We study the quantum phase transition of $U(1)$ charged Dirac fermions Yukawa coupled to the Kekul\'e valence-bond-solid order parameter with ${Z}_{3}$ symmetry of the honeycomb lattice. The symmetry allows for the presence of the term in the action which is cubic in the Kekul\'e order parameter and which is expected to prevent the quantum phase transition in question from being continuous. The Gross-Neveu-Yukawa theory for the transition is investigated using a perturbative renormalization group and within the $\ensuremath{\epsilon}$ expansion close to four space-time dimensions. For a vanishing $U(1)$ charge we show that quantum fluctuations may render the phase transition continuous only sufficiently far away from 3+1 dimensions, where the validity of the conclusions based on the leading-order $\ensuremath{\epsilon}$ expansion appears questionable. In the presence of a fluctuating gauge field, on the other hand, we find quantum critical behavior even at weak coupling to appear close to 3+1 dimensions, that is, within the domain of validity of the perturbation theory. We also determine the renormalization-group scaling of the cubic coupling at higher-loop orders and for a large number of Dirac fermions for vanishing charge.We study the quantum phase transition of $U(1)$ charged Dirac fermions Yukawa coupled to the Kekul\'e valence-bond-solid order parameter with ${Z}_{3}$ symmetry of the honeycomb lattice. The symmetry allows for the presence of the term in the action which is cubic in the Kekul\'e order parameter and which is expected to prevent the quantum phase transition in question from being continuous. The Gross-Neveu-Yukawa theory for the transition is investigated using a perturbative renormalization group and within the $\ensuremath{\epsilon}$ expansion close to four space-time dimensions. For a vanishing $U(1)$ charge we show that quantum fluctuations may render the phase transition continuous only sufficiently far away from 3+1 dimensions, where the validity of the conclusions based on the leading-order $\ensuremath{\epsilon}$ expansion appears questionable. In the presence of a fluctuating gauge field, on the other hand, we find quantum critical behavior even at weak coupling to appear close to 3+1 dimensions, that is, within the domain of validity of the perturbation theory. We also determine the renormalization-group scaling of the cubic coupling at higher-loop orders and for a large number of Dirac fermions for vanishing charge.

