arXiv:1711.11042 · Physical review. B./Physical review. B

Compatible orders and fermion-induced emergent symmetry in Dirac systems

Lukas Janssen, Igor F. Herbut, Michael M. Scherer

Published Jan 29, 2018• 31 citations

Abstract

We study the quantum multicritical point in a (2+1)-dimensional Dirac system between the semimetallic phase and two ordered phases that are characterized by anticommuting mass terms with $\mathrm{O}({N}_{1})$ and $\mathrm{O}({N}_{2})$ symmetries, respectively. Using $\ensuremath{\epsilon}$ expansion around the upper critical space-time dimension of four, we demonstrate the existence of a stable renormalization-group fixed point, enabling a direct and continuous transition between the two ordered phases directly at the multicritical point. This point is found to be characterized by an emergent $\mathrm{O}({N}_{1}+{N}_{2})$ symmetry for arbitrary values of ${N}_{1}$ and ${N}_{2}$ and fermion flavor numbers ${N}_{\mathrm{f}}$ as long as the corresponding representation of the Clifford algebra exists. Small $\mathrm{O}(N)$-breaking perturbations near the chiral $\mathrm{O}(N)$ fixed point are therefore irrelevant. This result can be traced back to the presence of gapless Dirac degrees of freedom at criticality, and it is in clear contrast to the purely bosonic $\mathrm{O}(N)$ fixed point, which is stable only when $N<3$. As a by-product, we obtain predictions for the critical behavior of the chiral $\mathrm{O}(N)$ universality classes for arbitrary $N$ and fermion flavor number ${N}_{\mathrm{f}}$. Implications for critical Weyl and Dirac systems in 3+1 dimensions are also briefly discussed.

DOI: 10.1103/physrevb.97.041117

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Itinerant quantum multicriticality of two-dimensional Dirac fermions

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

SO(4) multicriticality of two-dimensional Dirac fermions

Igor F. Herbut, Michael M. Scherer

arXiv:2206.04073Physical review. B./Physical review. BSep 21, 20227 citations

Absence of SO(4) quantum criticality in Dirac semimetals at two-loop order

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Simulation of fermionic and bosonic critical points with emergent SO(5) symmetry

Toshihiro Sato, Zhenjiu Wang, Yuhai Liu +4 more

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Fermionic multicriticality near Kekulé valence-bond ordering on a honeycomb lattice

Bitan Roy, Vladimir Juričić

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Competition of density waves and quantum multicritical behavior in Dirac materials from functional renormalization

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arXiv:1711.11042 · Physical review. B./Physical review. B

Compatible orders and fermion-induced emergent symmetry in Dirac systems

Lukas Janssen, Igor F. Herbut, Michael M. Scherer

Published Jan 29, 2018• 31 citations

arXiv HTML PDF

Abstract

We study the quantum multicritical point in a (2+1)-dimensional Dirac system between the semimetallic phase and two ordered phases that are characterized by anticommuting mass terms with

\mathrm{O}({N}_{1})

and

\mathrm{O}({N}_{2})

symmetries, respectively. Using

\ensuremath{\epsilon}

expansion around the upper critical space-time dimension of four, we demonstrate the existence of a stable renormalization-group fixed point, enabling a direct and continuous transition between the two ordered phases directly at the multicritical point. This point is found to be characterized by an emergent

\mathrm{O}({N}_{1}+{N}_{2})

symmetry for arbitrary values of

{N}_{1}

and

{N}_{2}

and fermion flavor numbers

{N}_{\mathrm{f}}

as long as the corresponding representation of the Clifford algebra exists. Small

\mathrm{O}(N)

-breaking perturbations near the chiral

\mathrm{O}(N)

fixed point are therefore irrelevant. This result can be traced back to the presence of gapless Dirac degrees of freedom at criticality, and it is in clear contrast to the purely bosonic

\mathrm{O}(N)

fixed point, which is stable only when

N<3

. As a by-product, we obtain predictions for the critical behavior of the chiral

\mathrm{O}(N)

universality classes for arbitrary

N

and fermion flavor number

{N}_{\mathrm{f}}

. Implications for critical Weyl and Dirac systems in 3+1 dimensions are also briefly discussed.

