Non-Abelian quantum Hall states at strong Landau level mixing: finding anyons with first-principles AI

Pierre-Antoine Graham; Ahmed Abouelkomsan; Liang Fu

cond-mat.mes-hallOct 13, 2026

Non-Abelian quantum Hall states at strong Landau level mixing: finding anyons with first-principles AI

Authors: Pierre-Antoine Graham, Ahmed Abouelkomsan, Liang Fu

Organizations: Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Abstract

Non-Abelian anyons, strange particles forming the basis of fault-tolerant topological quantum computing, are predicted to emerge in fractional quantum Hall states at even-denominator filling factors. Identifying which topological order is realized has proven to be a challenging problem that requires an accurate, non-perturbative treatment of realistic Landau level mixing, which has so far been missing from theoretical studies. Here we present the first complete solution of non-Abelian quantum Hall problem that includes all electrons and all Landau levels, for spin-polarized electrons at filling factor $ν= 3/2$ . We develop a momentum-resolved neural-network variational wave function, optimized solely on the microscopic Hamiltonian, which gives direct access to the topologically degenerate ground states. Without any physics input about the expected outcome, it discovers a degenerate ground-state manifold whose entanglement reveals anyons with quantum dimension $\sqrt{2}$ , the hallmark of Ising-type non-Abelian order. By directly computing the overlap with trial wave functions, we further show that Landau level mixing favors the anti-Pfaffian over other candidate states. Our results demonstrate the power of neural-network wave functions in finding and characterizing non-Abelian anyons from first principles, in regimes of strong Landau level mixing, relevant to many recent experiments.

Figures & tables

Figure 1: Schematic illustration of the momentum-resolved MagNet ansatz. Electron coordinates are mapped to high-dimensional features

\{\boldsymbol{h}_{i}\}

that are periodic on the torus and then passed through

L

layers combining self-attention and multilayer perceptrons. The network outputs the periodic amplitudes

F^{n}_{j}

and the winding maps

\eta^{(\alpha,n)}_{j}

, which parametrize the zeros (crosses) of the quasiperiodic factor

\chi^{n}_{j}

, shown on the supercell with its phase encoded in color. Together,

F^{n}_{j}

and

\chi^{n}_{j}

form the many-body orbitals of the variational wave function

\Psi_{\theta}

in Eq. ( 2 ). The momentum eigenstate

\Phi_{\mathbf{k}}

is then obtained by summing the translated copies of

\Psi_{\theta}

over the

N_{\phi}

center-of-mass magnetic translations

T(\mathbf{a})

, as in Eq. ( 5 ).

Figure 2: (a) Lowest energy in each momentum sector

\mathbf{k}=(k_{1},k_{2})

, labeled by

k_{1}+N_{1}k_{2}

, for

N=21

electrons and

\kappa=1.0

, obtained from NN-VMC and from ED restricted to the LLL and 1LL. We use a hexagonal supercell spanned by

\mathbf{L}_{1}=N_{1}\mathbf{a}_{1}

and

\mathbf{L}_{2}=N_{2}\mathbf{a}_{2}

with

N_{1}=7

and

N_{2}=2

, so that

k_{i}=0,\ldots,N_{i}-1

. The NN-VMC energies lie well below the ED energies in all sectors, and both show a twofold degenerate ground state at

\mathbf{k}=(0,0)

and

(0,1)

. (b) Pair correlation function

g(\mathbf{r})

of the two NN-VMC ground states.

Figure 3: (a) Lowest energies in each momentum sector

\mathbf{k}=(k_{1},k_{2})

, labeled by

k_{1}+N_{1}k_{2}

, for

N=18

electrons and

\kappa=1.0

, obtained from NN-VMC and from ED restricted to the LLL and 1LL. We use a hexagonal supercell spanned by

\mathbf{L}_{1}=N_{1}\mathbf{a}_{1}

and

\mathbf{L}_{2}=N_{2}\mathbf{a}_{2}

with

N_{1}=4

and

N_{2}=3

. Numbers indicate twofold degenerate ED levels within a sector. The blue labeled states are orthogonal ground states obtained with NN-VMC. (b) Differences

\Delta S_{n}

between the second Rényi entropies of the six MESs and the smallest one, for three inequivalent noncontractible cuts of the torus (bottom). For each cut, four MESs have

\Delta S_{n}\approx 0

, consistent with Abelian sectors, while the remaining two lie close to

\log 2

(dashed line), as expected for

d_{\sigma}=\sqrt{2}

Figure 4: (a) Occupation $\nu_{n}$ of Landau level $n$ , on a logarithmic scale, for $N=18$ and $N=21$ electrons at $\kappa=1,2$ and $5$ . Inset: deviation $\Delta\nu_{n}$ from the unmixed occupations $\nu_{0}=1$ , $\nu_{1}=1/2$ , and $\nu_{n\geq 2}=0$ . Both the LLL and the 1LL are depleted, with the weight transferred to higher levels. (b) Pair correlation function computed at $\kappa=5$ and $20$ (see [ 56 ] for more details).

