Determining the phase diagram of systems consisting of smaller subsystems 'connected' via a tunable coupling is a challenging task relevant for a variety of physical settings. A general question is whether new phases, not present in the uncoupled limit, may arise. We use machine learning and a suitable quasidistance between different points of the phase diagram to study layered spin models, in which the spin variables constituting each of the uncoupled systems (to which we refer as layers) are coupled to each other via an interlayer coupling. In such systems, in general, composite order parameters involving spins of different layers may emerge as a consequence of the interlayer coupling. We focus on the layered Ising and Ashkin–Teller models as a paradigmatic case study, determining their phase diagram via the application of a machine learning algorithm to the Monte Carlo data. Remarkably our technique is able to correctly characterize all the system phases also in the case of hidden order parameters, i.e. order parameters whose expression in terms of the microscopic configurations would require additional preprocessing of the data fed to the algorithm. We correctly retrieve the three known phases of the Ashkin–Teller model with ferromagnetic couplings, including the phase described by a composite order parameter. For the bilayer and trilayer Ising models the phases we find are only the ferromagnetic and the paramagnetic ones. Within the approach we introduce, owing to the construction of convolutional neural networks, naturally suitable for layered image-like data with arbitrary number of layers, no preprocessing of the Monte Carlo data is needed, also with regard to its spatial structure. The physical meaning of our results is discussed and compared with analytical data, where available. Yet, the method can be used without any a priori knowledge of the phases one seeks to find and can be applied to other models and structures.
New Journal of Physics
We thank Gesualdo Delfino, Michele Fabrizio, Piero Ferrarese, Robert Konik, Christoph Lampert and Mikhail Lemeshko for stimulating discussions at various stages of this work. WR has received funding from the EU Horizon 2020 program under the Marie Skłodowska-Curie Grant Agreement No. 665385 and is a recipient of a DOC Fellowship of the Austrian Academy of Sciences. GB acknowledges support from the Austrian Science Fund (FWF), under project No. M2641-N27. ND acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Collaborative Research Center SFB 1225 (ISOQUANT)--project-id 273811115--and under Germany's Excellence Strategy 'EXC-2181/1-390900948' (the Heidelberg STRUCTURES Excellence Cluster).
Rzadkowski W, Defenu N, Chiacchiera S, Trombettoni A, Bighin G. Detecting composite orders in layered models via machine learning. New Journal of Physics. 2020;22(9). doi:10.1088/1367-2630/abae44
Rzadkowski, W., Defenu, N., Chiacchiera, S., Trombettoni, A., & Bighin, G. (2020). Detecting composite orders in layered models via machine learning. New Journal of Physics. IOP Publishing. https://doi.org/10.1088/1367-2630/abae44
Rzadkowski, Wojciech, N Defenu, S Chiacchiera, A Trombettoni, and Giacomo Bighin. “Detecting Composite Orders in Layered Models via Machine Learning.” New Journal of Physics. IOP Publishing, 2020. https://doi.org/10.1088/1367-2630/abae44.
W. Rzadkowski, N. Defenu, S. Chiacchiera, A. Trombettoni, and G. Bighin, “Detecting composite orders in layered models via machine learning,” New Journal of Physics, vol. 22, no. 9. IOP Publishing, 2020.
Rzadkowski W, Defenu N, Chiacchiera S, Trombettoni A, Bighin G. 2020. Detecting composite orders in layered models via machine learning. New Journal of Physics. 22(9), 093026.
Rzadkowski, Wojciech, et al. “Detecting Composite Orders in Layered Models via Machine Learning.” New Journal of Physics, vol. 22, no. 9, 093026, IOP Publishing, 2020, doi:10.1088/1367-2630/abae44.
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