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arXiv:2604.04435 · Physical Review Research

Neural-network quantum states for solving few-body problems: Application to Efimov physics

Sora Yokoi, Shimpei Endo, Hiroki Saito

Published Aug 28, 2026• 1 citation
arXivHTMLPDF

Abstract

Neural-network quantum states have been developed as an efficient method for solving quantum many-body problems, not only in lattice systems but also in systems of particles in continuous space. Here, we apply this approach to strongly interacting few-body problems in continuous space at unitarity: the Efimov states and associated few-body bound states. We extend the previous work [], in which the ground states of few-boson systems were obtained, to the first excited states with a projection method, and also to a mass-imbalanced fermionic system consisting of two identical fermions and a third particle. The obtained energies of the ground and first excited states of these systems agree well with previously reported results. Furthermore, the proposed approach also reproduces the discrete scale invariance between the ground and first excited states and the critical-mass behavior in mass-imbalanced fermionic systems.

DOI
10.1103/31mv-7tz7

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Neural-network quantum states have been developed as an efficient method for solving quantum many-body problems, not only in lattice systems but also in systems of particles in continuous space. Here, we apply this approach to strongly interacting few-body problems in continuous space at unitarity: the Efimov states and associated few-body bound states. We extend the previous work [], in which the ground states of few-boson systems were obtained, to the first excited states with a projection method, and also to a mass-imbalanced fermionic system consisting of two identical fermions and a third particle. The obtained energies of the ground and first excited states of these systems agree well with previously reported results. Furthermore, the proposed approach also reproduces the discrete scale invariance between the ground and first excited states and the critical-mass behavior in mass-imbalanced fermionic systems.
We apply a deep-neural-network (DNN) ansatz to the symmetrized spectator vector of the subtracted three-body Faddeev integral equation for identical bosons near the Efimov limit. The network is trained by minimizing the residual of the discretized integral equation, while the positive binding scale associated with the three-body energy is treated as a trainable parameter. Deterministic diagonalization of the same discretized kernel is used only as an a posteriori numerical benchmark. As preliminary validation, the neural-solver strategy is tested on the analytically solvable hydrogen radial problem. At unitarity, the DNN reproduces the Efimov ground-state binding scale with a DNN--deterministic deviation of 0.022%0.022\%0.022%, while the first excited state is recovered to 0.002%0.002\%0.002%. The deterministic solver recovers the universal Efimov scaling ratio e2π/s0≃515.03e^{2π/s_0}\simeq 515.03e2π/s0​≃515.03, and the neural method traces the bound-state branches as a function of the inverse scattering length 1/a1/a1/a by continuation from the unitary solution. These results indicate that DNN-based residual minimization can provide a compact and differentiable representation of a renormalized few-body integral-equation solution in a regime governed by discrete scale invariance.
Neural networks have shown to be a powerful tool to represent the ground state of quantum many-body systems, including fermionic systems. However, efficiently integrating lattice symmetries into neural representations remains a significant challenge. In this work, we introduce a framework for embedding lattice symmetries in fermionic wavefunctions and demonstrate its ability to target both ground states and low-lying excitations. Using group-equivariant neural backflow transformations, we study the t-V model on a square lattice away from half-filling. Our symmetry-aware backflow significantly improves ground-state energies and yields accurate low-energy excitations for lattices up to 10 × 10. We also compute accurate two-point density-correlation functions and the structure factor to identify phase transitions and critical points. These findings introduce a symmetry-aware framework important for studying quantum materials and phase transitions. In this manuscript, the authors introduce a method for numerically modelling fermionic systems using a neural network wavefunction ansatz. They demonstrate that this method can efficiently and accurately find ground and low-lying excited states in two-dimensional models, outperforming existing approaches.
Solving the many-body Schr\"odinger equation in continuous spaces with the presence of strong correlations is an extremely important and challenging issue in quantum physics. In this work, we propose the functional tensor network (FTN) approach to solve the many-body Schr\"odinger equation. Provided the orthonormal functional bases, we represent the coefficients of the many-body wave function as a tensor network. The observables, such as energy, can be calculated simply by tensor contractions. Simulating the ground state becomes solving a minimization problem defined by the tensor network. An efficient gradient-decent algorithm based on automatically differentiable tensors is proposed. We here take the matrix product state (MPS), whose complexity scales only linearly with the system size, as an example. We apply our approach to solve the ground state of coupled harmonic oscillators and achieve high accuracy by comparing our results with the exact solutions. Reliable results are also given in the presence of three-body interactions, where the system cannot be decoupled from isolated oscillators. Our approach is simple and, with well-controlled error, essentially different from highly nonlinear neural-network solvers. Our work extends the applications of the tensor network from quantum lattice models to systems in continuous space. The FTN can be used as a general solver of the differential equations with many variables. The MPS exemplified here can be generalized to, e.g., fermionic tensor networks to solve the electronic Schr\"odinger equation.
Neural-network quantum states have been successfully used to study a variety of lattice and continuous-space problems. Despite a great deal of general methodological developments, representing fermionic matter is however still early research activity. Here we present an extension of neural-network quantum states to model interacting fermionic problems. Borrowing techniques from quantum simulation, we directly map fermionic degrees of freedom to spin ones, and then use neural-network quantum states to perform electronic structure calculations. For several diatomic molecules in a minimal basis set, we benchmark our approach against widely used coupled cluster methods, as well as many-body variational states. On some test molecules, we systematically improve upon coupled cluster methods and Jastrow wave functions, reaching chemical accuracy or better. Finally, we discuss routes for future developments and improvements of the methods presented.
The mass-imbalanced fermionic mixture is studied, where $N\ensuremath{\le}5$ identical fermions interact resonantly with an impurity, a distinguishable atom. The shell structure is explored, and the physics of a dynamic light-impurity is shown to be different from that of the static heavy-impurity case. The energies in a harmonic trap at unitarity are calculated and extrapolated to the zero-range limit. In doing so, the scale factor of the ground state, as well as of a few excited states, is calculated. In the $2\ensuremath{\le}N\ensuremath{\le}4$ systems, pure (N+1)(N+1)(N+1) Efimov states exist for large enough mass ratio. However, no sign for a six-body Efimov state in the (5+1)(5+1)(5+1) system is found in the mass ratio explored, $M/m\ensuremath{\le}12$.