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
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
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.
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 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) 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 Nf. 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.
We study the Gross-Neveu-Yukawa field theory for the 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 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, 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 and Nf,c2. Most importantly, we observe a sharp decrease in Nf,c2 from its one-loop value, which brings it closer to the point Nf=1 relevant to the standard Gross-Neveu model. Some three-loop results are also presented and discussed.
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 na=nb>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 Nf. This suggests the replacement of the critical behavior by a runaway flow in the physical case na=nb=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 na=3 and nb=1 in D=2+1 are provided.
We investigate nematic quantum phase transitions in two different Dirac fermion models. The models feature twofold and fourfold, respectively, lattice rotational symmetries that are spontaneously broken in the ordered phase. Using negative-sign-free quantum Monte Carlo simulations and an ε-expansion renormalization group analysis, we show that both models exhibit continuous phase transitions. In contrast to generic Gross-Neveu dynamical mass generation, the quantum critical regime is characterized by large velocity anisotropies, with fixed-point values being approached very slowly. Both experimental and numerical investigations will not be representative of the infrared fixed point, but of a quasiuniversal regime where the drift of the exponents tracks the velocity anisotropy.
Motivated by the physics of spin-orbital liquids, we study a model of interacting Dirac fermions on a bilayer honeycomb lattice at half filling, featuring an explicit global SO(3)×U(1) symmetry. Using large-scale auxiliary- field quantum Monte Carlo (QMC) simulations, we locate two zero-temperature phase transitions as function of increasing interaction strength. First, we observe a continuous transition from the weakly-interacting semimetal to a different semimetallic phase in which the SO(3) symmetry is spontaneously broken and where two out of three Dirac cones acquire a mass gap. The associated quantum critical point can be understood in terms of a Gross-Neveu-SO(3) theory. Second, we subsequently observe a transition towards an insulating phase in which the SO(3) symmetry is restored and the U(1) symmetry is spontaneously broken. While strongly first order at the mean-field level, the QMC data is consistent with a direct and continuous transition. It is thus a candidate for a new type of deconfined quantum critical point that features gapless fermionic degrees of freedom.