arXiv:2606.00657

Supersymmetric quantum criticality with discrete symmetry

Teng-Yue Wang, Shuai Yin

Published May 30, 2026• 0 citations

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, $ξ'$, alongside the usual correlation length, $ξ$. By tracking mass thresholds along symmetry-broken renormalization group trajectories, we extract the exponents $ν'$ and $ν$ without imposing prior scaling assumptions. For the $Z_4$, $Z_5$, and $Z_6$ models, our results support the scaling relation $ν'/ν= 1+|y_n|/p$ with $p=2$ in the isotropic framework used here.

DOI: 10.48550/arxiv.2606.00657

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arXiv:2606.00657

Supersymmetric quantum criticality with discrete symmetry

Teng-Yue Wang, Shuai Yin

Published May 30, 2026• 0 citations

arXiv HTML PDF

Abstract

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,

ξ'

, alongside the usual correlation length,

ξ

. By tracking mass thresholds along symmetry-broken renormalization group trajectories, we extract the exponents

ν'

and

ν

without imposing prior scaling assumptions. For the

Z_4

Z_5

, and

Z_6

models, our results support the scaling relation

ν'/ν= 1+|y_n|/p

with

p=2

in the isotropic framework used here.

DOI: 10.48550/arxiv.2606.00657

Related papers

Emergence of supersymmetry at a critical point of a lattice model

Sung-Sik Lee

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Bitan Roy, Pallab Goswami, Vladimir Juričić

We analyze emergent quantum multicriticality for strongly interacting, massless Dirac fermions in two spatial dimensions ( $d=2$ ) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break $\text{O}({S}_{1})$ and $\text{O}({S}_{2})$ symmetries in the ordered phase. Performing a renormalization-group analysis by using the $\ensuremath{\epsilon}=(3\ensuremath{-}d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an enlarged $\text{O}({S}_{1}+{S}_{2})$ chiral symmetry. Such a fixed point acts as an exotic quantum multicritical point (MCP), governing the continuous semimetal-insulator as well as insulator-insulator (for example, antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either $\text{O}({S}_{1})$ or $\text{O}({S}_{2})$ chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power-law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher-dimensional Dirac fermions is also outlined.We analyze emergent quantum multicriticality for strongly interacting, massless Dirac fermions in two spatial dimensions ($d=2$) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break $\text{O}({S}_{1})$ and $\text{O}({S}_{2})$ symmetries in the ordered phase. Performing a renormalization-group analysis by using the $\ensuremath{\epsilon}=(3\ensuremath{-}d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an enlarged $\text{O}({S}_{1}+{S}_{2})$ chiral symmetry. Such a fixed point acts as an exotic quantum multicritical point (MCP), governing the continuous semimetal-insulator as well as insulator-insulator (for example, antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either $\text{O}({S}_{1})$ or $\text{O}({S}_{2})$ chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power-law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher-dimensional Dirac fermions is also outlined.

arXiv:1712.05400Physical review. B./Physical review. BMay 11, 201822 citations

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,

ξ'

, alongside the usual correlation length,

ξ

. By tracking mass thresholds along symmetry-broken renormalization group trajectories, we extract the exponents

ν'

and

ν

without imposing prior scaling assumptions. For the

Z_4

Z_5

, and

Z_6

models, our results support the scaling relation

ν'/ν= 1+|y_n|/p

with

p=2

in the isotropic framework used here.

(2+1)

-dimensional lattice model which exhibits an emergent space-time supersymmetry at a quantum critical point. It is shown that there is only one relevant perturbation at the supersymmetric critical point in the

ϵ

expansion and the critical theory is the two copies of the Wess-Zumino theory with four supercharges. Exact critical exponents are predicted.

2+1

dimensions. We also confirm the emergence of supersymmetric field theories at four loops for the chiral Ising and the chiral XY models with

N=1/4

and

N=1/2

fermions, respectively. Furthermore, applications of our results relevant to various quantum transitions in the context of Dirac and Weyl semimetals are discussed, including interaction-induced transitions in graphene and surface states of topological insulators.

We analyze emergent quantum multicriticality for strongly interacting, massless Dirac fermions in two spatial dimensions (

d=2

) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break

\text{O}({S}_{1})

and

\text{O}({S}_{2})

symmetries in the ordered phase. Performing a renormalization-group analysis by using the $\ensuremath{\epsilon}=(3\ensuremath{-}d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an enlarged

\text{O}({S}_{1}+{S}_{2})

chiral symmetry. Such a fixed point acts as an exotic quantum multicritical point (MCP), governing the continuous semimetal-insulator as well as insulator-insulator (for example, antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either

\text{O}({S}_{1})

\text{O}({S}_{2})

chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power-law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher-dimensional Dirac fermions is also outlined.