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We study the ground state of the doped Hubbard model on honeycomb latticewith both nearest ($t$) and next-nearest neighboring hoppings ($t'$) in thesmall doping and strongly interacting region. Previous study on the modelwithout $t'$ showed the ground state is a half-filled stripe. We employ densitymatrix renormalization group and extrapolate the results with truncation errorsin the converged region. In the $t' < 0$ side, we find the half-filled stripephase at $t' = 0$ is stable against the frustration of $t'$ until a criticalpoint $-0.4 < t'_c < -0.3$, beyond which the ground state switches toanti-ferromagnetic N$\acute e$el phase with charge modulation. With furtherincrease of $t'$ to $-0.7$, the ground state becomes paramagnetic. In the $t' >0$ side, the half-filled stripe stretches to $t' \approx 0.7$. We don't findobvious enhancement of pairing for the range of $t'$ studied. We study width-4cylinders in this work but the results for spin, charge, and pairingcorrelation agree qualitatively for periodic and anti-periodic boundaryconditions in the half-filled stripe and anti-ferromagnetic N$\acute e$elphases, suggesting the results are likely to be representative for truetwo-dimensional systems. The half-filled stripe to anti-ferromagnetic N$\acutee$el phase transition can be realized on real materials or ultra-cold atomplatform.

Half-filled stripe to N$\acute e$el antiferromagnetism transition in the $t'$-Hubbard model on honeycomb lattice