# Gradient optimization of fermionic projected entangled pair states on directed lattices

## TL;DR

This paper extends a stochastic gradient optimization method to fermionic PEPS, introducing a fermi arrow notation, and demonstrates its effectiveness in simulating complex fermionic models with improved accuracy and scalability.

## Contribution

The authors develop a fermionic PEPS gradient optimization method with a new fermi arrow notation, enabling efficient simulation of fermionic systems with larger bond dimensions.

## Key findings

- Gradient optimization improves simple update results.
- Larger bond dimensions are needed for convergence.
- Method offers lower scaling than direct contraction methods.

## Abstract

The recently developed stochastic gradient method combined with Monte Carlo sampling techniques [PRB {\bf 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-J models. The numerical calculations show that the gradient optimization greatly improves the results of simple update method. Furthermore, much larger virtual bond dimensions ($D$) and truncation dimensions ($D_c$) than those of boson and spin systems are necessary to converge the results. The method therefore offer a powerful tool to simulate fermion systems because it has much lower scaling than the direct contraction methods.