# Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model

## TL;DR

This paper introduces a combined variational Monte Carlo and tensor network approach for fermionic models, demonstrating improved accuracy and revealing complex ground states in the doped Hubbard model.

## Contribution

It develops a novel wave function combining pair product states, tensor networks, and symmetry projections, enhancing fermionic model simulations.

## Key findings

- Significant accuracy improvement over individual methods.
- Observation of stripe order coexisting with weak d-wave superconductivity.
- Identification of metastable superconducting states at low doping.

## Abstract

The conventional tensor-network states employ real-space product states as reference wave functions. Here, we propose a many-variable variational Monte Carlo (mVMC) method combined with tensor networks by taking advantages of both to study fermionic models. The variational wave function is composed of a pair product wave function operated by real space correlation factors and tensor networks. Moreover, we can apply quantum number projections, such as spin, momentum and lattice symmetry projections, to recover the symmetry of the wave function to further improve the accuracy. We benchmark our method for one- and two-dimensional Hubbard models, which show significant improvement over the results obtained individually either by mVMC or by tensor network. We have applied the present method to hole doped Hubbard model on the square lattice, which indicates the stripe charge/spin order coexisting with a weak $d$-wave superconducting order in the ground state for the doping concentration less than 0.3, where the stripe oscillation period gets longer with increasing hole concentration. The charge homogeneous and highly superconducting state also exists as a metastable excited state for the doping concentration less than 0.25.