[Paper Review] Neural-network quantum states for ultra-cold Fermi gases

Jane Kim, Gabriel Pescia+6 more|arXiv (Cornell University)|May 15, 2023

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper introduces a Pfaffian-Jastrow neural-network quantum state with message-passing architecture and backflow transformations to efficiently model strong pairing correlations in ultra-cold Fermi gases. It achieves lower ground-state energies than diffusion Monte Carlo benchmarks and enables stable, transfer learning–accelerated exploration of the BCS-BEC crossover, including near-unitarity regimes.

ABSTRACT

Ultra-cold Fermi gases display diverse quantum mechanical properties, including the transition from a fermionic superfluid BCS state to a bosonic superfluid BEC state, which can be probed experimentally with high precision. However, the theoretical description of these properties is challenging due to the onset of strong pairing correlations and the non-perturbative nature of the interaction among the constituent particles. This work introduces a novel Pfaffian-Jastrow neural-network quantum state that includes backflow transformation based on message-passing architecture to efficiently encode pairing, and other quantum mechanical correlations. Our approach offers substantial improvements over comparable ansätze constructed within the Slater-Jastrow framework and outperforms state-of-the-art diffusion Monte Carlo methods, as indicated by our lower ground-state energies. We observe the emergence of strong pairing correlations through the opposite-spin pair distribution functions. Moreover, we demonstrate that transfer learning stabilizes and accelerates the training of the neural-network wave function, enabling the exploration of the BCS-BEC crossover region near unitarity. Our findings suggest that neural-network quantum states provide a promising strategy for studying ultra-cold Fermi gases.

Motivation & Objective

• To develop a flexible, symmetry-respecting neural-network quantum state that captures strong pairing and correlation effects in ultra-cold Fermi gases without relying on biased physical ansätze.
• To overcome limitations of traditional variational Monte Carlo and diffusion Monte Carlo methods, particularly the fixed-node approximation and dependence on initial wave function choice.
• To enable accurate and stable simulation of the BCS-BEC crossover, especially near the unitary limit, using transfer learning to avoid local minima.
• To provide a scalable and generalizable framework applicable to strongly correlated fermionic systems, including nuclear matter and molecules.
• To demonstrate that Pfaffian-Jastrow neural-network states can encode non-perturbative correlations more effectively than Slater-Jastrow or geminal-based approaches.

Proposed method

• The proposed method uses a Pfaffian-Jastrow (PJ) neural-network quantum state (PJ-BF) that combines a Jastrow correlation factor with a Pfaffian determinant to describe pairing correlations.
• A message-passing neural network (MPNN) is employed to compute backflow transformations, which dynamically reconfigure single-particle orbitals based on many-body correlations.
• The MPNN architecture encodes spatial and spin-dependent correlations by propagating information between particles through a graph-based representation of the system.
• The wave function is optimized using variational Monte Carlo (VMC) with stochastic reconfiguration, ensuring efficient and stable training.
• Transfer learning is applied by pre-training on systems with smaller particle numbers or known physics, then fine-tuning for larger or more complex systems, especially near unitarity.
• Symmetries such as translational invariance, parity, and time-reversal are explicitly enforced in the neural network architecture to ensure physical consistency.

Experimental results

Research questions

• RQ1 Can a neural-network quantum state with backflow and message-passing architecture outperform standard Slater-Jastrow and diffusion Monte Carlo methods in simulating ultra-cold Fermi gases?
• RQ2 How effectively can transfer learning stabilize and accelerate training in strongly correlated fermionic systems, particularly near the unitary limit?
• RQ3 To what extent can the Pfaffian-Jastrow ansatz capture non-perturbative pairing correlations without prior assumptions about orbital structure?
• RQ4 Can the proposed method be generalized to other strongly correlated systems, such as nuclear matter or isospin-asymmetric systems?
• RQ5 How does the inclusion of backflow via MPNN improve the description of the BCS-BEC crossover compared to single-particle picture approaches?

Key findings

• The Pfaffian-Jastrow neural-network state with backflow (PJ-BF) achieves lower ground-state energies than the diffusion Monte Carlo benchmark, indicating superior accuracy.
• Transfer learning enables stable and efficient training across the BCS-BEC crossover, particularly near unitarity, reducing the risk of convergence to local minima.
• The opposite-spin pair distribution function reveals strong pairing correlations, confirming the model’s ability to capture non-perturbative pairing behavior.
• The MPNN-based backflow transformation significantly improves correlation description over standard single-particle backflow, especially in deep pairing regimes.
• The method outperforms comparable Slater-Jastrow ansätze with identical MPNN architectures, demonstrating the advantage of the Pfaffian structure in encoding pairing.
• The framework is generalizable: unpolarized systems use a single architecture, while $N\pm1$ systems can be treated by adding a single FNN for the unpaired orbital, enabling direct gap calculations.