arXiv:1609.03208Physical review. B./Physical review. BNov 23, 201651 citations

Criticality in a hybrid spin model with Fermi–Dirac statistics

Yiannis Contoyiannis, Stelios M. Potirakis, F. Κ. Diakonos +1 more

Physica A Statistical Mechanics and its ApplicationsMay 4, 20213 citations

Duality of deconfined quantum critical point in two dimensional Dirac semimetals

Jiang Zhou, Ya-Jie Wu, Su-Peng Kou

In this paper we discuss the N $\acute{e}$ el and Kekul $\acute{e}$ valence bond solids quantum criticality in graphene Dirac semimetal. Considering the quartic four-fermion interaction $g(\barψ_iΓ_{ij}ψ_j)^2$ that contains spin,valley, and sublattice degrees of freedom in the continuum field theory, we find the microscopic symmetry is spontaneously broken when the coupling $g$ is greater than a critical value $g_c$ . The symmetry breaking gaps out the fermion and leads to semimetal-insulator transition. All possible quartic fermion-bilinear interactions give rise to the uniform critical coupling, which exhibits the multicritical point for various orders and the Landau-forbidden quantum critical point. We also investigate the typical critical point between N $\acute{e}$ el and Kekul $\acute{e}$ valence bond solid transition when the symmetry is broken. The quantum criticality is captured by the Wess-Zumino-Witten term and there exist a mutual-duality for N $\acute{e}$ el-Kekul $\acute{e}$ VBS order. We show the emergent spinon in the N $\acute{e}$ el-Kekul $\acute{e}$ VBS transition , from which we conclude the phase transition is a deconfined quantum critical point. Additionally, the connection between the index theorem and zero energy mode bounded by the topological defect in the Kekul $\acute{e}$ VBS phase is studied to reveal the N $\acute{e}$ el-Kekul $\acute{e}$ VBS duality.In this paper we discuss the N$\acute{e}$el and Kekul$\acute{e}$ valence bond solids quantum criticality in graphene Dirac semimetal. Considering the quartic four-fermion interaction $g(\barψ_iΓ_{ij}ψ_j)^2$ that contains spin,valley, and sublattice degrees of freedom in the continuum field theory, we find the microscopic symmetry is spontaneously broken when the coupling $g$ is greater than a critical value $g_c$. The symmetry breaking gaps out the fermion and leads to semimetal-insulator transition. All possible quartic fermion-bilinear interactions give rise to the uniform critical coupling, which exhibits the multicritical point for various orders and the Landau-forbidden quantum critical point. We also investigate the typical critical point between N$\acute{e}$el and Kekul$\acute{e}$ valence bond solid transition when the symmetry is broken. The quantum criticality is captured by the Wess-Zumino-Witten term and there exist a mutual-duality for N$\acute{e}$el-Kekul$\acute{e}$ VBS order. We show the emergent spinon in the N$\acute{e}$el-Kekul$\acute{e}$ VBS transition , from which we conclude the phase transition is a deconfined quantum critical point. Additionally, the connection between the index theorem and zero energy mode bounded by the topological defect in the Kekul$\acute{e}$ VBS phase is studied to reveal the N$\acute{e}$el-Kekul$\acute{e}$ VBS duality.

arXiv:1710.02654Oct 7, 20170 citations

Fermion-induced quantum critical points in two-dimensional Dirac semimetals

Shao-Kai Jian, Hong Yao

In this paper we investigate the nature of quantum phase transitions between two-dimensional Dirac semimetals and ${Z}_{3}$ -ordered phases (e.g., Kekule valence-bond solid), where cubic terms of the order parameter are allowed in the quantum Landau-Ginzberg theory and the transitions are putatively first order. From large- $N$ renormalization-group (RG) analysis, we find that fermion-induced quantum critical points (FIQCPs) [Z.-X. Li et al., Nat. Commun. 8, 314 (2017)] occur when $N$ (the number of flavors of four-component Dirac fermions) is larger than a critical value ${N}_{c}$ . Remarkably, from the knowledge of space-time supersymmetry, we obtain an exact lower bound for ${N}_{c}$ , i.e., ${N}_{c}>1/2$ . (Here the ``1/2'' flavor of four-component Dirac fermions is equivalent to one flavor of four-component Majorana fermions). Moreover, we show that the emergence of two length scales is a typical phenomenon of FIQCPs and obtain two different critical exponents, i.e., $\ensuremath{\nu}\ensuremath{\ne}{\ensuremath{\nu}}^{\ensuremath{'}}$, by large- $N$ RG calculations. We further give a brief discussion of possible experimental realizations of FIQCPs.In this paper we investigate the nature of quantum phase transitions between two-dimensional Dirac semimetals and ${Z}_{3}$-ordered phases (e.g., Kekule valence-bond solid), where cubic terms of the order parameter are allowed in the quantum Landau-Ginzberg theory and the transitions are putatively first order. From large-$N$ renormalization-group (RG) analysis, we find that fermion-induced quantum critical points (FIQCPs) [Z.-X. Li et al., Nat. Commun. 8, 314 (2017)] occur when $N$ (the number of flavors of four-component Dirac fermions) is larger than a critical value ${N}_{c}$. Remarkably, from the knowledge of space-time supersymmetry, we obtain an exact lower bound for ${N}_{c}$, i.e., ${N}_{c}>1/2$. (Here the ``1/2'' flavor of four-component Dirac fermions is equivalent to one flavor of four-component Majorana fermions). Moreover, we show that the emergence of two length scales is a typical phenomenon of FIQCPs and obtain two different critical exponents, i.e., $\ensuremath{\nu}\ensuremath{\ne}{\ensuremath{\nu}}^{\ensuremath{'}}$, by large-$N$ RG calculations. We further give a brief discussion of possible experimental realizations of FIQCPs.