Figure 5: Overlap

\mathcal{O}

between the NN-VMC ground-state manifold and the Moore–Read Pfaffian, the anti-Pfaffian, the PH-Pfaffian, and the exact Coulomb ground state of the 1LL-projected Hamiltonian, as a function of Landau level mixing

\kappa

, for

N=18

electrons. Open symbols with dashed lines show overlaps with the bare trial states, defined within the LLL and 1LL, and filled symbols with solid lines show overlaps with dressed trial states, which include a cloud of virtual inter-Landau-level excitations [ 56 ] . In both cases, the overlap with the anti-Pfaffian exceeds that with the other trial states at all

\kappa

Architecture		Optimization		Monte Carlo
Layers $L$	4	Optimizer	MinSR	Walkers	1024
Heads	4	Learning rate	0.02	Steps/sample	10
Attention dim.	32	Iterations	$10^{3}$ to $10^{4}$	Move width	$\ell_{B}$
Perceptron dim.	128			Burn-in	300
Perceptrons/layer	2
Determinants	2

Table 1: Hyperparameters used in this work.

Figure S1: Supercell tilings and fine-projected spectra, one row per system. (a) The tiling of

N=18

(

4\times 3

cells) and (d) of

N=21

(

7\times 2

cells). The one-flux cell vectors

\mathbf{a}_{1}

\mathbf{a}_{2}

are in orange, the fine lattice

\Lambda_{\phi}

in grey, and the maximal commuting set of translations in blue. In one cell the generator

T(\mathbf{a}_{2}/n_{2})

of the fine projection is drawn with its

n_{2}

points in dark blue. The unused

\mathbf{a}_{1}/3

N=18

is dashed. (b, c) NN-VMC spectra of

N=18

\kappa=2

and

\kappa=5

. Each bar is the energy of the fine-projected state

\Phi_{\mathbf{k},l}

of Eq. ( S25 ) in one momentum sector, with its statistical error. Energies are relative to the mean

E_{0}

of the sixfold ground multiplet,

E_{0}=-4.9736\pm 0.0001\,\hbar\omega_{c}

(b) and

-37.1050\pm 0.0002\,\hbar\omega_{c}

(c). The sectors

(1,0)

and

(3,0)

hold two states: the second is drawn in blue, shorter, over the first. (e, f) The same for

N=21

\kappa=2

and

\kappa=5

. Here

E_{0}

is the mean of the ground doublet (

k_{1}+N_{1}k_{2}=0,7

E_{0}=-6.5028\pm 0.0003\,\hbar\omega_{c}

(e) and

-45.0552\pm 0.0003\,\hbar\omega_{c}

(f).

Figure S2: Wigner crystals at $N=21$ , $\kappa=20$ . (a, b) The two $21$ -site triangular lattices $A$ and $B$ of the $7\times 2$ torus, rotated by $\pm 10.9^{\circ}$ from $\mathbf{L}_{1}$ , and the fine translation that preserves each of them: $T_{A}=T(\mathbf{a}_{1})\,T(\mathbf{a}_{2}/7)^{4}$ , by $\mathbf{a}_{1}+4\mathbf{a}_{2}/7$ , and $T_{B}=T(\mathbf{a}_{1})\,T(\mathbf{a}_{2}/7)^{8}$ , by $\mathbf{a}_{1}+8\mathbf{a}_{2}/7$ (arrows). (c) The fine sectors $(k_{1},l)$ of each sector $k_{2}=0,1$ , both labels modulo $7$ . A rigid crystal of orientation $A$ ( $B$ ) can exist only on the red (blue) line, Eq. ( S31 ); the lines cross only at the origin $k_{1}=0$ , $l=3k_{2}$ (black). (d) Spectrum relative to $E_{0}=-252.819\,\hbar\omega_{c}$ , the mean energy of the two ground states $G$ : in each sector the lowest fine-projected state on the $A$ line (red) and on the $B$ line (blue); at the origin, columns $0$ and $7$ , the cat pair, the ground state $G$ (A $+$ B) and its penalty partner $P$ (A $-$ B), in black, and the energies of the De-Cat states $\Psi_{\pm}=(G\pm e^{-i\phi_{A}}P)/\sqrt{2}$ (open diamonds). (g) The sign flip in the sector $k_{2}=0$ : the phase of $P(x)/G(x)$ , measured from its value on the $A$ side, against the orientation $O(x)=[S_{A}(x)-S_{B}(x)]/[S_{A}(x)+S_{B}(x)]$ of the sampled configurations $x$ , with $S_{X}$ of Eq. ( S32 ). (e) Pair correlation $g(\mathbf{r})$ and Bragg weights $S_{A}$ , $S_{B}$ of $G$ and $P$ : both superpose the two orientations. (f) The same as (e) for $\Psi_{+}$ and $\Psi_{-}$ : single crystals $A$ and $B$ .

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Foundation models for ground states in spin-1/2 systems are a promising method for problems ranging from quantum chemistry to identifying new phase diagrams. Nearly all such models are currently pure-states that condition on the Hamiltonian's parameters, whose Monte Carlo samples give energy estimates according to the variational principle. In this contribution, we show that this representation is topologically obstructed. For any gapped Hamiltonian family whose ground-state bundle is non-trivial, every continuous normalized state-vector model has zero fidelity with the ground state at some parameter value in the Hamiltonian family. For that value, the energy is at least one spectral gap,

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