Beyond Variational Bias: Resolving Intertwined Orders in the Hubbard Model

TL;DR

This paper demonstrates that different variational ansatzes can lead to qualitatively different conclusions about the Hubbard model's ground state, but improved methods reveal a consistent physical picture of coexisting orders.

Contribution

It introduces three Transformer-based fermionic wave functions and shows that systematic improvements unify their physical predictions, emphasizing the importance of correlation functions.

Findings

Different ansatzes yield distinct correlation patterns.

Systematic improvements lead to a consistent physical picture.

Variational energy alone is insufficient to determine the ground state.

Abstract

The two-dimensional Hubbard model at finite doping hosts competing or intertwined orders, resulting in conflicting conclusions from different computational approaches regarding its ground state. We show that a key source of such discrepancies is the bias encoded in the variational ansatz. We consider three different Transformer backflow fermionic wave functions based on a Slater determinant, its particle-hole counterpart, and a Pfaffian, initialized without any mean-field pretraining. We show that, despite achieving nearly degenerate, state-of-the-art variational energies, each ansatz converges to a state with qualitatively different spin, charge, and pairing correlations. Upon improving accuracy via symmetry restoration and variance reduction, however, all three converge to the same physical picture: coexisting superconducting and stripe orders. These results demonstrate that variational energy alone is insufficient to identify the ground state in the presence of competing phases, and highlight the importance of tracking how correlation functions evolve as the wave function is systematically improved before drawing physical conclusions.