Learning phase transitions by confusion (1610.02048v1)

Abstract: Classifying phases of matter is a central problem in physics. For quantum mechanical systems, this task can be daunting owing to the exponentially large Hilbert space. Thanks to the available computing power and access to ever larger data sets, classification problems are now routinely solved using machine learning techniques. Here, we propose to use a neural network based approach to find phase transitions depending on the performance of the neural network after training it with deliberately incorrectly labelled data. We demonstrate the success of this method on the topological phase transition in the Kitaev chain, the thermal phase transition in the classical Ising model, and the many-body-localization transition in a disordered quantum spin chain. Our method does not depend on order parameters, knowledge of the topological content of the phases, or any other specifics of the transition at hand. It therefore paves the way to a generic tool to identify unexplored phase transitions.

• The paper introduces a novel machine-learning method that uses intentionally mislabeled data to detect quantum phase transitions.
• It integrates supervised and unsupervised techniques to generate a distinct W-shaped performance curve that pinpoints critical points.
• The approach successfully identifies transitions in models like the Kitaev chain, Ising model, and many-body localized systems, showcasing broad applicability.

Insightful Overview of "Learning Phase Transitions by Confusion"

The paper "Learning phase transitions by confusion," authored by Evert P.L. van Nieuwenburg, Ye-Hua Liu, and Sebastian D. Huber, introduces a novel method for classifying phase transitions in physical systems using machine learning, particularly neural networks (NN). The central challenge addressed in this work is the classification of phases of matter in quantum mechanical systems, a task complicated by the vastness of the Hilbert space. The authors propose a unique approach that leverages a neural network trained on intentionally mislabeled data to identify phase transitions.

Methodology

The authors examined three distinct scenarios to validate their method: the topological phase transition in the Kitaev chain, the thermal phase transition in the classical Ising model, and the many-body localization transition in a disordered quantum spin chain. The core idea is to employ a combination of supervised and unsupervised learning techniques. They implement a confusion scheme, where the neural network is trained with deliberately incorrect labels. This technique revolves around mislabeling data across a guessed critical point, evaluating the NN's performance as a function of this critical point to infer the actual phase transition.

A significant contribution of this work is the development of a "W-shape" performance function for the NN, which assists in pinpointing the transition accurately. When analyzed in the context of this shape, the peak signifies the most probable location of the phase transition, thus offering a systematic approach to predict phase boundaries.

Numerical Experiments

The NN's training involved two output nodes, representing the classification into two different phases, while the input consisted of processed data in the form of entanglement spectra. Key numerical experiments highlighted include:

1. Kitaev Chain: The NN demonstrated an ability to predict transition points accurately despite data blanking around the transition. This was affirmed through the analysis of entanglement spectra.
2. Classical Ising Model: Application of the confusion scheme identified the thermal transition with reasonable precision, aligning with known results in the thermodynamic limit. The W-shape performance function effectively highlighted the correct transition temperature.
3. Many-Body Localization (MBL): The MBL transition in a disordered spin chain offered a robust challenge. Using unsupervised learning like PCA proved insufficient, but the confusion-based NN approach identified the transition accurately around the expected value, similar to prior studies.

Theoretical and Practical Implications

The proposed method dispenses with the need for predefined order parameters or prior domain knowledge of the system’s phases, making it broadly applicable. It paves the way for recognizing unexplored phase transitions in quantum systems, strikingly useful in fields dealing with complex datasets where phase characteristics or transitions are not well understood.

Further, by focusing on the entanglement spectrum, the method adeptly compresses the quantum wavefunction, thus addressing computational challenges associated with exponentially large data spaces. The use of framed confusion thus marks a significant departure from classical approaches, potentially impacting research paradigms in condensed matter physics, where distinctions between phases often rely on subtle quantum mechanical data.

Speculation on Future Developments

Looking ahead, this approach opens avenues for identifying not only binary phase transitions but also more complex, multi-phase systems. Future work may extend the framework to adaptively label data in scenarios where multiple transition points or intermediate phases exist. The resilience of this method against changes in network parameters also suggests robustness that could be harnessed in emerging areas like dynamic quantum phases or non-equilibrium systems.

Conclusion

The "Learning phase transitions by confusion" paper presents a compelling, systematic advance in the application of machine learning to physical sciences. By eschewing traditional reliance on order parameters, it provides a promising toolset for phase transition detection in quantum mechanical systems, with broader implications for data-rich scientific disciplines.