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Human-machine collaboration: Ordering mechanism of rank-2 spin liquid on breathing pyrochlore lattice

Nicolas Sadoune1,2, Ke Liu1,2,3,4, Han Yan5,6,7,8, Ludovic D. C. Jaubert9, Nic Shannon5, and Lode Pollet1,2

  • 1Arnold Sommerfeld Center for Theoretical Physics, LMU Munich, Theresienstr. 37, 80333 München, Germany
  • 2Munich Center for Quantum Science and Technology (MCQST), 80799 München, Germany
  • 3Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
  • 4Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China
  • 5Theory of Quantum Matter Unit, Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa 904-0412, Japan
  • 6Institute for Solid State Physics, University of Tokyo, Kashiwa, 277-8581 Chiba, Japan
  • 7Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
  • 8Smalley-Curl Institute, Rice University, Houston, Texas 77005, USA
  • 9CNRS, Université de Bordeaux, LOMA, UMR 5798, 33400 Talence, France

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Phys. Rev. Research 7, 033061 – Published 14 July, 2025

DOI: https://doi.org/10.1103/c6z1-wh6l

Abstract

Machine-learning algorithms thrive on large data sets of good quality. Here we show that they can also excel in a typical research setting with little data of limited quality, through an interplay of insights coming from machine and human researchers. The question we address is the unsolved problem of ordering out of a spin-liquid phase described by an emergent rank-2 U(1) gauge theory, as described by H. Yan et al. [Phys. Rev. Lett. 124, 127203 (2020)]. Published Monte Carlo simulations for this problem are consistent with a strong first-order phase transition, but were too noisy for the form of low-temperature order to be identified. Using a highly interpretable machine-learning approach based on a support vector machine with a tensorial kernel, we reanalyze these Monte Carlo data, gaining new information about the form of order that could in turn be interpreted by traditionally trained physicists. We find that the low-temperature ordered phase is a form of magnetic order analogous to a smectic liquid crystal. This arises due to a subtle thermal order-by-disorder mechanism, that can be understood from the fluctuations of the tensor electric field of the parent rank-2 gauge theory. These results were obtained by a back-and-forth process which closely resembles a collaboration between human researchers and machines. We argue that this “collaborative” approach may provide a blueprint for solving other problems that have not yielded to human insights alone.

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Phys. Rev. Research 7, 033061– Published 14 July, 2025
  • Received 16 February 2024
  • Accepted 19 May 2025
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