Neural Networks Map Magnetic Disorder, Revealing New Phase of Matter

The behaviour of magnetic materials in disordered environments continues to challenge condensed matter physics, with the quest to fully understand the emergence of spin glass phases remaining a central focus. These phases, characterised by randomly frozen magnetic moments, exhibit complex behaviour and are sensitive to even minor fluctuations. Now, Luciano Loris Viteritti from the Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Riccardo Rende from SISSA, and colleagues report findings concerning the two-dimensional frustrated Heisenberg model with bond disorder, utilising a novel computational approach. Their research, detailed in the article “Quantum Spin Glass in the Two-Dimensional Disordered Heisenberg Model via Foundation Neural-Network Quantum States”, demonstrates the stability of the spin glass phase against fluctuations, a result differing from classical models and offering new insights into the behaviour of these complex magnetic systems. The team employed Foundation Neural-Network States, a recently developed framework enabling efficient calculation of disorder-averaged properties, to simulate the model on large lattices and reveal an extended region where long-range magnetic order vanishes, while the spin glass order persists.

Spin glasses represent a complex state of matter characterised by disordered magnetic moments and frustrated interactions, first observed in metallic alloys during the 1970s. These materials exhibit a unique combination of randomness and competing magnetic forces, preventing simple alignment of spins and extending study beyond condensed matter physics to complex systems and optimisation problems. Quantum spin glasses emerge when quantum fluctuations become significant, prompting investigation into whether spin-glass order persists at the quantum level and how this impacts the system’s properties.

Theoretical approaches to spin glasses often encounter challenges, particularly in lower dimensions, as averaging over inherent disorder requires complex mathematical techniques such as replica symmetry breaking or the introduction of auxiliary fields. Numerical simulations also face limitations when studying quantum spin glasses, especially in two dimensions, as quantum Monte Carlo algorithms are hampered by the ‘sign problem’ in disordered systems, restricting system size and complexity. Researchers are therefore exploring alternative approaches like exact diagonalisation or tensor network techniques.

Recent research focuses on developing new computational methods to accurately model quantum spin glasses, with the Foundation Neural-Network States framework recently introduced as a technique aiming to efficiently calculate disorder-averaged observables. This approach allows exploration of larger systems and a deeper understanding of the interplay between disorder, quantum fluctuations, and magnetic order, streamlining the process of averaging over numerous disorder configurations and enabling simulations on larger lattices. The system under investigation is the frustrated Heisenberg model, where nearest-neighbour interactions are randomly varied, introducing disorder and allowing for the exploration of complex magnetic phases.

Researchers are employing this novel computational technique to investigate magnetic disorder in two-dimensional materials, identifying the conditions under which long-range magnetic order emerges or vanishes, and determining the stability of any emergent magnetic phase. They discovered an extensive region within the model’s phase diagram where long-range magnetic order is suppressed, while simultaneously observing a finite overlap order parameter, a quantity signifying the presence of spin glass states, characterised by frozen, random spin orientations. This coexistence of suppressed long-range order and finite spin glass order is a key finding, suggesting a distinct magnetic phase beyond simple ferromagnetic or antiferromagnetic behaviour.

To corroborate these computational findings, the team performed a semiclassical analysis, utilising a large-spin expansion, providing independent verification of the results and strengthening confidence in the observed phase behaviour. The agreement between computational and analytical results is noteworthy, suggesting the observed phase is not merely an artefact of the numerical method. Importantly, these findings contrast with classical models, which indicate the spin glass phase is unstable at any finite temperature, demonstrating that the spin glass phase is, in fact, stable against fluctuations, at least within the parameters explored.

Simulations reveal a substantial region within the phase diagram where long-range magnetic order fails to persist as the system size increases, and crucially, the research demonstrates the stability of the spin glass phase, a state characterised by randomly frozen magnetic moments. Unlike classical models predicting the disappearance of this phase at any finite temperature, these simulations show a finite overlap order parameter, indicating sustained spin glass order, quantifying the degree of similarity between different spin configurations and confirming the robustness of the spin glass state against thermal fluctuations.

Researchers employed a semiclassical analysis, building upon a large-spin expansion, to corroborate these findings, providing independent validation of the simulation results and strengthening the evidence for a stable spin glass phase. The combination of advanced computational techniques and analytical methods provides a robust understanding of the system’s behaviour, challenging previous understandings of spin glass behaviour, particularly in two-dimensional systems. The demonstrated stability of the spin glass phase suggests that disorder plays a crucial role in maintaining magnetic order, even in the presence of thermal fluctuations, contributing to a deeper understanding of disordered magnetic materials and their potential applications in areas such as data storage and spintronics.

This research establishes that quantum mechanics stabilises the spin glass phase in two-dimensional frustrated Heisenberg models with bond disorder, a departure from classical predictions where such phases are unstable at any finite temperature. The study employs the Foundation Neural-Network States framework, a computational technique allowing for efficient calculation of disorder-averaged properties, to simulate these complex systems on large lattices, circumventing the limitations of traditional methods when dealing with disorder and quantum fluctuations. Simulations reveal an extended region within the phase diagram where long-range magnetic order diminishes as the system size increases, indicating a loss of conventional magnetism, while simultaneously, the overlap order parameter remains finite within this region.

This coexistence suggests a stable spin glass phase, characterised by frozen, disordered spins, even in the presence of quantum fluctuations and competing interactions, and the findings are further supported by a semiclassical analysis, utilising a large-spin expansion, which corroborates the stability of the spin glass phase. This analytical approach provides independent validation of the numerical results, strengthening the conclusion that quantum effects play a crucial role in maintaining order within the disordered system, providing a robust understanding of the system’s behaviour through the combination of computational and analytical techniques.

Future work should focus on extending this analysis to three-dimensional systems, which are more representative of real materials and present greater computational challenges, and investigating the influence of different types of disorder, beyond bond disorder, would also broaden the understanding of spin glass behaviour. Furthermore, exploring the dynamic properties of the spin glass phase, such as the response to external fields or temperature changes, could reveal additional insights into its stability and functionality, and a key area for future research involves comparing these theoretical predictions with experimental observations on real materials, potentially leading to the development of novel magnetic materials with tailored properties.