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arXiv:1808.00549 · Physical review. B./Physical review. B

Critical behavior of the QED3 -Gross-Neveu-Yukawa model at four loops

Nikolai Zerf, Peter Marquard, Rufus Boyack, Joseph Maciejko

Published Oct 16, 2018• 34 citations
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Abstract

We study the universal critical properties of the ${\mathrm{QED}}_{3}$-Gross-Neveu-Yukawa model with $N$ flavors of four-component Dirac fermions coupled to a real scalar order parameter at four-loop order in the $\ensuremath{\epsilon}$ expansion below four dimensions. For $N=1$, the model is conjectured to be infrared dual to the $\text{SU}(2)$-symmetric noncompact $\mathbb{C}{\mathrm{P}}^{1}$ model, which describes the deconfined quantum critical point of the N\'eel-valence-bond-solid transition of spin-1/2 quantum antiferromagnets on the two-dimensional square lattice. For $N=2$, the model describes a quantum phase transition between an algebraic spin liquid and a chiral spin liquid in the spin-1/2 kagome antiferromagnet. For general $N$ we determine the order-parameter anomalous dimension, the correlation length exponent, the stability critical exponent, as well as the scaling dimensions of $\text{SU}(N)$ singlet and adjoint fermion mass bilinears at the critical point. We use Pad\'e approximants to obtain estimates of critical properties in 2+1 dimensions.

DOI
10.1103/physrevb.98.165125

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We study the universal critical properties of the QED3{\mathrm{QED}}_{3}QED3​-Gross-Neveu-Yukawa model with NNN flavors of four-component Dirac fermions coupled to a real scalar order parameter at four-loop order in the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion below four dimensions. For N=1N=1N=1, the model is conjectured to be infrared dual to the SU(2)\text{SU}(2)SU(2)-symmetric noncompact CP1\mathbb{C}{\mathrm{P}}^{1}CP1 model, which describes the deconfined quantum critical point of the N'eel-valence-bond-solid transition of spin-1/2 quantum antiferromagnets on the two-dimensional square lattice. For N=2N=2N=2, the model describes a quantum phase transition between an algebraic spin liquid and a chiral spin liquid in the spin-1/2 kagome antiferromagnet. For general NNN we determine the order-parameter anomalous dimension, the correlation length exponent, the stability critical exponent, as well as the scaling dimensions of SU(N)\text{SU}(N)SU(N) singlet and adjoint fermion mass bilinears at the critical point. We use Pad'e approximants to obtain estimates of critical properties in 2+1 dimensions.
We study the critical properties of the QED3{\mathrm{QED}}_{3}QED3​-Gross-Neveu model with 2N2N2N flavors of two-component Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For N=1N=1N=1, this theory has recently been suggested to be dual to the SU(2) noncompact CP1{\mathrm{CP}}^{1}CP1 model that describes the deconfined phase transition between the N\'eel antiferromagnet and the valence bond solid on the square lattice. For N=2N=2N=2, the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin-1/21/21/2 systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the QED3{\mathrm{QED}}_{3}QED3​-Gross-Neveu model for all values of NNN. This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain QED3{\mathrm{QED}}_{3}QED3​ (without scalar-field coupling) unstable at low values of NNN. The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter $\ensuremath{\epsilon}=4\ensuremath{-}D$. We compute the scalar-field anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, the correlation-length exponent $\ensuremath{\nu}$, as well as the scaling dimension of the flavor-symmetry-breaking bilinear $\overline{\ensuremath{\psi}}{\ensuremath{\sigma}}^{z}\ensuremath{\psi}$ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.
The critical properties of the QED3{\mathrm{QED}}_{3}QED3​ Gross-Neveu-Yukawa (GNY) model in 2 + 1 dimensions with NNN flavors of two-component Dirac fermions are computed to first order in the 1/N1/N1/N expansion. For the specific case of N=2N=2N=2, the critical point is conjectured to be dual to the N\'eel-to-valence-bond-solid (VBS) deconfined critical point of quantum antiferromagnets on the square lattice. It is found that Aslamazov-Larkin diagrams, missed by previous $\ensuremath{\epsilon}$- and 1/N1/N1/N-expansion studies with four-component fermions, give important contributions to the scaling dimensions of various operators. With the inclusion of these diagrams, the resummed scaling dimensions of the adjoint fermion bilinear and scalar field at the QED3{\mathrm{QED}}_{3}QED3​ GNY critical point are in reasonable agreement with numerical studies of the N\'eel-to-VBS transition, in support of the duality conjecture.
We study the chiral Ising, the chiral XY, and the chiral Heisenberg models at four-loop order with the perturbative renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and compute critical exponents for the Gross-Neveu-Yukawa fixed points to order $\mathcal{O}({\ensuremath{\epsilon}}^{4})$. Further, we provide Pad\'e estimates for the correlation length exponent, the boson and fermion anomalous dimension, as well as the leading correction to scaling exponent in 2+12+12+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=1/4N=1/4N=1/4 and N=1/2N=1/2N=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 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.
Dirac and Weyl fermions appear as quasiparticle excitations in many different condensed-matter systems. They display various quantum transitions which represent unconventional universality classes related to the variants of the Gross-Neveu model. In this paper we study the bosonized version of the standard Gross-Neveu model---the Gross-Neveu-Yukawa theory---at three-loop order, and compute critical exponents in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions for a general number of fermion flavors. Our results fully encompass the previously known two-loop calculations, and agree with the known three-loop results in the purely bosonic limit of the theory. We also find the exponents to satisfy the emergent superscaling relations in the limit of a single-component fermion, order by order up to three loops. Finally, we apply the computed series for the exponents and their Pad\'e approximants to several phase transitions of current interest: metal-insulator transitions of spin-1/2 and spinless fermions on the honeycomb lattice, emergent supersymmetric surface field theory in topological phases, as well as the disorder-induced quantum transition in Weyl semimetals. Comparison with the results of other analytical and numerical methods is discussed.
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