Fractional quantum Hall states host emergent anyons with exotic exchange statistics, but obtaining direct access to their topological properties in real systems remains a challenge. Neural network wave functions provide a flexible computational approach, as they can represent highly correlated states without requiring a tailored basis. Here, we use the neural network variational Monte Carlo method to study the fractional quantum Hall effect on the torus and find the three degenerate ground states at filling factor ν = 1 / 3 . From these, we extract the modular S -matrix via entanglement interferometry, a technique previously only applied to lattice models. The resulting S -matrix encodes the quantum dimensions, fusion rules, and exchange statistics of the emergent anyons, providing a direct numerical demonstration of the topological order. The calculated anyon properties match the well-known theoretical and experimental results. Our work establishes neural network wave functions as a powerful tool for investigating anyonic properties.
Fractional quantum Hall states host emergent anyons with exotic exchange statistics, but obtaining direct access to their topological properties in real systems remains a challenge. Neural network wave functions provide a flexible computational approach, as they can represent highly correlated states without requiring a tailored basis. Here, we use the neural network variational Monte Carlo method to study the fractional quantum Hall effect on the torus and find the three degenerate ground states at filling factor ν = 1 / 3 . From these, we extract the modular S -matrix via entanglement interferometry, a technique previously only applied to lattice models. The resulting S -matrix encodes the quantum dimensions, fusion rules, and exchange statistics of the emergent anyons, providing a direct numerical demonstration of the topological order. The calculated anyon properties match the well-known theoretical and experimental results. Our work establishes neural network wave functions as a powerful tool for investigating anyonic properties.
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 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.
Topologically ordered states are among the most interesting quantum phases of matter that host emergent quasi-particles having fractional charge and obeying fractional quantum statistics. Theoretical study of such states is however challenging owing to their strong-coupling nature that prevents conventional mean-field treatment. Here, we demonstrate that an attention-based deep neural network provides an expressive variational wavefunction that discovers fractional Chern insulator ground states purely through energy minimization without prior knowledge and achieves remarkable accuracy. We introduce an efficient method to extract ground state topological degeneracy -- a hallmark of topological order -- from a single optimized real-space wavefunction in translation-invariant systems by decomposing it into different many-body momentum sectors. Our results establish neural network variational Monte Carlo as a versatile tool for discovering strongly correlated topological phases.
Strong correlation brings a rich array of emergent phenomena, as well as a daunting challenge to theoretical physics study. In condensed matter physics, the fractional quantum Hall effect is a prominent example of strong correlation, with Landau level mixing being one of the most challenging aspects to address using traditional computational methods. Deep learning real-space neural network wave function methods have emerged as promising architectures to describe electron correlations in molecules and materials, but their power has not been fully tested for exotic quantum states. In this work, we employ real-space neural network wave function techniques to investigate fractional quantum Hall systems. On both 1/3 and 2/5 filling systems, we achieve energies consistently lower than exact diagonalization results which only consider the lowest Landau level. We also demonstrate that the real-space neural network wave function can naturally capture the extent of Landau level mixing up to a very high level, overcoming the limitations of traditional methods. Our work underscores the potential of neural networks for future studies of strongly correlated systems and opens new avenues for exploring the rich physics of the fractional quantum Hall effect.
Fractional quantum Hall systems host strongly correlated topological phases with potential applications in topological quantum computation, yet identifying their microscopic ground states remains challenging because competing phases are often nearly degenerate and Landau-level mixing is difficult to treat. Here, we introduce a symmetry-adapted neural-network wavefunction that describes Abelian and non-Abelian states while incorporating all Landau levels. The method accurately captures composite-fermion liquids and the emergence of Wigner crystallization at low filling. When the first Landau level is partially filled, finite-size overlaps and entanglement spectra provide evidence consistent with 2ˉk1k+1-type parton phases at ν=4/3 and 7/5, while the ν=3/2 state exhibits anti-Pfaffian-like entanglement structure. These results demonstrate that neural-network wavefunctions offer a flexible framework for studying competing topological orders in fractional quantum Hall systems.
Given a microscopic lattice Hamiltonian for a topologically ordered phase, we propose a numerical approach to characterize its emergent anyon model and, in a chiral phase, also its gapless edge theory. First, a tensor network representation of a complete, orthonormal set of ground states on a cylinder of infinite length and finite width is obtained through numerical optimization. Each of these ground states is argued to have a different anyonic flux threading through the cylinder. Then a quasiorthogonal basis on the torus is produced by chopping off and reconnecting the tensor network representation on the cylinder. From these two bases, and by using a number of previous results, most notably the recent proposal of Y. Zhang et al. [Phys. Rev. B 85, 235151 (2012)] to extract the modular U and S matrices, we obtain (i) a complete list of anyon types i, together with (ii) their quantum dimensions d(i) and total quantum dimension D, (iii) their fusion rules N(ij)(k), (iv) their mutual statistics, as encoded in the off-diagonal entries S(ij) of S, (v) their self-statistics or topological spins θ(i), (vi) the topological central charge c of the anyon model, and, in a chiral phase (vii) the low energy spectrum of each sector of the boundary conformal field theory. As a concrete application, we study the hard-core boson Haldane model by using the two-dimensional density matrix renormalization group. A thorough characterization of its universal bulk and edge properties unambiguously shows that it realizes a ν=1/2 bosonic fractional quantum Hall state.
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer. The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE. An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions. One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality. Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus. Having explicit torus wave functions allows us to use the methods developed by Read and Read \\& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information. We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.