DOI: 10.1103/physrevb.97.041117

Related papers

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.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

SO(4) multicriticality of two-dimensional Dirac fermions

Igor F. Herbut, Michael M. Scherer

We study quantum multicritical behavior in a $(2+1)\text{\ensuremath{-}}\mathrm{dimensional}$ Gross-Neveu-Yukawa field theory with eight-component Dirac fermions coupled to two triplets of order parameters that act as Dirac masses and transform as $(1,0)+(0,1)$ representation under the $\mathrm{S}\mathrm{O}(4)\ensuremath{\simeq} \mathrm{S}\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)$ symmetry group. This field theory is relevant to spin-1/2 fermions on honeycomb or $\ensuremath{\pi}$-flux lattices, for example, near the transition points between an $s$ -wave superconductor and a charge-density wave, on one side, and N\'eel order, on the other. Two triplets of such order parameters always allow for a common pair of two other order parameters that would complete them to the maximal set of compatible (anticommuting) orders of five. We first derive a unitary transformation in the Nambu (particle-hole) space which maps any two such triplets, possibly containing some superconducting orders, onto purely insulating order parameters. This allows one to consider a universal SO(4) Gross-Neveu-Yukawa description of the multicriticality without any Nambu doubling. We then proceed to derive the renormalization-group flow of the coupling constants at one-loop order in $4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions, allowing also a more general set of order parameters transforming under $\mathrm{S}\mathrm{O}({n}_{a})\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{O}({n}_{b})$. While for ${n}_{a}={n}_{b}>2$ in the bosonic sector and with fermions decoupled there is a stable fixed point of the flow, the Yukawa coupling to fermions quickly leads to its elimination by a generic fixed-point collision in the relevant range of fermion flavor numbers ${N}_{f}$ . This suggests the replacement of the critical behavior by a runaway flow in the physical case ${n}_{a}={n}_{b}=3$ . The structure of the renormalization group flow at ${n}_{a}\ensuremath{\ne}{n}_{b}$ is also discussed, and some nonperturbative arguments in favor of the stability of the decoupled critical point when ${n}_{a}=3$ and ${n}_{b}=1$ in $D=2+1$ are provided.We study quantum multicritical behavior in a $(2+1)\text{\ensuremath{-}}\mathrm{dimensional}$ Gross-Neveu-Yukawa field theory with eight-component Dirac fermions coupled to two triplets of order parameters that act as Dirac masses and transform as $(1,0)+(0,1)$ representation under the $\mathrm{S}\mathrm{O}(4)\ensuremath{\simeq} \mathrm{S}\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)$ symmetry group. This field theory is relevant to spin-1/2 fermions on honeycomb or $\ensuremath{\pi}$-flux lattices, for example, near the transition points between an $s$-wave superconductor and a charge-density wave, on one side, and N\'eel order, on the other. Two triplets of such order parameters always allow for a common pair of two other order parameters that would complete them to the maximal set of compatible (anticommuting) orders of five. We first derive a unitary transformation in the Nambu (particle-hole) space which maps any two such triplets, possibly containing some superconducting orders, onto purely insulating order parameters. This allows one to consider a universal SO(4) Gross-Neveu-Yukawa description of the multicriticality without any Nambu doubling. We then proceed to derive the renormalization-group flow of the coupling constants at one-loop order in $4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions, allowing also a more general set of order parameters transforming under $\mathrm{S}\mathrm{O}({n}_{a})\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{O}({n}_{b})$. While for ${n}_{a}={n}_{b}>2$ in the bosonic sector and with fermions decoupled there is a stable fixed point of the flow, the Yukawa coupling to fermions quickly leads to its elimination by a generic fixed-point collision in the relevant range of fermion flavor numbers ${N}_{f}$. This suggests the replacement of the critical behavior by a runaway flow in the physical case ${n}_{a}={n}_{b}=3$. The structure of the renormalization group flow at ${n}_{a}\ensuremath{\ne}{n}_{b}$ is also discussed, and some nonperturbative arguments in favor of the stability of the decoupled critical point when ${n}_{a}=3$ and ${n}_{b}=1$ in $D=2+1$ are provided.

