A second-order phase transition for the three-dimensional Gross-Neveu model is established for one fermion species, $N=1.$ This transition breaks a paritylike discrete symmetry. It constitutes its peculiar universality class with critical exponent $\ensuremath{\nu}=0.63$ and scalar and fermionic anomalous dimensions ${\ensuremath{\eta}}_{\ensuremath{\sigma}}=0.31$ and ${\ensuremath{\eta}}_{\ensuremath{\psi}}=0.11,$ respectively. We also compute critical exponents for other N. Our results are based on exact renormalization-group equations.
A second-order phase transition for the three-dimensional Gross-Neveu model is established for one fermion species, N=1. This transition breaks a paritylike discrete symmetry. It constitutes its peculiar universality class with critical exponent \ensuremathν=0.63 and scalar and fermionic anomalous dimensions \ensuremathη\ensuremathσ=0.31 and \ensuremathη\ensuremathψ=0.11, respectively. We also compute critical exponents for other N. Our results are based on exact renormalization-group equations.
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.
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 study the critical properties of the QED3-Gross-Neveu model with 2N flavors of two-component Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For N=1, this theory has recently been suggested to be dual to the SU(2) noncompact CP1 model that describes the deconfined phase transition between the N\'eel antiferromagnet and the valence bond solid on the square lattice. For N=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-1/2 systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the QED3-Gross-Neveu model for all values of N. This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain QED3 (without scalar-field coupling) unstable at low values of N. 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.
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.
We study the universal critical properties of the QED3-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 SU(2)-symmetric noncompact 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=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 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.
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.