Exotic Magnetic States Mapped in Complex Crystal Structures Using AI Techniques

Researchers are investigating the complex low-temperature behaviour of three-dimensional frustrated magnets, a notoriously difficult area due to computational limitations. Rajah P. Nutakki and Filippo Vicentini, both from CPHT, LIX, CNRS, Inria, Ecole Polytechnique, Institut Polytechnique de Paris, and Collège de France, Université PSL, alongside their colleagues, have employed neural states to map the ground-state phase diagrams of spin models on the body-centered cubic lattice. Their work is significant because it addresses the potential for exotic states of matter, such as spin liquids, in real materials and provides crucial insights into the magnetic properties of compounds like NaCaCu(VO)₄. The team’s findings reveal a first-order phase transition in the antiferromagnetic model and, surprisingly, demonstrate that a tetragonally-distorted variant does not replicate the observed low-temperature magnetism of NaCaCu(VO)₄, suggesting additional factors govern its behaviour.

This work addresses a longstanding challenge in condensed matter physics, namely accurately modelling frustrated quantum magnets which exhibit peculiar magnetic properties.

Researchers calculated the ground-state phase diagrams of frustrated spin models on the body-centred cubic lattice, employing a novel computational approach to overcome limitations inherent in traditional methods. The study utilised neural networks as physically unbiased variational ansatze, enabling the simulation of systems containing up to 288 spins, a substantial increase in scale for this type of calculation.
Initially, investigations focused on the antiferromagnetic J1, J2 model, revealing a direct first-order phase transition between Néel and collinear long-range-ordered phases at J2/J1 = 0.705 ±0.005. This result aligns with previous studies and validates the accuracy of the employed methodology. Subsequently, the research extended to a tetragonally-distorted variant of the model, designed to mimic the properties of the material NaCa2Cu2(VO4)3.

Surprisingly, calculations revealed no evidence of a quantum paramagnetic ground state, instead identifying a first-order phase transition between Néel and chain phases at J2ab/J1 = 1.0375 ±0.0125. This finding indicates that the simplified model fails to fully capture the low-temperature magnetic behaviour observed in NaCa2Cu2(VO4)3, suggesting the need to incorporate additional interactions or effects for a complete understanding.
The research demonstrates the power of neural quantum states in tackling complex three-dimensional quantum systems, offering a valuable tool for interpreting experimental observations and guiding the search for exotic states of matter like quantum spin liquids. The ability to accurately map ground-state phase diagrams of minimal models will assist in rationalizing experimental findings and potentially unlocking new technological applications.

Variational Monte Carlo with factored vision transformer neural networks for frustrated magnetism

A factored vision transformer wavefunction underpinned the variational Monte Carlo simulations performed to investigate the ground-state phase diagrams of frustrated spin models on the body-centred cubic lattice. The research employed neural quantum states to simulate lattices containing up to 288 spins, addressing the challenges posed by the sign problem and finite-size effects inherent in studying three-dimensional frustrated magnets.

Network hyperparameters were carefully defined, including a depth of 4, a feature dimension of 64, 8 attention heads, and a factored-attention mechanism spanning all patches, resulting in approximately 105 parameters within the network. The wavefunction was projected to a specific momentum k, building upon previously established methods, and point group symmetries were enforced through quantum number projection to ensure symmetry of the wavefunction with respect to the lattice space group.

Calculations were restricted to the P i Sz i = 0 sector using a magnetization-conserving sampling scheme, maintaining total spin conservation throughout the simulations. Periodic cubic clusters, defined by (Lx, Ly, Lz) unit cells with a lattice constant of 1, were utilised, mirroring the two-site cubic unit-cell of the body-centred cubic lattice.

MinSR optimisation was implemented to refine the variational wavefunction, and the static structure factor S(q) = 1/N Σi,j eiq·(ri−rj)⟨Si · Sj⟩ was computed to elucidate ground-state properties. Associated correlation ratios, R(Q) = 1 −S(Q + δq) / S(Q), were also calculated, using ordering wavevectors QNeel = (2π, 2π, 2π), QCollinear = (π, π, π), and QChain = (π, π, 0) to characterise the different magnetic phases. The antiferromagnetic J1 −J2 model exhibited a first-order phase transition at (J2/J1)c = 0.705 ±0.005, aligning with prior investigations, while the tetragonally-distorted variant showed a first-order transition between Néel and chain phases at Jab = 1.0375 ±0.0125, without evidence of a paramagnetic ground state.

Neural network determination of phase transitions in frustrated body-centred cubic magnets

Simulating the low-temperature properties of three-dimensional frustrated magnets presents computational challenges, yet experimental evidence suggests these materials may host exotic states of matter. Calculations of ground-state phase diagrams for frustrated spin models on the body-centred cubic lattice were performed using neural states.

Initial study of the antiferromagnetic model revealed a first-order phase transition between Néel and collinear long-range-ordered phases occurring at a critical value of, aligning with prior investigations. A tetragonally-distorted variant, proposed as a model for NaCaCu(VO)₄, exhibited a first-order phase transition between Néel and chain phases at, however, no paramagnetic ground state was observed.

Investigation of the cubic-symmetric antiferromagnetic J1 −J2 model, Hcub, identified a first-order phase transition between long-range-ordered Néel and collinear phases at (J2/J1)c = 0.705 ±0.005. This transition point is slightly above the 0.67 value found in corresponding classical Ising models and remains consistent with previously published results.

Analysis of the experimentally relevant tetragonally-symmetric variant, Htet, with parameters J1 = 1, J2c = −4.9 and J2ab = 1.7, did not support a quantum paramagnetic ground state. Instead, a first-order phase transition between Néel and chain phases was found at Jab = 1.0375 ±0.0125, suggesting additional factors are needed to fully explain the low-temperature magnetic behaviour of NaCaCu(VO)₄.

Variational Monte Carlo calculations utilised a factored vision transformer wavefunction with 10⁵ parameters. The wavefunction was projected to a specific momentum k and symmetrized using point group symmetries within the PSz = 0 sector. Static structure factor S(q) and correlation ratios R(Q) were computed to elucidate ground-state properties.

For a N = 32 lattice, relative errors against exact ground-state energies computed by exact diagonalization were less than 10⁻⁴ for all values of J2. The energy exhibited a discontinuity at Jc 2, characteristic of a first-order phase transition, while R(Q) increased with system size, confirming long-range order.

Neural quantum state analysis of magnetic phase transitions in frustrated lattices

Researchers have employed neural quantum states to investigate the ground-state phase diagrams of frustrated spin models on the body-centred cubic lattice, addressing a longstanding challenge in simulating low-temperature magnetism. Calculations were performed on both a cubic antiferromagnetic model and a tetragonally-distorted variant proposed as a simplified representation of the material NaCaCu(VO)₄.

The cubic model exhibited a first-order phase transition between Néel and collinear long-range ordered phases at a critical value of approximately 0.705, aligning with previous computational findings. However, analysis of the tetragonally-distorted model did not support the existence of a paramagnetic ground state, instead revealing a first-order phase transition between Néel and chain phases at a different critical value of 1.0375.

This indicates that the minimal model currently used does not fully capture the complex magnetic behaviour observed experimentally in NaCaCu(VO)₄, suggesting the need for additional considerations. The authors acknowledge limitations related to the complexity of learning ground-state sign structures in more geometrically frustrated lattices, potentially restricting the broader application of their neural network approach. Future research could explore the inclusion of ferromagnetic interactions to potentially induce a quantum paramagnetic phase, and expand the methodology to other frustrated lattice structures, provided techniques to address the sign problem are developed.