Many electromagnetic properties of graphene can be described by the Hubbard model on a honeycomb lattice. However, this system suffers strongly from the sign problem if a chemical potential is included. Tensor network methods are not affected by this problem. We use the imaginary time evolution of a fermionic projected entangled pair state, which allows to simulate both parity sectors independently. Incorporating the fermionic nature on the level of the tensor network allows to fix the particle number to be either even or odd. This way we can access the states at half filling and with one additional electron. We calculate the energy and other observables of both states, which was not possible before with Monte Carlo methods.
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 take advantage of recent improvements in the grand canonical hybrid Monte Carlo 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 ${U}_{c}/\ensuremath{\kappa}=3.834(14)$ and the critical exponent $z\ensuremath{\nu}=1.185(43)$. Under the assumption that this corresponds to the expected antiferromagnetic Mott transition, we are also able to provide a preliminary estimate $\ensuremath{\beta}=1.095(37)$ for the critical exponent of the order parameter. 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 hybrid Monte Carlo algorithm, and possible extensions of our work to carbon nanotubes, fullerenes, and topological insulators.
The Hubbard model on the honeycomb lattice undergoes a quantum phase transition from a semimetallic to a Mott insulating phase and from a disordered to an anti-ferromagnetically phase. We show that these transitions occur simultaneously and we calculate the critical coupling
Uc=3.835(14) as well as the critical exponents
ν=1.181(43) and
β=0.898(37) which are expected to fall into the
SU(2) Gross-Neveu universality class. For this we employ Hybrid Monte Carlo simulations, extrapolate the single particle gap and the spin structure factors to the thermodynamic and continuous time limits, and perform a data collapse fit. We also determine the zero temperature values of single particle gap and staggered magnetisation on both sides of the phase transition.
Tensor networks are a powerful tool to simulate a variety of different physical models, including those that suffer from the sign problem in Monte Carlo simulations. The Hubbard model on the honeycomb lattice with nonzero chemical potential is one such problem. Our method is based on projected entangled pair states using imaginary-time evolution. We demonstrate that it provides accurate estimators for the ground state of the model, including cases where Monte Carlo simulations fail miserably. In particular, it shows near to optimal, that is linear, scaling in lattice size. We also present an approach to directly simulate the subspace with an odd number of fermions. It allows to independently determine the ground state in both sectors. Without a chemical potential, this corresponds to half-filling and the lowest-energy state with one additional electron or hole. We identify several stability issues, such as degenerate ground states and large single-particle gaps, and provide possible fixes.
The Hubbard model encapsulates the physics of strongly correlated quantum systems in its most basic form. It has been studied intensively in the context of the high-temperature superconductivity. A number of novel phases were recently proposed for Hubbard-like models on the honeycomb lattice, the structure of graphene. We analyzed the Hubbard model of spin-
21 fermions on the honeycomb lattice at half-filling using large-scale quantum Monte Carlo simulations. We find that the weak coupling semimetal and the antiferromagnetic Mott insulator at strong interaction are separated by an extended gapped phase in an intermediate coupling regime. Exploring excitation gaps, various correlation functions as well as probing for flux quantization, we conclude that a quantum spin liquid, lacking any conventional order, emerges with local charge and spin correlations, best described by a resonating valence bonds state.