We present one- and two-body measurements for the Hubbard model on the honeycomb (graphene) lattice from ab-initio quantum monte carlo simulations. Of particular interest is excitons, which are particle/hole excitations in low-dimensional systems. They are analogous to the pion in QCD, but without confinement, the question of whether they are bound and stable is of great interest in the condensed matter arena. By measuring one- and two-body correlators across various spin and isospin channels we can compute two-body energies relative to their thresholds, ultimately allowing us to check for stable states.
We study the one-band Hubbard model on the honeycomb lattice using a combination of quantum Monte Carlo (QMC) simulations and static as well as dynamical mean-field theory (DMFT). This model is known to show a quantum phase transition between a Dirac semimetal and the antiferromagnetic insulator. The aim of this paper is to provide a detailed comparison between these approaches by computing static properties, notably ground-state energy, single-particle gap, double occupancy, and staggered magnetization, as well as dynamical quantities such as the single-particle spectral function. At the static mean-field level, local moments cannot be generated without breaking the SU(2) spin symmetry. The DMFT approximation accounts for temporal fluctuations and thus captures both the evolution of the double occupancy and the resulting local moment formation in the paramagnetic phase. As a consequence, the DMFT approximation is found to be very accurate in the Dirac semimetallic phase where local moment formation is present and the spin correlation length small. However, in the vicinity of the fermion quantum critical point, the spin correlation length diverges and the spontaneous SU(2) symmetry breaking leads to low-lying Goldstone modes in the magnetically ordered phase. The impact of these spin fluctuations on the single-particle spectral function---waterfall features and narrow spin-polaron bands---is only visible in the lattice QMC approach.
We take advantage of recent improvements in the grand canonical Hybrid Monte Carlo (HMC) algorithm, to perform a precision study of the single-particle gap in the hexagonal Hubbard model, with on-site electron-electron interactions. After carefully controlled analyses of the Trotter error, the thermodynamic limit, and finite-size scaling with inverse temperature, we find a critical coupling of Uc/κ=3.835(14) and the critical exponent ν=1.181(43) for the semimetal-antiferromagnetic Mott insulator quantum phase transition in the hexagonal Hubbard Model. Based on these results, we provide a unified, comprehensive treatment of all operators that contribute to the anti-ferromagnetic, ferromagnetic, and charge-density-wave structure factors and order parameters of the hexagonal Hubbard Model. We expect our findings to improve the consistency of Monte Carlo determinations of critical exponents. We perform a data collapse analysis and determine the critical exponent β=0.898(37). We consider our findings in view of the SU(2) Gross-Neveu, or chiral Heisenberg, universality class. We also discuss the computational scaling of the HMC algorithm. Our methods are applicable to a wide range of lattice theories of strongly correlated electrons. The Ising model, a simple statistical model for ferromagnetism, is one such theory. There are analytic solutions for low dimensions and very efficient Monte Carlo methods, such as cluster algorithms, for simulating this model in special cases. However most approaches do not generalise to arbitrary lattices and couplings. We present a formalism that allows one to apply HMC simulations to the Ising model, demonstrating how a system with discrete degrees of freedom can be simulated with continuous variables.
We present a systematic study of the quantum phase transition from a Dirac semimetal to an antiferromagnetic Mott insulator in the honeycomb-lattice Hubbard model, using finite-temperature auxiliary-field quantum Monte Carlo (AFQMC) simulations. Given the dynamical critical exponent z=1 at criticality, we scale the inverse temperature β with the linear system size L, i.e., βt=L (with t as the hopping amplitude), to approach the thermodynamic and zero-temperature limits simultaneously. On the algorithmic front, we generalize the fast Fourier transform in path propagation to multi-sublattice systems and employ the delayed update technique in configuration updates to accelerate the AFQMC simulations. These advances enable us to access unprecedented system sizes of up to 3528 sites (with L=42 and βt=42). We additionally use twist-averaged boundary conditions to reduce the finite-size effects in computed observables. On the physical side, we determine the Dirac quantum criticality via finite-size scaling analysis of the spin correlations and off-diagonal single-particle Green's function, and locate the critical point at Uc/t=3.731(4) with the critical exponents ν=1.097(7), ηϕ=0.72(2) and ηψ=0.155(6). We further corroborate the critical point by examining several relevant quantities, including the quasiparticle weight, Fermi-liquid parameter, charge compressibility, U-derivative of double occupancy, and fidelity susceptibility. Our work establishes an independent benchmark for the chiral Heisenberg Gross-Neveu-Yukawa criticality in the honeycomb Hubbard model, offering a complementary perspective to previous ground-state simulations.