arXiv:2206.04073Physical review. B./Physical review. BSep 21, 20227 citations

Absence of SO(4) quantum criticality in Dirac semimetals at two-loop order

Max Uetrecht, Igor F. Herbut, Emmanuel Stamou +1 more

Evidence for relativistic quantum criticality of antiferromagnetism and superconductivity in two-dimensional Dirac fermion systems has been found in large-scale quantum Monte Carlo simulations. However, the corresponding ( $2+1$ )-dimensional Gross--Neveu--Yukawa field theory with ${N}_{f}=2$ four-component Dirac fermions coupled to two triplets of order parameters does not exhibit a renormalization group fixed point at one-loop order. Instead, the theory only features a critical point for a large or very small fractional number of fermion flavors ${N}_{f}$ , which disappears for a broad range of flavor numbers around the physical case, ${N}_{f}=2$ , due to fixed-point annihilation. This raises the question on how to explain the observed scaling collapse in the quantum Monte Carlo data. Here, we extend previous renormalization-group analyses by studying a generalized model at two-loop order in $4\ensuremath{-}\ensuremath{\epsilon}$ spacetime dimensions. We determine the $\ensuremath{\epsilon}$ correction to the upper and lower critical flavor numbers for the fixed-point annihilation and find that they both go towards the physical case ${N}_{f}=2$ . However, this only happens very slowly, such that an extrapolation to $\ensuremath{\epsilon}=1$ still suggests the absence of criticality in $2+1$ dimensions. Thereby, we consolidate the finding that the continuum field theory does not feature a stable renormalization-group fixed point and no true quantum criticality would be expected for the considered system. We briefly discuss a possible reconciliation in terms of a complex conformal field theory. Further, we also explore the fixed-point structure in an enlarged theory space and identify a candidate stable fixed-point solution.Evidence for relativistic quantum criticality of antiferromagnetism and superconductivity in two-dimensional Dirac fermion systems has been found in large-scale quantum Monte Carlo simulations. However, the corresponding ($2+1$)-dimensional Gross--Neveu--Yukawa field theory with ${N}_{f}=2$ four-component Dirac fermions coupled to two triplets of order parameters does not exhibit a renormalization group fixed point at one-loop order. Instead, the theory only features a critical point for a large or very small fractional number of fermion flavors ${N}_{f}$, which disappears for a broad range of flavor numbers around the physical case, ${N}_{f}=2$, due to fixed-point annihilation. This raises the question on how to explain the observed scaling collapse in the quantum Monte Carlo data. Here, we extend previous renormalization-group analyses by studying a generalized model at two-loop order in $4\ensuremath{-}\ensuremath{\epsilon}$ spacetime dimensions. We determine the $\ensuremath{\epsilon}$ correction to the upper and lower critical flavor numbers for the fixed-point annihilation and find that they both go towards the physical case ${N}_{f}=2$. However, this only happens very slowly, such that an extrapolation to $\ensuremath{\epsilon}=1$ still suggests the absence of criticality in $2+1$ dimensions. Thereby, we consolidate the finding that the continuum field theory does not feature a stable renormalization-group fixed point and no true quantum criticality would be expected for the considered system. We briefly discuss a possible reconciliation in terms of a complex conformal field theory. Further, we also explore the fixed-point structure in an enlarged theory space and identify a candidate stable fixed-point solution.

arXiv:2308.12426Physical review. B./Physical review. BDec 12, 20236 citations

Simulation of fermionic and bosonic critical points with emergent SO(5) symmetry

Toshihiro Sato, Zhenjiu Wang, Yuhai Liu +4 more

We introduce a model of Dirac fermions in $2+1$ dimensions with a semimetallic, a quantum spin-Hall insulating (QSHI), and an $s$ -wave superconducting (SSC) phase. The phase diagram features a multicritical point at which all three phases meet as well as a QSHI-SSC deconfined critical point. The QSHI and SSC orders correspond to mutually anticommuting mass terms of the Dirac Hamiltonian. Based on this algebraic property, SO(5) symmetric field theories have been put forward to describe both types of critical points. Using quantum Monte Carlo simulations, we directly study the operator that rotates between QSHI and SSC states. The results suggest that it commutes with the low-energy effective Hamiltonian at criticality but has a gap in the ordered phases. This implies an emergent SO(5) symmetry at both the multicritical and the deconfined critical points.We introduce a model of Dirac fermions in $2+1$ dimensions with a semimetallic, a quantum spin-Hall insulating (QSHI), and an $s$-wave superconducting (SSC) phase. The phase diagram features a multicritical point at which all three phases meet as well as a QSHI-SSC deconfined critical point. The QSHI and SSC orders correspond to mutually anticommuting mass terms of the Dirac Hamiltonian. Based on this algebraic property, SO(5) symmetric field theories have been put forward to describe both types of critical points. Using quantum Monte Carlo simulations, we directly study the operator that rotates between QSHI and SSC states. The results suggest that it commutes with the low-energy effective Hamiltonian at criticality but has a gap in the ordered phases. This implies an emergent SO(5) symmetry at both the multicritical and the deconfined critical points.

