Fixed-point stability] The statement that the scaling relation holds 'in the isotropic framework used here' (abstract) appears to presuppose that the isotropic fixed point remains stable once the anisotropy is declared irrelevant; a quantitative check that the isotropic fixed-point coordinates themselves are stable against the inclusion of the anisotropy operator at higher truncation order is missing.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it stated the precise range of n for which results are presented and the number of independent runs or regulator choices employed.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our work and for the detailed, constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [FRG truncation and fixed-point analysis] The central claim that the Z_n anisotropy is irrelevant for n>3 (and therefore that the fixed point is the N=2 Wess-Zumino theory) rests entirely on the sign of the eigenvalue y_n obtained from the linearized FRG flow around the putative supersymmetric fixed point. The manuscript provides no convergence test against an enlarged truncation that includes additional four-fermion or higher-derivative operators that could mix into the anisotropy channel; without such a check the sign of y_n remains truncation-dependent.

    Authors: We agree that an explicit convergence test with a systematically enlarged truncation would strengthen the result. Our truncation includes all symmetry-allowed operators up to fourth order in the fields within the derivative expansion, which is the standard level used in FRG studies of Gross-Neveu-Yukawa models. Within this controlled truncation the sign of y_n is robustly negative for n>3 and remains unchanged under moderate regulator variations. We will add an explicit discussion of the truncation scheme, its limitations, and the regulator checks in the revised manuscript. revision: partial

  2. Referee: [Ordered-phase analysis and exponent extraction] The extraction of ν' and ν by tracking mass thresholds along symmetry-broken RG trajectories, and the subsequent verification of the relation ν'/ν = 1 + |y_n|/p, likewise depends on the same truncation. The paper reports support for the relation but does not tabulate the individual numerical values of y_n, ν and ν' or demonstrate regulator independence, so it is impossible to judge whether cutoff artifacts in the broken-phase flow could alter the reported ratio.

    Authors: We will include a new table in the revised manuscript that lists the individual numerical values of y_n, ν and ν' for the Z_4, Z_5 and Z_6 cases together with the resulting ratios. We will also add a short paragraph on regulator dependence in the broken-phase flows, confirming that the reported ratio remains stable under the regulator variations we have performed. revision: yes

  3. Referee: [Fixed-point stability] The statement that the scaling relation holds 'in the isotropic framework used here' (abstract) appears to presuppose that the isotropic fixed point remains stable once the anisotropy is declared irrelevant; a quantitative check that the isotropic fixed-point coordinates themselves are stable against the inclusion of the anisotropy operator at higher truncation order is missing.

    Authors: The isotropic fixed point is obtained in the absence of anisotropy; the relevance of the anisotropy operator is then assessed by linearization around that point. To address the stability concern we have performed additional flows in which a small anisotropy is retained from the ultraviolet and the fixed-point coordinates are tracked; they remain consistent with the isotropic values to linear order in the anisotropy. We will clarify this procedure and include the quantitative comparison in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; exponents extracted independently and relation checked post-hoc

full rationale

The abstract states that exponents ν' and ν are extracted by tracking mass thresholds along symmetry-broken RG trajectories without imposing prior scaling assumptions. The relation ν'/ν = 1+|y_n|/p is then reported as supported by those results for specific n, rather than imposed as a fit or definition. No quoted equation reduces the extracted quantities to the same inputs by construction, and the provided text contains no self-citation chains, ansatz smuggling, or renaming of known results that would force the central claims. The derivation chain is therefore self-contained against the paper's own stated procedure.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of a particular FRG truncation and on the assumption that mass thresholds along broken-symmetry flows correctly identify the two correlation lengths; no new particles or forces are introduced.

free parameters (1)
  • FRG truncation order and regulator parameters
    The functional renormalization group calculation necessarily truncates the effective action and chooses a regulator; these choices function as free parameters whose effect on the fixed-point stability and exponent values is not quantified in the abstract.
axioms (2)
  • domain assumption The (2+1)D Gross-Neveu-Yukawa action with Z_n anisotropy is the correct low-energy description of the microscopic lattice model.
    The paper begins from this continuum field theory; its validity is presupposed rather than derived.
  • standard math Standard mathematical properties of the functional renormalization group flow equation hold without additional anomalies or non-perturbative effects.
    The method relies on the existence and regularity of the Wetterich equation and its numerical integration.

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discussion (0)

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Reference graph

Works this paper leans on

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    Flow Equation for the Effective Potential By definition of the effective potential, the projection of Eq. (11) onto a constant bosonic background gives V ∂tUk = 1 2STr Γ(2) k +R k −1 ∂tRk ,(23) whereVis the total volume. The quantum correction to the flow equation for the effective potential has the same structure for different Zn anisotropies, while the ...

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    Flow Equation for the Yukawa Coupling The Yukawa coupling is projected from the fermion- boson three-point vertex: ∂th=− i Nf dγ Tr γ5 δ δ∆ϕ2 δ δψ δ δψ ∂tΓk ∆ϕi=0 p=0 .(25) Evaluating this projection yields ∂th2 =(D−4 +η ϕ + 2ηψ)h2 + 8vDh4 l(F B1) 11 −l (F B2) 11 −16v D √ 2κh4ω221l(F B1B2) 111 . (26) The form of this flow is independent of the explicit ch...

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    Notation for the SSB Phase To perform the expansion, the notation needs to be clarified. The bosonic fields are decomposed as ϕi =ϕ i,0 + ∆ϕi, i= 1,2,(A1) where ϕi,0 denotes the background value and ∆ϕi denotes the fluctuation. At the end of the calculation, we will set all fluctuations and fermion fields to zero. We will also choose coordinates such that...

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