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arXiv:2608.06168

Neural Flux Attachment: From Bose Condensates to Chiral Topological Matter

Rudik Badalyan, Khachatur G. Nazaryan, Tigran A. Sedrakyan

cond-mat.str-elcond-mat.dis-nnquant-phSubmitted Aug 6, 2026• Announced Aug 7, 2026• 0 citations
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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.

Comments
12 pages, 4 figures

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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=20N=20N=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.
Neural networks (NNs) have great potential in solving the ground state of various many-body problems. However, several key challenges remain to be overcome before NNs can tackle problems and system sizes inaccessible with more established tools. Here, we present a general and efficient method for learning the NN representation of an arbitrary many-body complex wave function from its N-particle probability density and probability current density and successfully test on (non-Abelian) fractional quantum Hall states and chiral BCS wavefunction. Having reached overlaps as large as 99.9%, we employ our neural wave function for pre-training to effortlessly solve the fractional quantum Hall problem with Coulomb interactions and realistic Landau-level mixing for as many as 25 particles and uncover distinctive features of the edge. Our work demonstrates efficient, scalable and accurate simulation of highly-entangled quantum matter using general-purpose deep NNs enhanced with physics-informed initialization.
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Gauge theories form a large class of interacting conformal field theories in 3D, among which critical Chern-Simons-matter theories play a prominent role. Here, we study a simple instance: a gapless complex scalar coupled to a U(1)_{2} Chern-Simons gauge field. This theory is conjectured to be dual to three other Chern-Simons-matter theories and to describe the transition between a Kalmeyer-Laughlin chiral spin liquid (a ν=1/2 bosonic Laughlin state) and a trivially gapped phase. Using the fuzzy-sphere regularization, we realize the corresponding quantum phase transition between a ν_{f}=2 fermionic integer quantum Hall state and a ν_{b}=1/2 bosonic fractional quantum Hall state. We show that the transition is continuous and governed by a conformal field theory with SO(3) global symmetry. The operator spectrum contains only one relevant Lorentz scalar, the SO(3) singlet with scaling dimension Δ_{S}=1.52(18), which tunes the transition. Our nonperturbative results identify the universality class naturally associated with the U(1)_{2} Chern-Simons-Higgs theory and show that the conjectured dualities remain self-consistent.
Chiral spin liquids realize a topological state whose universal response is a fractional spin Hall conductance νsν_sνs​. The three quantities that determine this response, the integer Chern number of the fractionalized spinons, the level of the emergent Chern--Simons gauge field, and the physically measured spin pump, are related but distinct, and their relation is often stated only schematically. Here we derive it from a single object: the parity-odd determinant of a gapped Dirac cone on a spatial cylinder, resummed exactly to all orders in the compact holonomy. This determinant fixes the map from spinon topology to measurable response, and proves that finite-size corrections to the topological pump are strictly exponential, with no universal 1/L1/L1/L term. We test the resulting predictions on the kagome chiral spin liquid at three independent levels: the exact one-loop field theory, a parton band-structure calculation (C=−1C=-1C=−1, converging exponentially over cylinders four to twelve sites wide), and an interacting density-matrix renormalization group flux pump on the explicitly chiral JJJ--JχJ_χJχ​ Hamiltonian (νs=−0.500±0.011ν_s=-0.500\pm0.011νs​=−0.500±0.011). All three agree with the analytic prediction without adjustable parameters, providing a fully quantitative bridge between microscopic topology and observable fractional response.