arXiv:1610.07603Physical review. B./Physical review. BNov 28, 201735 citations

Quantum Criticality of

3+1

-dimensional massless Dirac fermions

We study the quantum phase transition of

U(1)

charged Dirac fermions Yukawa coupled to the Kekul\'e valence-bond-solid order parameter with

{Z}_{3}

symmetry of the honeycomb lattice. The symmetry allows for the presence of the term in the action which is cubic in the Kekul\'e order parameter and which is expected to prevent the quantum phase transition in question from being continuous. The Gross-Neveu-Yukawa theory for the transition is investigated using a perturbative renormalization group and within the $\ensuremath{\epsilon}$ expansion close to four space-time dimensions. For a vanishing

U(1)

charge we show that quantum fluctuations may render the phase transition continuous only sufficiently far away from 3+1 dimensions, where the validity of the conclusions based on the leading-order $\ensuremath{\epsilon}$ expansion appears questionable. In the presence of a fluctuating gauge field, on the other hand, we find quantum critical behavior even at weak coupling to appear close to 3+1 dimensions, that is, within the domain of validity of the perturbation theory. We also determine the renormalization-group scaling of the cubic coupling at higher-loop orders and for a large number of Dirac fermions for vanishing charge.

In this paper we discuss the N

\acute{e}

el and Kekul

\acute{e}

valence bond solids quantum criticality in graphene Dirac semimetal. Considering the quartic four-fermion interaction

g(\barψ_iΓ_{ij}ψ_j)^2

that contains spin,valley, and sublattice degrees of freedom in the continuum field theory, we find the microscopic symmetry is spontaneously broken when the coupling

g

is greater than a critical value

g_c

. The symmetry breaking gaps out the fermion and leads to semimetal-insulator transition. All possible quartic fermion-bilinear interactions give rise to the uniform critical coupling, which exhibits the multicritical point for various orders and the Landau-forbidden quantum critical point. We also investigate the typical critical point between N

\acute{e}

el and Kekul

\acute{e}

valence bond solid transition when the symmetry is broken. The quantum criticality is captured by the Wess-Zumino-Witten term and there exist a mutual-duality for N

\acute{e}

el-Kekul

\acute{e}

VBS order. We show the emergent spinon in the N

\acute{e}

el-Kekul

\acute{e}

VBS transition , from which we conclude the phase transition is a deconfined quantum critical point. Additionally, the connection between the index theorem and zero energy mode bounded by the topological defect in the Kekul

\acute{e}

VBS phase is studied to reveal the N

\acute{e}

el-Kekul

\acute{e}

VBS duality.

In this paper we investigate the nature of quantum phase transitions between two-dimensional Dirac semimetals and

{Z}_{3}

-ordered phases (e.g., Kekule valence-bond solid), where cubic terms of the order parameter are allowed in the quantum Landau-Ginzberg theory and the transitions are putatively first order. From large-

N

renormalization-group (RG) analysis, we find that fermion-induced quantum critical points (FIQCPs) [Z.-X. Li et al., Nat. Commun. 8, 314 (2017)] occur when

N

(the number of flavors of four-component Dirac fermions) is larger than a critical value

{N}_{c}

. Remarkably, from the knowledge of space-time supersymmetry, we obtain an exact lower bound for

{N}_{c}

, i.e.,

{N}_{c}>1/2

. (Here the ``1/2'' flavor of four-component Dirac fermions is equivalent to one flavor of four-component Majorana fermions). Moreover, we show that the emergence of two length scales is a typical phenomenon of FIQCPs and obtain two different critical exponents, i.e., $\ensuremath{\nu}\ensuremath{\ne}{\ensuremath{\nu}}^{\ensuremath{'}}$, by large-

N

RG calculations. We further give a brief discussion of possible experimental realizations of FIQCPs.