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The continuous imaginary-time quantum Monte Carlo method with the worm updatealgorithm is applied to explore the ground state properties of the spin-1/2Heisenberg 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-dimensionalzigzag spin chains to a two-dimensional magnetic ordered state. In absence ofan external field, the system is in a stripe order phase. In presence ofuniform and staggered fields, the uniform and staggered out-of-planemagnetizations appear while the stripe order keeps in $xy$ plane, and asecond-order quantum phase transition (QPT) at a critical staggered field isobserved. The critical exponents of correlation length for QPTs induced by astaggered 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 ina staggered field is analyzed, and the ground state phase diagrams in the planeof coupling ratio and staggered field are presented for two cases. Thetemperature dependence of susceptibility and specific heat of both systems inexternal magnetic fields is also discussed.

physical review. e

Quantum Monte Carlo Study on the Spin-1/2 Honeycomb Heisenberg Model  with Mixing Antiferromagnetic and Ferromagnetic Interactions in External  Magnetic Fields