Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2510.23725 · Physical review. D/Physical review. D.

Spontaneous symmetry breaking of SO ( 2 N ) in Gross-Neveu theory from 2 + ε expansion

Bilal Hawashin, Max Uetrecht

Published Feb 25, 2026• 0 citations
arXivHTMLPDF

Abstract

It was recently established that the paradigmatic Gross-Neveu model with N copies of two-dimensional Dirac fermions features an SO ( 2 N ) symmetry if certain interactions are suppressed. This becomes evident when the theory is rewritten in terms of 2 N copies of two-dimensional Majorana fermions. Mean-field theory for the SO ( 2 N ) model predicts, besides the chiral Ising transition at g c 1 , a second critical point g c 2 where SO ( 2 N ) is broken down to SO ( N ) × SO ( N ) . A subsequent Wilsonian renormalization group analysis directly in d = 3 supports its existence in a generalized theory, where N f copies of the 4 N -component Majorana fermions are introduced. This allows us to track the evolution of a (i) quantum anomalous Hall Gross-Neveu-Ising, (ii) symmetric-tensor, and (iii) adjoint-nematic fixed point separately. However, it turns out that (ii) and (iii) lose their criticality when approaching N f = 1 , suggesting that the transition is first order. In this work, we approach the problem from the lower-critical dimension of two. We construct a Fierz-complete renormalizable Lagrangian, compute the leading order β functions, fermion anomalous dimension, as well as the order parameter anomalous dimensions, and resolve the three universality classes corresponding to (i)–(iii). Before becoming equal to the Gaussian fixed point at N f = 1 , (ii) remains critical for all values of N f > N f , c ST ( N ) ≈ 0.56 + 1.48 N + O ( ε ) , which compares well with the estimate of previous studies. We further find that (iii) becomes equal to (i) when approaching N f = 1 . An instability is, however, only present in the susceptibility corresponding to the Gross-Neveu-Ising order parameter.

DOI
10.1103/7yb8-7pk5

Related papers

Spontaneous breaking of the SO(2N) symmetry in the Gross-Neveu model

SangEun Han, Igor F. Herbut

The canonical Gross-Neveu model for N two-component Dirac fermions in 2+1 dimensions suffers a continuous phase transition at a critical interaction gc1∼1/N at large N , at which its continuous symmetry SO(2N) is preserved and a discrete (Ising) symmetry becomes spontaneously broken. A recent mean-field calculation, however, points to an additional transition at a different critical gc2∼−Ngc1 , at which SO(2N)→SO(N)×SO(N) . To study the latter phase transition we rewrite the Gross-Neveu interaction g(ψ¯ψ)2 in terms of three different quartic terms for the single ( L=1 ) 4N -component real (Majorana) fermion, and then extend the theory to L>1 . This allows us to track the evolution of the fixed points of the renormalization group transformation starting from L≫1 , where one can discern three distinct critical points which correspond to continuous phase transitions into (1) SO(2N) -singlet mass-order-parameter, (2) SO(2N) -symmetric-tensor mass-order-parameters, and (3) SO(2N) -adjoint nematic-order-parameters, down to L=1 value that is relevant to the standard Gross-Neveu model. Below the critical value of Lc(N)≈0.35N for N≫1 only the Gross-Neveu critical point (1) still implies a diverging susceptibility for its corresponding ( SO(2N) -singlet) order parameter, whereas the two new critical points that existed at large L ultimately become equivalent to the Gaussian fixed point at L=1 . We interpret this metamorphosis of the SO(2N) -symmetric-tensor fixed point from critical to spurious as an indication that the transition at gc2 in the original Gross-Neveu model is turned first-order by fluctuations. Published by the American Physical Society 2024

