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arXiv:2507.10705 · Proceedings of the National Academy of Sciences

Neural network–augmented Pfaffian wave-functions for scalable simulations of interacting fermions

Ao Chen, Zhou‐Quan Wan, Anirvan M. Sengupta, Antoine Georges, Christopher Roth

Published Aug 28, 2026• 3 citations
arXivHTMLPDF

Abstract

Developing accurate numerical methods for strongly interacting fermions is crucial for improving our understanding of various quantum many-body phenomena, especially unconventional superconductivity. Recently, neural quantum states have emerged as a promising approach for studying correlated fermions, highlighted by the hidden fermion and backflow methods, which use neural networks to model corrections to fermionic quasiparticle orbitals. In this work, we expand these ideas to the space of Pfaffians, a wave-function that naturally expresses superconducting pairings, and propose the hidden fermion Pfaffian state (HFPS), which flexibly represents both unpaired and superconducting phases and scales to large systems with favorable asymptotic complexity. In our numerical experiments, HFPS provides state-of-the-art variational accuracy in different regimes of both the attractive and repulsive Hubbard models. We show that the HFPS is able to capture general phases with and without pairing, and therefore may be a useful tool for modeling phases with unconventional superconductivity.

DOI
10.1073/pnas.2535288123

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Developing accurate numerical methods for strongly interacting fermions is crucial for improving our understanding of various quantum many-body phenomena, especially unconventional superconductivity. Recently, neural quantum states have emerged as a promising approach for studying correlated fermions, highlighted by the hidden fermion and backflow methods, which use neural networks to model corrections to fermionic quasiparticle orbitals. In this work, we expand these ideas to the space of Pfaffians, a wave-function that naturally expresses superconducting pairings, and propose the hidden fermion Pfaffian state (HFPS), which flexibly represents both unpaired and superconducting phases and scales to large systems with favorable asymptotic complexity. In our numerical experiments, HFPS provides state-of-the-art variational accuracy in different regimes of both the attractive and repulsive Hubbard models. We show that the HFPS is able to capture general phases with and without pairing, and therefore may be a useful tool for modeling phases with unconventional superconductivity.
We introduce a systematically improvable family of variational wave functions for the simulation of strongly correlated fermionic systems. This family consists of Slater determinants in an augmented Hilbert space involving "hidden" additional fermionic degrees of freedom. These determinants are projected onto the physical Hilbert space through a constraint that is optimized, together with the single-particle orbitals, using a neural network parameterization. This construction draws inspiration from the success of hidden-particle representations but overcomes the limitations associated with the mean-field treatment of the constraint often used in this context. Our construction provides an extremely expressive family of wave functions, which is proved to be universal. We apply this construction to the ground-state properties of the Hubbard model on the square lattice, achieving levels of accuracy that are competitive with those of state-of-the-art variational methods.
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.
Among the variational wave functions for fermionic Hamiltonians, neural network backflow (NNBF) and hidden fermion determinant states (HFDS) are two prominent classes that provide accurate approximations to the ground state. Here we develop a unifying view of fermionic neural quantum states casting them all in the framework of NNBF. NNBF wave functions have configuration-dependent single-particle orbitals (SPO) which are parameterized by a neural network. We show that HFDS can be written as a NNBF wave function with a restricted low-rank rrr additive correction to the SPO times a neural-network generated determinant of an $r\ifmmode\times\else\texttimes\fi{}r$ matrix. Furthermore, we show that in NNBF wave functions, this $r\ifmmode\times\else\texttimes\fi{}r$ determinant can generically be removed when rrr is less than or equal to the number of fermions, at the cost of further complicating the additive SPO correction increasing its rank by rrr. We numerically and analytically compare additive SPO corrections generated by the product of two matrices with inner dimension rrr. We find that larger rrr wave functions span a larger space and give evidence that simpler and more direct updates to the SPO's tend to be more expressive and better energetically. These suggest the standard NNBF approach is preferred amongst other related choices. Finally, we uncover that the row selection used to select single-particle orbitals allows significant sign and amplitude modulation between nearby configurations and is partially responsible for the quality of NNBF and HFDS wave functions.
We propose a general framework for finding the ground state of many-body fermionic systems by using feed-forward neural networks. The anticommutation relation for fermions is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N3)O(N^3)O(N3) for NNN particles. We bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost to O(N2)O(N^2)O(N2) or less. We show that the accuracy of the approximation can be improved by optimizing the variance of the energy simultaneously with the energy itself. We also find that a reweighting method in Monte Carlo sampling can stabilize the calculation. These improvements can be applied to other approaches based on variational Monte Carlo methods. Moreover, we show that the accuracy can be further improved by using the symmetry of the system, the representative states, and an additional neural network implementing a generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the method by applying it to a two-dimensional Hubbard model.
Strongly correlated materials exhibit complex emergent phenomena -- such as unconventional high-temperature superconductivity -- that cannot be explained within a single or weakly-interacting particle description, making understanding their physics a central challenge in condensed matter physics. This thesis develops and applies novel approaches to strongly correlated electron systems, focusing on neural quantum states (NQS) and quantum simulation with cold atoms in optical lattices or Rydberg tweezer arrays. A key contribution of this work is the advancement of neural-network-based representations of quantum many-body states. By employing state-of-the-art network architectures like Transformers and combining them with physical constructions such as determinants, Pfaffians and Gutzwiller-projection, NQS are extended to simulate large two-dimensional fermionic single- and multi-band systems with high accuracy and beyond the capabilities of conventional methods. In addition, the thesis explores how quantum simulators based on ultracold atoms in optical lattices can be leveraged to study models directly relevant to strongly correlated materials -- e.g. the multi-band model of cuprate superconductors -- as well as proposes strategies to probe superconductivity in this setting. Furthermore, hybrid approaches that combine NQS with data from quantum simulators are developed. This synergy allows for improved optimization and direct comparison between theory and experiment. These methodological developments are applied to investigate two fundamental questions in high-temperature superconductivity: On the one hand, the work proposes a microscopic mechanism based on Feshbach resonances and tests this hypothesis extensively in a model system relevant to cuprates, nickelates and heavy fermion materials. On the other hand, it develops systematic approaches to derive effective low-energy models from more complex systems like multi-band models or electron-phonon coupled systems, shining light on the timely question which single-band models capture the physics of high-temperature superconductors like cuprate or nickelate materials. Overall, the thesis demonstrates how combining machine learning and quantum simulation provides powerful tools for understanding strongly correlated quantum matter.
We propose a general framework for finding the ground state of many-body fermionic systems by using feed-forward neural networks. The anticommutation relation for fermions is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N3)O({N}^{3})O(N3) for NNN particles. We bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost from O(N3)O({N}^{3})O(N3) to O(N2)O({N}^{2})O(N2) or less, while the overall cost depends on the number of Monte Carlo samples and the number of parameters in neural networks. We show that the accuracy of the approximation can be improved by optimizing the ``variance'' of the energy simultaneously with the energy itself. We also find that a reweighting method in Monte Carlo sampling can stabilize the calculation. These improvements can be applied to other approaches based on variational Monte Carlo methods. Moreover, we show that the accuracy can be further improved by using the symmetry of the system, the representative states, data preprocessing, and an additional neural network implementing a generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the method by applying it to a two-dimensional Hubbard model.