Learning pairing correlations

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Learning pairing correlations
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Pairing phenomena are widespread in quantum systems, manifesting in various forms such as Cooper pairing in superconductors, electron pairing in atomic orbitals, and nucleon pairing in atomic nuclei. This ubiquity extends to Fermi gases, where attractive interactions among constituent fermions can induce superfluid behavior at extremely low temperatures. While ultra-cold Fermi gases can be studied in the laboratory with meticulous precision, their non-perturbative interactions pose theoretical challenges for quantum many-body methods. 
In particular, the transition between the Bardeen-Cooper-Schrieffer (BCS) phase of loosely-bound Cooper pairs and the Bose-Einstein condensate (BEC) phase of tightly-bound dimers is marked by a crossover region dominated by strongly-interacting pairs. The point of maximum interaction strength, known as the "unitary limit," is especially interesting for studying pairing phenomena due to connections to seemingly unrelated systems, like the crusts of neutron stars. Ultra-cold Fermi gases near the unitary limit also serve as an important testing ground for developing new computational methods, as the strong pairing correlations render traditional single-particle descriptions insufficient. 

The BCS-BEC crossover.  In the BCS regime (left), the attractive interactions among the fermions are weak, leading to loosely-bound Cooper pairs. In the BEC regime (right), the attractive interactions are so strong that the pairs condense into tightly-bound bosonic dimers. Due to the Pauli exclusion principle of the constituent fermions, the net interactions among the composite bosons are weak. The unitary Fermi gas (middle) represents the strongest-interacting case in the entire BCS-BEC crossover. Our work focuses on ultra-cold Fermi gases at and around this unitary limit.

In our recent paper, we introduce a Pfaffian neural-network quantum state to learn pairing correlations in the BCS-BEC crossover near the unitary limit. Much like how determinants enforce the antisymmetry of fermionic wave functions constructed from single-particle orbitals, Pfaffians serve the same role for wave functions constructed from a pairing orbital. However, our ansatz differs from previous Pfaffian wave functions because we represent the pairing orbital with a neural network, enabling our model to learn arbitrary relationships between the pairs. To further enhance the flexibility of our method, we incorporate a message-passing neural network that iteratively builds backflow correlations into new pair features that are later fed into the pairing orbital. Then, our ansatz is trained by minimizing the expectation value of the Hamiltonian, in which the interaction potential is taken to be very strong and short-ranged. 

Before arriving at our Pfaffian ansatz, we experimented with different neural-network quantum states based on the single-particle picture, i.e. Slater determinants. This proved to be difficult. For one, the presence of non-perturbative interactions led to substantial numerical instabilities. In response, we adopted a transfer learning strategy, initially training our ansatz on softer interactions before proceeding to harder ones. While transfer learning helped stabilize the training, even the softest interaction yielded ground-state energies that fell short of our expectations, particularly when compared to benchmark energies from state-of-the-art diffusion Monte Carlo calculations. This initial outcome was disappointing, considering our ambition to develop an ansatz capable of learning the ground state of any continuous-space fermionic system. Nonetheless, it prompted us to explore an alternative path centered around the Pfaffian. This pivot revealed immediate benefits, with our model consistently outperforming benchmark energies across various configurations of the message-passing neural network—see Figure 1 of the original paper. 

One of the most notable features of our Pfaffian ansatz is its independence from particle number, presenting further opportunities for leveraging the transfer learning approach. Specifically, the number of trainable parameters depends solely on the spatial dimension and the hyperparameters chosen for the individual feedforward neural networks. In Supplementary Table 1 of the original paper, we explicitly provide the total number of parameters, assuming an unpolarized system and rectangular-shaped feedforward neural networks. To progress towards the thermodynamic limit, one has the option to take a trained Pfaffian ansatz as an initial point for systems with larger number of particles, rather than training the larger system from scratch. However, there is one small modification that is necessary to handle odd particle numbers: introducing an unpaired single-particle orbital. This orbital is represented by another feedforward neural network, while all other pieces of the ansatz remain the same. Thus, the pairing orbital from the even systems can be transferred to the odd systems, exploiting prior knowledge and saving computational resources. 

The inherent strength of using neural networks as wave functions lies in their remarkable ability to adapt across a wide range of problem domains. Given the particularly extreme pairing correlations observed in ultra-cold Fermi gases, the success of the Pfaffian ansatz is anticipated to translate to other strongly-paired quantum systems. Aside from changing the interaction potential, only minor adjustments of the ansatz would be necessary, such as modifying the input degrees of freedom or enforcing different symmetries and boundary conditions. Moreover, we show how transfer learning can be used to more easily navigate strong interactions, explore the BCS-BEC crossover, and approach the thermodynamic limit. Utilizing the Pfaffian ansatz, along with the transfer learning strategy demonstrated in our work, holds great promise for advancing our understanding of superfluidity in fermionic systems and reaching larger systems than ever before.

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Quantum Physics
Physical Sciences > Physics and Astronomy > Quantum Physics
Theoretical, Mathematical and Computational Physics
Physical Sciences > Physics and Astronomy > Theoretical, Mathematical and Computational Physics

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