arXiv:2403.09627•Physical review. D/Physical review. D.•May 20, 2024•6 citations

Gross-Neveu-Yukawa SO(2) and SO(3) tensorial criticality

SangEun Han, Shouryya Ray, Igor F. Herbut

We investigate the relativistic SO(2)- and SO(3)-invariant Gross-Neveu-Yukawa field theories for real, rank-two, symmetric, traceless tensor order parameters coupled to ${N}_{\text{f}}$ flavors of two-component Dirac fermions. These field theories arise as an effective description of fractionalized spin-orbital liquids. The two theories are the simplest and special cases of the more general class of field theories with $\mathrm{SO}(N$) symmetric tensor order parameter coupled to Dirac fermions, in which the symmetry is low enough to allow only one and not the usual two quartic self-interaction terms. Using a two-loop renormalization group near the upper critical dimension, we demonstrate that the theory exhibits an exotic critical fixed point for $N=3$ and the concomitant continuous phase transition for any value of ${N}_{\text{f}}$. For $N=2$ the theory is equivalent to the chiral XY model. We discuss the crucial role of the symmetry-allowed sextic self-interactions in the selection of the ground state configuration in the case of SO(3). The universal quantities such as the anomalous dimensions of order parameters and fermions, the correlation length exponent, and the mass gap ratio between order parameter and fermion masses are computed up to ${\ensuremath{\epsilon}}^{2}$ order.

arXiv:2411.16842•Physical review. B./Physical review. B•Mar 12, 2025•3 citations

Fractionalized quantum criticality in spin-orbital liquids from field theory beyond the leading order

Shouryya Ray, Bernhard Ihrig, Daniel Kruti +3 more

Two-dimensional spin-orbital magnets with strong exchange frustration have recently been predicted to facilitate the realization of a quantum critical point in the Gross-Neveu-SO(3) universality class. In contrast to previously known Gross-Neveu-type universality classes, this quantum critical point separates a Dirac semimetal and a long-range-ordered phase, in which the fermion spectrum is only partially gapped out. Here, we characterize the quantum critical behavior of the Gross-Neveu-SO(3) universality class by employing three complementary field-theoretical techniques beyond their leading orders. We compute the correlation-length exponent $\ensuremath{\nu}$, the order-parameter anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, and the fermion anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\psi}}$ using a three-loop $\ensuremath{\epsilon}$ expansion around the upper critical space-time dimension of four, a second-order large-$N$ expansion (with the fermion anomalous dimension obtained even at the third order), as well as a functional renormalization group approach in the improved local potential approximation. For the physically relevant case of $N=3$ flavors of two-component Dirac fermions in $2+1$ space-time dimensions, we obtain the estimates $1/\ensuremath{\nu}=1.03(15), {\ensuremath{\eta}}_{\ensuremath{\phi}}=0.42(7)$, and ${\ensuremath{\eta}}_{\ensuremath{\psi}}=0.180(10)$ from averaging over the results of the different techniques, with the displayed uncertainty representing the degree of consistency among the three methods.

arXiv:2101.10335•Physical review. B./Physical review. B•Apr 30, 2021•20 citations

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

SangEun Han, Igor F. Herbut

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.

arXiv:2506.20710•Physical review. B./Physical review. B•Oct 7, 2025•2 citations

Itinerant quantum multicriticality of two-dimensional Dirac fermions

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.

arXiv:1712.05400•Physical review. B./Physical review. B•May 11, 2018•22 citations

Wilson-Fisher fixed points in the presence of Dirac fermions

Igor F. Herbut

Wilson–Fisher expansion near upper critical dimension has proven to be an invaluable conceptual and computational tool in our understanding of the universal critical behavior in the [Formula: see text] field theories that describe low-energy physics of the canonical models, such as Ising, XY, and Heisenberg. Here, I review its application to a class of the Gross–Neveu–Yukawa (GNY) field theories, which emerge as possible universal description of a number of quantum phase transitions in electronic two-dimensional systems such as graphene and d-wave superconductors. GNY field theories may be viewed as minimal modifications of the [Formula: see text] field theories in which the order parameter is coupled to relativistic Dirac fermions through Yukawa term and which still exhibit critical fixed points in the suitably formulated Wilson–Fisher [Formula: see text]-expansion. I discuss the unified GNY field theory for a set of different symmetry-breaking patterns, with focus on the semimetal-Néel-ordered-Mott insulator quantum phase transition in the half-filled Hubbard model on the honeycomb lattice, for which a comparison between the state-of-the-art [Formula: see text]-expansion, quantum Monte Carlo, large N, and functional renormalization-group calculations can be made.

