We perform large-scale quantum Monte Carlo simulations of SLAC fermions on a two-dimensional square lattice at half filling with a single Dirac cone with N=2 spinor components and repulsive on-site interactions. Despite the presence of a sign problem, we accurately identify the critical interaction strength U_{c}=7.28±0.02 in units of the hopping amplitude, for a continuous quantum phase transition between a paramagnetic Dirac semimetal and a ferromagnetic insulator. Using finite-size scaling, we extract the critical exponents for the corresponding N=2 chiral Ising Gross-Neveu universality class: the inverse correlation length exponent ν^{-1}=1.19±0.03, the order parameter anomalous dimension η_{ϕ}=0.31±0.01, and the fermion anomalous dimension η_{ψ}=0.136±0.005.
It is expected that the Gross-Neveu-Yukawa (GNY) chiral Ising transition of Dirac fermions coupled with a scalar field in
(2+1) dimensions will be the first fermionic quantum critical point that various methods, such as conformal bootstrap, perturbative renormalization group, and quantum Monte Carlo (QMC) simulations, would yield converged critical exponents---serving the same role as the Ising and
O(N) models in the textbooks of statistical and quantum physics. However, such an expectation has not been fully realized from the lattice QMC simulations due to the obstacles introduced by the UV finite-size effect. In this Letter, by means of the elective-momentum ultrasize (EMUS)-QMC method, we compute the critical exponents of the $O{(N/2)}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{2}$ GNY
N=8 chiral Ising transition on a two-dimensional $\ensuremath{\pi}$-flux fermion lattice model between Dirac semimetal and quantum spin Hall insulator phases. With the matching of fermionic and bosonic momentum transfer and collective update in momentum space, our QMC results provide fully consistent exponents with those obtained from the bootstrap and perturbative approaches. In this way, the EMUS now live happily on the
N=8 island and could explore the chiral Gross-Neveu-Yukawa archipelago with ease.
According to the Landau criterion, a phase transition should be first order when cubic terms of order parameters are allowed in its effective Ginzburg-Landau free energy. Recently, it was shown by renormalization-group (RG) analysis that continuous transition can happen at putatively first-order
Z3 transitions in two-dimensional (2D) Dirac semimetals and such non-Landau phase transitions were dubbed ``fermion-induced quantum critical points'' (FIQCPs) [Z.-X. Li, Y.-F. Jiang, S.-K. Jian, and H. Yao, Nat. Commun. 8, 314 (2017)]. The RG analysis, controlled by the
1/N expansion with
N the number of flavors of four-component Dirac fermions, shows that the FIQCP occurs for $N\ensuremath{\ge}{N}_{c}$. Previous quantum Monte Carlo simulations of a microscopic model of
SU(N) fermions on the honeycomb lattice showed that the FIQCP occurs at the transition between Dirac semimetals and Kekule valence bond solids for $N\ensuremath{\ge}2$. However, the precise value of the lower bound
Nc has not been established. Especially, the case of
N=1 has not been explored by studying microscopic models so far. Here, by introducing a generalized
SU(N) fermion model with
N=1 (namely, spinless fermions on the honeycomb lattice), we perform large-scale sign-problem-free Majorana quantum Monte Carlo simulations and find convincing evidence of the FIQCP for
N=1. Consequently, our results suggest that the FIQCP can occur in 2D Dirac semimetals for all positive integers $N\ensuremath{\ge}1$.
Abstract We investigate a quantum criticality of an antiferromagnetic phase transition in the Hubbard model on a square lattice with a d-wave pairing field by large-scale auxiliary-field quantum Monte Carlo simulations. Since the d-wave pairing filed induces Dirac cones in the non-interacting single-particle spectrum, the quantum criticality should correspond to the chiral Heisenberg universality class in terms of the Gross-Neveu theory, which is the same as those expected in the Hubbard model on the honeycomb lattice, despite the unit cells being different (e.g., they contain one and two sites, respectively). We show that both the two phase transitions, expected to occur on the square and on the honeycomb lattices, indeed have the same quantum criticality. We also argue that details of the models, i.e., the way of counting the total number N of fermion components and the anisotropy of the Dirac cones, do not change the critical exponents.
Spinless fermions on a honeycomb lattice provide a minimal realization of lattice Dirac fermions. Repulsive interactions between nearest neighbors drive a quantum phase transition from a Dirac semimetal to a charge-density-wave state through a fermionic quantum critical point, where the coupling of the Ising order parameter to the Dirac fermions at low energy drastically affects the quantum critical behavior. Encouraged by a recent discovery (Huffman and Chandrasekharan 2014 Phys. Rev. B 89 111101 ) of the absence of the fermion sign problem in this model, we study the fermionic quantum critical point using the continuous-time quantum Monte Carlo method with a worm-sampling technique. We estimate the transition point with the critical exponents and . Compatible results for the transition point are also obtained with infinite projected entangled-pair states.
We perform large-scale quantum Monte Carlo simulations of the Hubbard model at half filling with a single Dirac cone close to the critical point, which separates a Dirac semimetal from an antiferromagnetically ordered phase where SU(2) spin rotational symmetry is spontaneously broken. We discuss the implementation of a single Dirac cone in the SLAC formulation for eight Dirac components and the influence of dynamically induced long-range superexchange interactions. The finite-size behavior of dimensionless ratios and the finite-size scaling properties of the Hubbard model with a single Dirac cone are shown to be superior compared to the honeycomb lattice. We extract the critical exponent believed to belong to the chiral Heisenberg Gross-Neveu-Yukawa universality class: the critical exponent ν = 1.02 ( 3 ) coincides for the two lattice types once honeycomb lattices of linear dimension L > 15 are considered. In contrast to the SLAC formulation, where the anomalous dimensions are estimated to be η ϕ = 0.73 ( 1 ) and η ψ = 0.09 ( 1 ) , they remain less stable on honeycomb lattices, but tend towards the estimates from the SLAC formulation.