Neural Flux Attachment: From Bose Condensates to Chiral Topological Matter

Abstract

Can one neural wave function describe both a Bose condensate and a chiral topological liquid? We introduce ChernFormer, which combines a fermionic transformer with a fixed Chern-Simons phase that attaches one statistical vortex to every particle pair. Each factor changes sign under exchange, so their product is exactly bosonic. The fixed phase changes statistics but not probability, making every bosonic learning problem equivalent to a fermionic one with the same approximation error and overlap. With enough capacity, ChernFormer can approximate any normalizable bosonic wave function on the plane at fixed particle number. A finite, smooth network still vanishes when particles meet, yet this contact hole can shrink while the wave function and condensate fraction approach those of a nodeless condensate. Following the needle-in-a-haystack target-reconstruction benchmark introduced in \cite{NazaryanGaggioliTengFu2025}, we test ChernFormer on the Kalmeyer--Laughlin ground state and its first two chiral edge states. The overlap curves stay close to unity through their largest sampled sizes, while independent amplitude and phase maps at $N=20$ for all three states recover both local Laughlin vortices and the collective edge vortex. By contrast, a continuous, nonzero product of identical particle-wise factors misses these elementary edge sectors. ChernFormer therefore provides one variational language for conventional bosonic order and chiral topological matter.