arXiv:2304.07654•Modern Physics Letters B•Oct 8, 2024•16 citations
It was recently established that the paradigmatic Gross-Neveu model with N copies of two-dimensional Dirac fermions features an SO ( 2 N ) symmetry if certain interactions are suppressed. This becomes evident when the theory is rewritten in terms of 2 N copies of two-dimensional Majorana fermions. Mean-field theory for the SO ( 2 N ) model predicts, besides the chiral Ising transition at g c 1 , a second critical point g c 2 where SO ( 2 N ) is broken down to SO ( N ) × SO ( N ) . A subsequent Wilsonian renormalization group analysis directly in d = 3 supports its existence in a generalized theory, where N f copies of the 4 N -component Majorana fermions are introduced. This allows us to track the evolution of a (i) quantum anomalous Hall Gross-Neveu-Ising, (ii) symmetric-tensor, and (iii) adjoint-nematic fixed point separately. However, it turns out that (ii) and (iii) lose their criticality when approaching N f = 1 , suggesting that the transition is first order. In this work, we approach the problem from the lower-critical dimension of two. We construct a Fierz-complete renormalizable Lagrangian, compute the leading order β functions, fermion anomalous dimension, as well as the order parameter anomalous dimensions, and resolve the three universality classes corresponding to (i)–(iii). Before becoming equal to the Gaussian fixed point at N f = 1 , (ii) remains critical for all values of N f > N f , c ST ( N ) ≈ 0.56 + 1.48 N + O ( ε ) , which compares well with the estimate of previous studies. We further find that (iii) becomes equal to (i) when approaching N f = 1 . An instability is, however, only present in the susceptibility corresponding to the Gross-Neveu-Ising order parameter.
The canonical Gross-Neveu model for N two-component Dirac fermions in 2+1 dimensions suffers a continuous phase transition at a critical interaction gc1∼1/N at large N , at which its continuous symmetry SO(2N) is preserved and a discrete (Ising) symmetry becomes spontaneously broken. A recent mean-field calculation, however, points to an additional transition at a different critical gc2∼−Ngc1 , at which SO(2N)→SO(N)×SO(N) . To study the latter phase transition we rewrite the Gross-Neveu interaction g(ψ¯ψ)2 in terms of three different quartic terms for the single ( L=1 ) 4N -component real (Majorana) fermion, and then extend the theory to L>1 . This allows us to track the evolution of the fixed points of the renormalization group transformation starting from L≫1 , where one can discern three distinct critical points which correspond to continuous phase transitions into (1) SO(2N) -singlet mass-order-parameter, (2) SO(2N) -symmetric-tensor mass-order-parameters, and (3) SO(2N) -adjoint nematic-order-parameters, down to L=1 value that is relevant to the standard Gross-Neveu model. Below the critical value of Lc(N)≈0.35N for N≫1 only the Gross-Neveu critical point (1) still implies a diverging susceptibility for its corresponding ( SO(2N) -singlet) order parameter, whereas the two new critical points that existed at large L ultimately become equivalent to the Gaussian fixed point at L=1 . We interpret this metamorphosis of the SO(2N) -symmetric-tensor fixed point from critical to spurious as an indication that the transition at gc2 in the original Gross-Neveu model is turned first-order by fluctuations. Published by the American Physical Society 2024
We investigate the relativistic SO(2)- and SO(3)-invariant Gross-Neveu-Yukawa field theories for real, rank-two, symmetric, traceless tensor order parameters coupled to Nf{N}_{\text{f}}Nf​ flavors of two-component Dirac fermions. These field theories arise as an effective description of fractionalized spin-orbital liquids. The two theories are the simplest and special cases of the more general class of field theories with SO(N\mathrm{SO}(NSO(N) symmetric tensor order parameter coupled to Dirac fermions, in which the symmetry is low enough to allow only one and not the usual two quartic self-interaction terms. Using a two-loop renormalization group near the upper critical dimension, we demonstrate that the theory exhibits an exotic critical fixed point for N=3N=3N=3 and the concomitant continuous phase transition for any value of Nf{N}_{\text{f}}Nf​. For N=2N=2N=2 the theory is equivalent to the chiral XY model. We discuss the crucial role of the symmetry-allowed sextic self-interactions in the selection of the ground state configuration in the case of SO(3). The universal quantities such as the anomalous dimensions of order parameters and fermions, the correlation length exponent, and the mass gap ratio between order parameter and fermion masses are computed up to ${\ensuremath{\epsilon}}^{2}$ order.