arXiv:2212.11395Physical review. B./Physical review. BSep 21, 20233 citations

Fermionic multicriticality near Kekulé valence-bond ordering on a honeycomb lattice

Bitan Roy, Vladimir Juričić

We analyze the possibility of emergent quantum multicritical points (MCPs) with enlarged chiral symmetry, when strongly interacting gapless Dirac fermions acquire a comparable propensity toward the nucleation of Kekul\'e valence-bond solid (KVBS) and charge-density-wave ( ${N}_{b}=1$ ) or $s$ -wave pairing ( ${N}_{b}=2$ ) or antiferromagnet ( ${N}_{b}=3$ ) in a honeycomb lattice, where ${N}_{b}$ counts the number of bosonic order-parameter components. Besides the cubic terms present in the order-parameter description of KVBS due to the breaking of a discrete ${Z}_{3}$ symmetry, quantum fluctuations generate new cubic vertices near the high-symmetry MCPs. All cubic terms are strongly relevant at the bare level near three spatial dimensions, about which we perform a leading-order renormalization group analysis of coupled Gross-Neveu-Yukawa field theory. We show that due to nontrivial Yukawa interactions among gapless bosonic and fermionic degrees of freedom, all cubic terms ultimately become irrelevant at an $O(2+{N}_{b})$ symmetric MCP, at leas near two spatial dimensions, where ${N}_{b}=1,2,3$ . Therefore, MCPs with an enlarged $O(2+{N}_{b})$ symmetry near KVBS ordering are stable.We analyze the possibility of emergent quantum multicritical points (MCPs) with enlarged chiral symmetry, when strongly interacting gapless Dirac fermions acquire a comparable propensity toward the nucleation of Kekul\'e valence-bond solid (KVBS) and charge-density-wave (${N}_{b}=1$) or $s$-wave pairing (${N}_{b}=2$) or antiferromagnet (${N}_{b}=3$) in a honeycomb lattice, where ${N}_{b}$ counts the number of bosonic order-parameter components. Besides the cubic terms present in the order-parameter description of KVBS due to the breaking of a discrete ${Z}_{3}$ symmetry, quantum fluctuations generate new cubic vertices near the high-symmetry MCPs. All cubic terms are strongly relevant at the bare level near three spatial dimensions, about which we perform a leading-order renormalization group analysis of coupled Gross-Neveu-Yukawa field theory. We show that due to nontrivial Yukawa interactions among gapless bosonic and fermionic degrees of freedom, all cubic terms ultimately become irrelevant at an $O(2+{N}_{b})$ symmetric MCP, at leas near two spatial dimensions, where ${N}_{b}=1,2,3$. Therefore, MCPs with an enlarged $O(2+{N}_{b})$ symmetry near KVBS ordering are stable.

arXiv:1902.08616Physical review. B./Physical review. BJun 5, 201917 citations

Competition of density waves and quantum multicritical behavior in Dirac materials from functional renormalization

