Dressing composite fermions with artificial intelligence

TL;DR

This work tackles LL mixing in the fractional quantum Hall effect by introducing CF-Flow, a variational approach that dresses Jain CF wavefunctions with backflow corrections learned by symmetry-preserving neural networks. By embedding CF theory with a Feynman-Cohen–style backflow and implementing permutation and $O(3)$ symmetries via DeepSets and equivariant architectures, CF-Flow achieves accurate ground-state energies at Jain fillings with far fewer parameters and faster convergence than prior neural quantum states. It also provides access to excited states and quantifies the transport gap at $ u=1/3$, finding an exponential decay with LL mixing that remains finite in the large-mixing limit, indicating a first-order transition away from the FQH liquid. The framework enables transfer learning across system sizes and parameter ranges, offering a scalable tool for exploring nonperturbative FQHE physics, including potential extensions to disorder and FQHE in Chern bands.

Abstract

Recent variational studies have demonstrated that the strongly correlated ground states of the fractional quantum Hall (FQH) effect can be captured using machine learning approaches starting from no prior knowledge of the underlying physics. We introduce a complementary framework that instead starts from Jain's composite-fermion (CF) wavefunctions, which accurately describe FQH states as weakly interacting states of CFs at fillings $ν= n/(2pn+1)$ in an idealized limit. As we move away from this idealized limit to one more in line with experimental reality, we expect CFs to become dressed much like the electrons of a noninteracting system, which are dressed by neutral excitations as interaction is turned on adiabatically, as in Landau's Fermi-liquid theory. We model this dressing using a Feynman-Cohen-style backflow approach, implemented through symmetry-preserving neural networks-a framework we refer to as CF-Flow. CF-Flow achieves competitive accuracy with substantially greater computational efficiency and scales to systems of $\gtrsim 26$ electrons. At fillings $ν= 1/3$ and $2/5$, as a function of Landau-level mixing strength, CF-Flow produces ground-state energies with low local-energy variance that are nearly indistinguishable from those obtained using the fixed-phase diffusion Monte Carlo (fp-DMC) method, even though the latter constrains the wavefunction phase to that of the lowest Landau level-thereby providing insight into why fp-DMC has been successful in giving an accurate quantitative account of several experiments. Finally, the symmetry-preserving architecture of CF-Flow enables access to excited states and computation of the transport gap at $ν= 1/3$, where we find, unexpectedly, that it decays exponentially toward a finite value in the limit of large Landau-level mixing, suggesting a first-order transition from the FQH liquid to a non-FQH state.

Figures (6)

• Figure 1: Schematic illustration of CF-Flow. In the Feynman--Cohen backflow picture Feynman56, turning on interactions causes particles to shift from their noninteracting positions $\bm{r}_{i}$ to backflow positions $\bm{R}_{i}$. CF-Flow implements this idea by combining the backflow approach with composite-fermion (CF) wavefunctions defined on the Haldane sphere using symmetry-preserving machine-learning methods, enabling us to study regimes in which interactions between CFs become significant.
• Figure 2: Decomposition of the Aharonov-Bohm phase. The line integral $\int \bm{A}\cdot \bm{dl}$ along the path $\mathcal{P}$ between $\Omega$ and $\Omega^{\prime}$ (top) can always be decomposed into one along a canonical geodesic path $\mathcal{P}_{0}$ and along a closed loop path $\mathcal{L}$ (bottom), where the loop contribution depends only on the enclosed flux—a gauge invariant quantity, which consequently does not depend on the location of $\mathcal{L}$ (since the magnetic field $B$ is uniform everywhere).
• Figure 3: Schematic illustration of the CF-Flow architecture. The Ansatz (top) consists of a CF wavefunction $\Psi_{0}$ evaluated at backflow coordinates $\bm{R}_{i}$ and multiplied by four physically motivated factors. The model takes as input the physical parameters $\kappa$, $Q$, and $N$ (Landau-level mixing strength, monopole strength, and particle number) and constructs geometric features from electron coordinates $\bm{r}_{1}, \dots, \bm{r}_{N}$, pairwise distances $d_{ij}$ and signed volumes $\chi_{ijk}$. Using the DeepSets architecture Zaheer17, multilayer perceptrons (MLPs) learn $O(3)$-invariant scalars parameterizing each component: (i) "two-body" scalars $f_{ij}$ (many-body functions with pair indices) for an amplitude-modulating Jastrow factor (#2) and backflow coordinates (#5), and (ii) "three-body" scalars $\bm{f}_{a}^{ijk}$ ($a=i,j,k$; many-body functions with triplet indices) for a phase-modulating Jastrow factor (#3). The architecture also includes physically enforced terms: Kato's cusp (#1) with short-distance coefficient set by $\mathcal{M}$ (the relative angular momentum at small $d_{ij}$ in $\Psi_{0}$) and $\kappa$, gated at large distances; and the Aharonov-Bohm phase (#4) enforcing our gauge choice (in spinor representation, $\phi_{\rm AB}(U, V;u,v)=2Q\arg[\bar{U}u+\bar{V}v]$). By construction, the combined Jastrow term is $SO(3)$ invariant and transforms to its complex conjugate under inversion, while the full wavefunction is manifestly permutation equivariant.
• Figure 4: Per-particle energy $E/N$ (left) and local-energy standard deviation $\sigma(E)/N$ (right) as functions of the Landau-level mixing strength $\kappa$ during optimization of CF-Flow for $N=18$ electrons at $\nu=1/3$. Color indicates the training epoch. CF-Flow begins from Laughlin's wavefunction—the uniform ($L=M=0$) state of CFs filling the $n=0$$\Lambda$ level [see Eq. \ref{['eq:cf-ground-state']}]—and progressively learns backflow corrections through neural networks (see Fig. \ref{['fig:model-flowchart']}). Training proceeds in five stages: (i) enabling Kato's cusp term, (ii) adding the amplitude Jastrow factor $\exp[\tilde{\mathcal{J}}_{\rm re}(\mathbf{r})]$, (iii) adding the phase Jastrow factor $\exp[i\mathcal{J}_{\rm im}(\mathbf{r})]$, (iv) enabling backflow, and (v) fine-tuning with a reduced learning rate. The red curve marks the best model (lowest mean energy across $\kappa$ over all training runs). The total energy $E$ includes contributions from the uniform background charge and is corrected for the "density shift" inherent to spherical geometry Morf86.
• Figure 5: Per-particle energy $E/N$ for uniform (i.e., FQH ground) states at fillings $\nu = 1/3$ and $\nu = 2/5$ as functions of $1/N$ and the Landau-level mixing strength $\kappa$. Top: CF-Flow results compared with DeepHall Qian25—a recent Psiformer-based neural quantum state approach—and fixed-phase diffusion Monte Carlo (fp-DMC) Zhao18Qian25, which constrains the wavefunction phase to match Eq. \ref{['eq:cf-ground-state']} and provides the ground-state energy within this phase sector. Dashed lines denote linear fits in $1/N$ at fixed $\kappa$. Bottom: Thermodynamic-limit energies $E/N$ ($N \to \infty$) from CF-Flow obtained via linear $1/N$ extrapolation, compared with lowest-Landau-level ($\kappa = 0$) values from CF theory Jain07. Error bars are smaller than the marker size.