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Using first-principle Hybrid-Monte-Carlo (HMC) simulations, we carry out anunbiased study of the competition between spin-density wave (SDW) andcharge-density wave (CDW) order in the extended Hubbard model on the twodimensional hexagonal lattice at half filling. We determine the phase diagramin the space of on-site and nearest-neighbor couplings $U$ and $V$ in theregion $V<U/3$, which can be simulated without a fermion sign problem, and findthat a transition from semimetal to a SDW phase occurs at sufficiently large$U$ for basically all $V$. Tracing the corresponding phase boundary from $V=0$to the $V=U/3$ line, we find evidence for critical scaling in the Gross-Neveuuniversality class for the entire boundary. With rather high confidence we ruleout the existence of the CDW ordered phase anywhere in the range of parametersconsidered. We also discuss several improvements of the HMC algorithm which arecrucial to reach these conclusions, in particular the improved fermion actionwith exact sublattice symmetry and the complexification of theHubbard-Stratonovich field to ensure the ergodicity of the algorithm.

Hybrid-Monte-Carlo study of competing order in the extended fermionic Hubbard model on the hexagonal lattice