Laura Classen, Igor F. Herbut, Lukas Janssen +1 more

We study the competition of spin- and charge-density waves and their quantum multicritical behavior for the semimetal-insulator transitions of low-dimensional Dirac fermions. Employing the effective Gross-Neveu-Yukawa theory with two order parameters as a model for graphene and a growing number of other two-dimensional Dirac materials allows us to describe the physics near the multicritical point at which the semimetallic and the spin- and charge-density-wave phases meet. With the help of a functional renormalization group approach, we are able to reveal a complex structure of fixed points, the stability properties of which decisively depend on the number of Dirac fermions ${N}_{f}$ . We give estimates for the critical exponents and observe crucial quantitative corrections as compared to the previous first-order $\ensuremath{\epsilon}$ expansion. For small ${N}_{f}$ , the universal behavior near the multicritical point is determined by the chiral Heisenberg universality class supplemented by a decoupled, purely bosonic, Ising sector. At large ${N}_{f}$ , a novel fixed point with nontrivial couplings between all sectors becomes stable. At intermediate ${N}_{f}$ , including the graphene case $({N}_{f}=2)$ , no stable and physically admissible fixed point exists. Graphene's phase diagram in the vicinity of the intersection between the semimetal, antiferromagnetic, and staggered density phases should consequently be governed by a triple point exhibiting first-order transitions.We study the competition of spin- and charge-density waves and their quantum multicritical behavior for the semimetal-insulator transitions of low-dimensional Dirac fermions. Employing the effective Gross-Neveu-Yukawa theory with two order parameters as a model for graphene and a growing number of other two-dimensional Dirac materials allows us to describe the physics near the multicritical point at which the semimetallic and the spin- and charge-density-wave phases meet. With the help of a functional renormalization group approach, we are able to reveal a complex structure of fixed points, the stability properties of which decisively depend on the number of Dirac fermions ${N}_{f}$. We give estimates for the critical exponents and observe crucial quantitative corrections as compared to the previous first-order $\ensuremath{\epsilon}$ expansion. For small ${N}_{f}$, the universal behavior near the multicritical point is determined by the chiral Heisenberg universality class supplemented by a decoupled, purely bosonic, Ising sector. At large ${N}_{f}$, a novel fixed point with nontrivial couplings between all sectors becomes stable. At intermediate ${N}_{f}$, including the graphene case $({N}_{f}=2)$, no stable and physically admissible fixed point exists. Graphene's phase diagram in the vicinity of the intersection between the semimetal, antiferromagnetic, and staggered density phases should consequently be governed by a triple point exhibiting first-order transitions.

arXiv:1510.09003Physical review. B./Physical review. BMar 11, 201639 citations

We study the quantum multicritical point in a (2+1)-dimensional Dirac system between the semimetallic phase and two ordered phases that are characterized by anticommuting mass terms with

\mathrm{O}({N}_{1})

and

\mathrm{O}({N}_{2})

symmetries, respectively. Using

\ensuremath{\epsilon}

\mathrm{O}({N}_{1}+{N}_{2})

symmetry for arbitrary values of

{N}_{1}

and

{N}_{2}

and fermion flavor numbers

{N}_{\mathrm{f}}

as long as the corresponding representation of the Clifford algebra exists. Small

\mathrm{O}(N)

-breaking perturbations near the chiral

\mathrm{O}(N)

fixed point are therefore irrelevant. This result can be traced back to the presence of gapless Dirac degrees of freedom at criticality, and it is in clear contrast to the purely bosonic

\mathrm{O}(N)

fixed point, which is stable only when

N<3

. As a by-product, we obtain predictions for the critical behavior of the chiral

\mathrm{O}(N)

universality classes for arbitrary

N

and fermion flavor number

{N}_{\mathrm{f}}

. Implications for critical Weyl and Dirac systems in 3+1 dimensions are also briefly discussed.

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.

(1,0)+(0,1)

representation under the $\mathrm{S}\mathrm{O}(4)\ensuremath{\simeq} \mathrm{S}\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)$ symmetry group. This field theory is relevant to spin-1/2 fermions on honeycomb or $\ensuremath{\pi}$-flux lattices, for example, near the transition points between an

s

-wave superconductor and a charge-density wave, on one side, and N\'eel order, on the other. Two triplets of such order parameters always allow for a common pair of two other order parameters that would complete them to the maximal set of compatible (anticommuting) orders of five. We first derive a unitary transformation in the Nambu (particle-hole) space which maps any two such triplets, possibly containing some superconducting orders, onto purely insulating order parameters. This allows one to consider a universal SO(4) Gross-Neveu-Yukawa description of the multicriticality without any Nambu doubling. We then proceed to derive the renormalization-group flow of the coupling constants at one-loop order in $4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions, allowing also a more general set of order parameters transforming under $\mathrm{S}\mathrm{O}({n}_{a})\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{O}({n}_{b})$. While for

{n}_{a}={n}_{b}>2

in the bosonic sector and with fermions decoupled there is a stable fixed point of the flow, the Yukawa coupling to fermions quickly leads to its elimination by a generic fixed-point collision in the relevant range of fermion flavor numbers

{N}_{f}

. This suggests the replacement of the critical behavior by a runaway flow in the physical case

{n}_{a}={n}_{b}=3

. The structure of the renormalization group flow at ${n}_{a}\ensuremath{\ne}{n}_{b}$ is also discussed, and some nonperturbative arguments in favor of the stability of the decoupled critical point when

{n}_{a}=3

and

{n}_{b}=1

D=2+1

are provided.

