The recently developed stochastic gradient method combined with Monte Carlo sampling techniques [W. Y. Liu et al., Phys. Rev. B 95, 195154 (2017)] offers a low scaling and accurate method to optimize the projected entangled pair states (PEPS). We extended this method to the fermionic PEPS (fPEPS). To simplify the implementation, we introduce a Fermi arrow notation to specify the order of the fermion operators in the virtual entangled EPR pairs. By defining some local operation rules associated with the Fermi arrows, one can implement fPEPS algorithms very similar to that of standard PEPS. We benchmark the method for the interacting spinless fermion models and the $t\text{−}J$ models. The numerical calculations show that the gradient optimization greatly improves the results of simple update method. Furthermore, very large virtual bond dimensions ($D$) and truncation dimensions ($D_c$) are necessary to converge the results of these models. The method therefore offers a powerful tool to simulate fermion systems because it has much lower scaling than the direct contraction methods.

• Received 10 December 2018
• Revised 10 March 2019

DOI:https://doi.org/10.1103/PhysRevB.99.195153