Two-dimensional spin-orbital magnets with strong exchange frustration have recently been predicted to facilitate the realization of a quantum critical point in the Gross-Neveu-SO(3) universality class. In contrast to previously known Gross-Neveu-type universality classes, this quantum critical point separates a Dirac semimetal and a long-range-ordered phase, in which the fermion spectrum is only partially gapped out. Here, we characterize the quantum critical behavior of the Gross-Neveu-SO(3) universality class by employing three complementary field-theoretical techniques beyond their leading orders. We compute the correlation-length exponent $\ensuremath{\nu}$, the order-parameter anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, and the fermion anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\psi}}$ using a three-loop $\ensuremath{\epsilon}$ expansion around the upper critical space-time dimension of four, a second-order large-NNN expansion (with the fermion anomalous dimension obtained even at the third order), as well as a functional renormalization group approach in the improved local potential approximation. For the physically relevant case of N=3N=3N=3 flavors of two-component Dirac fermions in 2+12+12+1 space-time dimensions, we obtain the estimates $1/\ensuremath{\nu}=1.03(15), {\ensuremath{\eta}}_{\ensuremath{\phi}}=0.42(7)$, and ${\ensuremath{\eta}}_{\ensuremath{\psi}}=0.180(10)$ from averaging over the results of the different techniques, with the displayed uncertainty representing the degree of consistency among the three methods.
We study the Gross-Neveu-Yukawa field theory for the SO(N\mathrm{SO}(NSO(N) symmetric traceless rank-2 tensor order parameter coupled to Majorana fermions using the $\ensuremath{\epsilon}$-expansion around upper critical dimensions of 3+13+13+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 Nf{N}_{f}Nf​ of fermion flavors. For ${N}_{f}<{N}_{f,c1}\ensuremath{\approx}N/2$, no critical fixed point exists; for Nf,c1<Nf<Nf,c2{N}_{f,c1}<{N}_{f}<{N}_{f,c2}Nf,c1​<Nf​<Nf,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 Nf,c1{N}_{f,c1}Nf,c1​ and Nf,c2{N}_{f,c2}Nf,c2​. Most importantly, we observe a sharp decrease in Nf,c2{N}_{f,c2}Nf,c2​ from its one-loop value, which brings it closer to the point Nf=1{N}_{f}=1Nf​=1 relevant to the standard Gross-Neveu model. Some three-loop results are also presented and discussed.
We analyze emergent quantum multicriticality for strongly interacting, massless Dirac fermions in two spatial dimensions (d=2d=2d=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 O(S1)\text{O}({S}_{1})O(S1​) and O(S2)\text{O}({S}_{2})O(S2​) 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 O(S1+S2)\text{O}({S}_{1}+{S}_{2})O(S1​+S2​) 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 O(S1)\text{O}({S}_{1})O(S1​) or O(S2)\text{O}({S}_{2})O(S2​) 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.
Wilson–Fisher expansion near upper critical dimension has proven to be an invaluable conceptual and computational tool in our understanding of the universal critical behavior in the [Formula: see text] field theories that describe low-energy physics of the canonical models, such as Ising, XY, and Heisenberg. Here, I review its application to a class of the Gross–Neveu–Yukawa (GNY) field theories, which emerge as possible universal description of a number of quantum phase transitions in electronic two-dimensional systems such as graphene and d-wave superconductors. GNY field theories may be viewed as minimal modifications of the [Formula: see text] field theories in which the order parameter is coupled to relativistic Dirac fermions through Yukawa term and which still exhibit critical fixed points in the suitably formulated Wilson–Fisher [Formula: see text]-expansion. I discuss the unified GNY field theory for a set of different symmetry-breaking patterns, with focus on the semimetal-Néel-ordered-Mott insulator quantum phase transition in the half-filled Hubbard model on the honeycomb lattice, for which a comparison between the state-of-the-art [Formula: see text]-expansion, quantum Monte Carlo, large N, and functional renormalization-group calculations can be made.