2+1

)-dimensional Gross--Neveu--Yukawa field theory with

{N}_{f}=2

four-component Dirac fermions coupled to two triplets of order parameters does not exhibit a renormalization group fixed point at one-loop order. Instead, the theory only features a critical point for a large or very small fractional number of fermion flavors

{N}_{f}

, which disappears for a broad range of flavor numbers around the physical case,

{N}_{f}=2

, due to fixed-point annihilation. This raises the question on how to explain the observed scaling collapse in the quantum Monte Carlo data. Here, we extend previous renormalization-group analyses by studying a generalized model at two-loop order in $4\ensuremath{-}\ensuremath{\epsilon}$ spacetime dimensions. We determine the $\ensuremath{\epsilon}$ correction to the upper and lower critical flavor numbers for the fixed-point annihilation and find that they both go towards the physical case

{N}_{f}=2

. However, this only happens very slowly, such that an extrapolation to $\ensuremath{\epsilon}=1$ still suggests the absence of criticality in

2+1

dimensions. Thereby, we consolidate the finding that the continuum field theory does not feature a stable renormalization-group fixed point and no true quantum criticality would be expected for the considered system. We briefly discuss a possible reconciliation in terms of a complex conformal field theory. Further, we also explore the fixed-point structure in an enlarged theory space and identify a candidate stable fixed-point solution.

We introduce a model of Dirac fermions in

2+1

dimensions with a semimetallic, a quantum spin-Hall insulating (QSHI), and an

s

-wave superconducting (SSC) phase. The phase diagram features a multicritical point at which all three phases meet as well as a QSHI-SSC deconfined critical point. The QSHI and SSC orders correspond to mutually anticommuting mass terms of the Dirac Hamiltonian. Based on this algebraic property, SO(5) symmetric field theories have been put forward to describe both types of critical points. Using quantum Monte Carlo simulations, we directly study the operator that rotates between QSHI and SSC states. The results suggest that it commutes with the low-energy effective Hamiltonian at criticality but has a gap in the ordered phases. This implies an emergent SO(5) symmetry at both the multicritical and the deconfined critical points.

{N}_{b}=1

) or

s

-wave pairing (

{N}_{b}=2

) or antiferromagnet (

{N}_{b}=3

) in a honeycomb lattice, where

{N}_{b}

counts the number of bosonic order-parameter components. Besides the cubic terms present in the order-parameter description of KVBS due to the breaking of a discrete

{Z}_{3}

symmetry, quantum fluctuations generate new cubic vertices near the high-symmetry MCPs. All cubic terms are strongly relevant at the bare level near three spatial dimensions, about which we perform a leading-order renormalization group analysis of coupled Gross-Neveu-Yukawa field theory. We show that due to nontrivial Yukawa interactions among gapless bosonic and fermionic degrees of freedom, all cubic terms ultimately become irrelevant at an

O(2+{N}_{b})

symmetric MCP, at leas near two spatial dimensions, where

{N}_{b}=1,2,3

. Therefore, MCPs with an enlarged

O(2+{N}_{b})

symmetry near KVBS ordering are stable.

{N}_{f}

. We give estimates for the critical exponents and observe crucial quantitative corrections as compared to the previous first-order $\ensuremath{\epsilon}$ expansion. For small

{N}_{f}

, the universal behavior near the multicritical point is determined by the chiral Heisenberg universality class supplemented by a decoupled, purely bosonic, Ising sector. At large

{N}_{f}

, a novel fixed point with nontrivial couplings between all sectors becomes stable. At intermediate

{N}_{f}

, including the graphene case

({N}_{f}=2)

, no stable and physically admissible fixed point exists. Graphene's phase diagram in the vicinity of the intersection between the semimetal, antiferromagnetic, and staggered density phases should consequently be governed by a triple point exhibiting first-order transitions.