  The continuous imaginary-time quantum Monte Carlo method with the worm update algorithm is applied to explore the ground state properties of the spin-1/2 Heisenberg model with antiferromagnetic (AF) coupling $J>0$ and ferromagnetic (F) coupling $J^{\prime}<0$ along zigzag and armchair directions, respectively, on honeycomb lattice. It is found that by enhancing the F coupling $J^{\prime}$ between zigzag AF chains, the system is smoothly crossover from one-dimensional zigzag spin chains to a two-dimensional magnetic ordered state. In absence of an external field, the system is in a stripe order phase. In presence of uniform and staggered fields, the uniform and staggered out-of-plane magnetizations appear while the stripe order keeps in $xy$ plane, and a second-order quantum phase transition (QPT) at a critical staggered field is observed. The critical exponents of correlation length for QPTs induced by a staggered field for the cases with $J>0$, $J^{\prime}<0$ and $J<0$, $J^{\prime}>0$ are obtained to be $\nu=0.677(2)$ and $0.693(0)$, respectively, indicating that both cases belong to O(3) universality. The scaling behavior in a staggered field is analyzed, and the ground state phase diagrams in the plane of coupling ratio and staggered field are presented for two cases. The temperature dependence of susceptibility and specific heat of both systems in external magnetic fields is also discussed. 
Quantum Monte Carlo Study on the Spin-1/2 Honeycomb Heisenberg Model   with Mixing Antiferromagnetic and Ferromagnetic Interactions in External   Magnetic Fields