Superconductivity in the two-dimensional Hubbard model revealed by neural quantum states

TL;DR

This paper uses advanced neural quantum state methods to investigate the ground state phases of the square lattice Hubbard model, providing evidence for superconductivity at certain doping levels and elucidating the role of next-neighbor hopping.

Contribution

The study introduces a novel neural quantum state wavefunction extension using Pfaffians to accurately simulate large electron systems and resolve competing phases in the Hubbard model.

Findings

Superconductivity coexists with partially-filled stripes at strong coupling.

Uniform d-wave superconductivity is observed at U=4 with sufficient doping.

Negative next-neighbor hopping stabilizes superconductivity with stripes.

Abstract

Whether the ground state of the square lattice Hubbard model exhibits superconductivity remains a major open question, central to understanding high temperature cuprate superconductors and ultra-cold fermions in optical lattices. Numerical studies have found evidence for stripe-ordered states and superconductivity at strong coupling but the phase diagram remains controversial. Here, we show that one can resolve the subtle energetics of metallic, superconducting, and stripe phases using a new class of neural quantum state (NQS) wavefunctions that extend hidden fermion determinant states to Pfaffians. We simulate several hundred electrons using fast Pfaffian algorithms allowing us to measure off-diagonal long range order. At strong coupling and low hole-doping, we find that a non-superconducting filled stripe phase prevails, while superconductivity coexisting with partially-filled stripes is stabilized by a negative next neighbor hopping t-prime, with |t-prime| > 0.1. At larger doping levels, we introduce momentum-space correlation functions to mitigate finite size effects that arise from weakly-bound pairs. These provide evidence for uniform d-wave superconductivity at U = 4, even when t-prime = 0. Our results highlight the potential of NQS approaches, and provide a fresh perspective on superconductivity in the square lattice Hubbard model.