  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 $U_c/\kappa=3.835(14)$ and the critical exponent $\nu=1.181(43)$
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
$\beta=0.898(37)$. 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.
