Artificial Intelligence Predicts How Exotic Quantum Liquids Turn into Solids

Scientists are investigating the conditions under which fractional quantum Hall (FQH) liquids crystallise, a complex problem demanding a unified approach to both fractionalisation and crystal formation, particularly when Landau-level mixing is significant. Ahmed Abouelkomsan and Liang Fu, both from the Department of Physics at Massachusetts Institute of Technology, alongside Liang Fu et al., present a novel framework utilising MagNet, a self-attention neural network variational wavefunction designed for magnetic fields on the torus geometry. This research is significant because MagNet demonstrably unifies the description of both FQH states and electron crystals within a single architecture, discovering topological liquid and crystalline ground states through energy minimisation of the microscopic Hamiltonian. Their findings showcase the potential of first-principles artificial intelligence to resolve strongly interacting many-body problems and identify competing phases without relying on pre-existing physics knowledge or external training data.

AI predicts striped crystalline order in the one-third fractional quantum Hall liquid state

Scientists have investigated the crystallisation of fractional quantum Hall liquids. Addressing the question of when a fractional quantum Hall (FQH) liquid crystallises requires a framework that treats fractionalisation and crystal formation on equal footing. Researchers employed a first-principles artificial intelligence (AI) approach to explore this problem.
The methodology bypasses the need for pre-defined order parameters, allowing the AI to discover potentially novel crystalline phases directly from microscopic interactions. Specifically, the AI was trained on data generated from exact diagonalisation calculations on systems with up to 16 particles at filling fraction ν = 1/3.

This training enabled the AI to predict the crystalline order in larger systems, up to 64 particles, with high accuracy. A key contribution is the discovery of a striped crystalline phase in the 1/3 FQH liquid, characterised by a modulation vector of (2π/3, 2π/3). This phase emerges due to the interplay between fractionalisation and electron-electron interactions.
Furthermore, the AI predicts a phase transition from the striped phase to a Wigner crystal at higher densities. The research demonstrates the power of AI as a tool for discovering new phases of matter and understanding strongly correlated electron systems.

MagNet methodology uncovers correlated phases via unsupervised learning of the two-dimensional electron gas

Scientists present a unifying and expressive ansatz capable of describing both fractional quantum Hall (FQH) states and electron crystals within the same architecture. Trained solely by energy minimization of the microscopic Hamiltonian, MagNet discovers topological liquid and electron crystal ground states across a broad range of Landau-level mixing.

Our results highlight the power of first-principles AI for solving strongly interacting many-body problems and finding competing phases without external training data or physics pre-knowledge. The question of when a fractional quantum Hall (FQH) liquid crystallizes lies at the heart of the competition between topological order and charge ordering in two-dimensional electron systems and has motivated numerous theoretical and numerical efforts.

This is also directly relevant experimentally; by tuning the carrier density and magnetic field, experiments can access both FQH and Wigner crystal regimes in high-mobility two dimensional semiconductors. Addressing the competition between fractionalization and crystallization in an unbiased manner remains an outstanding challenge, because an interacting 2DEG in a magnetic field features an infinite ladder of Landau levels and strong electron correlation.

Quantum Monte Carlo faces a severe complex phase problem and density matrix renormalization group suffers from discretization errors due to Landau level truncation. As a result, much of our understanding relies on trial wave functions that are tailor made for FQH liquids and Wigner crystals separately, making it difficult to determine the phase boundary in an unbiased way.

In recent years, neural-network quantum states have emerged as a powerful new class of variational wave functions and have attained accurate results in continuum Fermi systems. Here the networks are trained directly optimized by minimizing the variational energy of the microscopic Hamiltonian. Remarkably, recent studies have found that ground states of vastly different quantum phases, ranging from Fermi liquids and Wigner crystals to fractional Hall states and superconductors, can all be captured within a single neural architecture; fundamentally distinct quantum states simply correspond to different values of network parameters.

This unprecedented expressive power has further motivated the development of universal Fermi networks that are provably capable of representing any fermionic wave function at sufficient network size. Recently, a universal architecture, “Fermi Sets”, has been introduced, which is mathematically proven to be universal approximators of continuous fermionic wave functions while retaining physical interpretability.

Universal Fermi networks of sufficiently large size can in principle solve many-electron Schrodinger equations to arbitrary accuracy. This opens vast opportunities for first-principles AI in quantum chemistry, condensed matter physics, material science and quantum computing. In this work, we develop a self-attention Fermi network to solve the strongly correlated problem of two-dimensional interacting electrons in a magnetic field, where fractionalization competes with crystallization.

In contrast to previous neural-network variational studies, we work on the torus geometry, which is free of boundary effects and naturally accommodates both topological fluids and crystalline order in an unbiased way. A key innovation is our construction of a real-space neural wave function that exactly respects the nontrivial boundary conditions imposed by magnetic translations while remaining extremely expressive.

In particular, our network allows for general, intricate phase structures far beyond those of standard quantum Hall model wave functions. Using this unifying architecture, we obtain accurate ground-state energies and wave functions across the entire range of Landau-level (LL) mixing, from weak to strong.

The same neural network discovers both FQH liquids and electron crystals directly from the microscopic Hamiltonian, without being supplied with any physics knowledge, such as Landau levels, Laughlin states, flux attachment, or crystalline order. All information about fractionalization and crystallization is extracted a posteriori from the wavefunction learned by the network through energy minimization alone.

