Trapped Fermions Through Kolmogorov-Arnold Wavefunctions

TL;DR

This work develops a variational Monte Carlo framework for a 1D trapped spin-1/2 fermion mixture using Kolmogorov-Arnol'd networks (KANs) to build universal neural-network wavefunctions. The authors augment the standard Slater-Jastrow form with a KAN-based cusp-aware Jastrow factor and an explicit short-distance term to capture delta-function interactions, achieving sub-percent accuracy across diverse particle numbers and coupling strengths. Training employs ADAM optimization with a spline-based representation, progressively increasing the spline knots to study convergence and demonstrating exponential improvement of precision with knot count. The results reproduce exact and known analytic benchmarks (Busch, McGuire, perturbation theory), reveal pairing gaps for attractive interactions, and suggest the approach functions as an effectively exact Monte Carlo method within the studied regime, with potential extensions to three dimensions and effective theories. Overall, the method offers a scalable, accurate route to ground-state properties in strongly interacting quantum many-body systems using universal neural-network wavefunctions.

Abstract

We investigate a variational Monte Carlo framework for trapped one-dimensional mixture of spin-$\frac{1}{2}$ fermions using Kolmogorov-Arnold networks (KANs) to construct universal neural-network wavefunction ansätze. The method can, in principle, achieve arbitrary accuracy, limited only by the Monte Carlo sampling and was checked against exact results at sub-percent precision. For attractive interactions, it captures pairing effects, and in the impurity case it agrees with known results. We present a method of systematic transfer learning in the number of network parameters, allowing for efficient training for a target precision. We vastly increase the efficiency of the method by incorporating the short-distance behavior of the wavefunction into the ansätz without biasing the method.

Figures (6)

• Figure 1: Diagram depicting the Kolmogorov-Arnold neural network representing the function $\kappa(X^\uparrow, X^\downarrow)$ in the ${N^\uparrow}=3, {N^\downarrow}=2$ case. Only some of the lines representing the functions $\phi_{ij}^{\uparrow\downarrow}, \rho_i$ are labelled.
• Figure 2: Quadratic spline specified by the values of $y$ at the knots (solid circles) and the new knots introduced during training.
• Figure 3: Energy as a function of training step for the ${N^\uparrow}={N^\downarrow}=5$ case. The coupling and the number of splines changes during the training at the steps shown by the dashed lines: $g=-1, K=10$ in steps $0$ to $200$, $g=-2, K=20$ in steps $200$ to $400$ and $g=-3, K=40$ in steps $400$ to $1200$. Error bars represent only the statistical noise and are not visible on this scale.
• Figure 4: left panel: Interaction energy per particle of the ground state $\mathcal{E}=E-E_{\text{osc}}$ as a function of the total number of particles ${N^\uparrow}+{N^\downarrow}$. right panel:$\Delta^2 \mathcal{E} = \mathcal{E}({N^\uparrow}, {N^\uparrow})-\frac{1}{2}(\mathcal{E}({N^\uparrow}+1, {N^\uparrow})+\mathcal{E}({N^\uparrow}-1, {N^\uparrow}))$, as a function of ${N^\uparrow}$ for $g=-3.0$. The even-odd pairing effect is evident. The error bars include only the statistical uncertainties and a one-sided $1\%$ theoretical error (since the variational bound guarantees a rigorous upper bound in the energy) at each value of the energy. Notice that the $\approx 1\%$ uncertainty is enhanced by the particular combination contributing to $\Delta^2\mathcal{E}$, especially at larger ${N^\uparrow}$.
• Figure 5: Left: Log plot of the relative error in the ground state energy of the $g=3.0$ case, $N_\uparrow=N_\downarrow=5$, (in red) and ${N^\uparrow}={N^\downarrow}=4$ (in blue) system. We take $E_{\text{asym}}$ as the value obtained at $K=320$ ($K=160$). The fits, shown as dashed lines, suggest an exponential increase of precision with the error scaling as $\sim e^{-K/35.9} (e^{-K/16.9})$. Right: Interaction energy $\mathcal{E}$ for $N_\uparrow=N_\downarrow=15$ particles plotted against the coupling strength $g$. The gray line is the result from first order perturbation theory and the dashed line represents the dimer limit. The error bars include the statistical uncertainty only.