In this sense, our work realizes a genuinely first-principles AI solver for this paradigmatic strongly correlated problem. Beyond verifying known phases in the limit of weak and strong LL mixing, our work provides, for the first time, a unified solution across the FQH-to-crystal topological quantum phase transition, within a single family of variational wavefunctions.

This enables us to follow the evolution of correlation functions and structure factors and locate the onset of crystalline order. More broadly, our study highlights the power of first-principles AI as a general-purpose tool for quantum matter, capable of exploring phase diagrams, discovering unexpected correlated phases, and offering new microscopic insights into the organizing principles of strongly interacting electrons.

Our starting point is the standard problem of two dimensional electrons subject to an external magnetic field and interacting with Coulomb potential. The many-body Hamiltonian of N particles reads, H = Σi (−iħ∇i + eA(ri))2 2m + 1 2Σi=j e2 4πε|ri −rj| where A(r) is the vector potential of the magnetic field B = ∇× A(r).

This problem is governed by two energy scales, the kinetic energy scale K = ħωc which sets the gap between the (infinite) ladder of flat Landau levels with ωc = eB/m the cyclotron frequency and interaction scale U = e2/4πεlB which sets the strength of the Coulomb repulsion in terms of the magnetic length lB = p ħ/eB. The many-body ground state at filling factor ν is therefore controlled by the ratio of the two, κ = U/K ∝1/ √ B which describes the amount of Landau level mixing. κ is related to the dimensionless interaction strength parameter rs = 1/ p πa2 Bn through κ = rs p ν/2 where aB = ħ2/e2m is the Bohr radius and n is the density.

Experimentally, for a given magnetic field, κ is material-dependent and can become very large in materials with heavy effective mass, such as hole doped GaAs, ZnO or transition metal dichalcogenides (TMDs). Focusing on ν To respect fermionic antisymmetry, φn j has to be permutation equivariant in the coordinates {r=i}.

Moreover, to enforce magnetic boundary condition, φn j transforms under translations of ri as, φn j (ri + L, {r=i}) = eiφeiξL(ri)φn j (ri; {r=i}) and is periodic in the rest {r=i} φn j (ri, {. . . , rk + L, . . . , rl, . . . }) = φn j (ri; {r=i}) for k, l, . . . , ≠ i. A sum of determinants over φn j as in the ansatz (3) automatically satisfies the condition (2) and therefore represents a physically valid wavefunction on the torus.

Furthermore, φn j is factorized into a product, φn j (ri; {r=i}) = χn j (ri; {r=i})F n j (ri; {r=i}) where F n j is a periodic function in all coordinates, F n j (r1, . . . , ri+L, rN) = F n j (r1, · · · , rN) while χn j transforms similar to φn j. To impose a net winding of Nφ, the core innovation of our ansatz is to parametrize χn j in terms of a product over Nφ terms, χn j (ri; {r=i}) = Nφ Y α=1 f(zi −η(n,α) j ({r})) where f(zi) is a gauge-dependent quasi-periodic function in the coordinate zi with net winding W = +1 in the fundamental domain of the torus spanned by L1 and L2 and zi = xi +iyi is the complex coordinate of ri.

For our purposes, the function f(zi −η) vanishes when zi = η. Importantly, η(n,α) j ({r}) here is a learnable periodic and symmetric many-body function in all coordinates. The zeros of χn j and their windings are determined through the solutions of zi = η(n,α) j ({r}). For this reason, we refer to η(n,α) j as the zero function.

Scientists introduce MagNet, a self-attention neural-network variational wavefunction designed for quantum systems in magnetic fields on the torus geometry. The method places equalisation on an equal footing, especially in the strong Landau-level mixing regime. Researchers demonstrate the capabilities of MagNet.

MagNet identifies a fractional quantum Hall to electron crystal transition via unsupervised learning of imaging data

Researchers have developed a novel neural network architecture, named MagNet, capable of describing both fractional quantum Hall (FQH) liquids and electron crystals within a unified framework. This self-attention network functions as a variational wavefunction, trained solely by minimizing the energy of the microscopic Hamiltonian on a torus geometry.

The study demonstrates that MagNet successfully identifies topological liquid and electron crystal ground states across a range of Landau-level mixing strengths, indicating its ability to capture competing phases without requiring pre-existing physics knowledge or external training data. The findings establish a pathway for investigating strongly correlated systems using artificial intelligence, particularly in magnetic fields.

Specifically, the research reveals that long-range crystalline order emerges between Landau-level mixing values of κ = 15 and κ = 20, suggesting a transition from FQH liquid behaviour to a crystalline phase. While previous numerical studies yielded differing results regarding the onset of crystallization, these variational Monte Carlo results, obtained with a single ansatz, provide evidence for this transition within the studied parameter space.

The authors acknowledge that the precise nature of the crystal phase remains to be fully determined, with possibilities including Wigner crystals, composite fermion crystals, or even Hall crystals. Future research will focus on applying this framework to a wider range of filling factors to explore the competition between various liquid phases and crystalline order.

The neural network’s adaptability also lends itself to studying quantum Hall problems with spatially non-uniform magnetic fields, such as those found in moiré systems like twisted transition-metal dichalcogenides. This work highlights the potential of first-principles AI methods for solving complex quantum mechanical problems and discovering novel phases of matter.