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Sun H, Panda RK, Verdel R, Rodriguez A, Dalmonte M, Bianconi G. Network science: Ising states of matter. Phys Rev E 2024; 109:054305. [PMID: 38907445 DOI: 10.1103/physreve.109.054305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2023] [Accepted: 02/26/2024] [Indexed: 06/24/2024]
Abstract
Network science provides very powerful tools for extracting information from interacting data. Although recently the unsupervised detection of phases of matter using machine learning has raised significant interest, the full prediction power of network science has not yet been systematically explored in this context. Here we fill this gap by providing an in-depth statistical, combinatorial, geometrical, and topological characterization of 2D Ising snapshot networks (IsingNets) extracted from Monte Carlo simulations of the 2D Ising model at different temperatures, going across the phase transition. Our analysis reveals the complex organization properties of IsingNets in both the ferromagnetic and paramagnetic phases and demonstrates the significant deviations of the IsingNets with respect to randomized null models. In particular percolation properties of the IsingNets reflect the existence of the symmetry between configurations with opposite magnetization below the critical temperature and the very compact nature of the two emerging giant clusters revealed by our persistent homology analysis of the IsingNets. Moreover, the IsingNets display a very broad degree distribution and significant degree-degree correlations and weight-degree correlations demonstrating that they encode relevant information present in the configuration space of the 2D Ising model. The geometrical organization of the critical IsingNets is reflected in their spectral properties deviating from the one of the null model. This work reveals the important insights that network science can bring to the characterization of phases of matter. The set of tools described hereby can be applied as well to numerical and experimental data.
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Affiliation(s)
- Hanlin Sun
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
- Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden
| | - Rajat Kumar Panda
- The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
- SISSA-International School of Advanced Studies, via Bonomea 265, 34136 Trieste, Italy
- INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
- Department of Physics, University of Trieste, 34127 Trieste, Italy
| | - Roberto Verdel
- The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
| | - Alex Rodriguez
- The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
- Dipartimento di Matematica e Geoscienze, Universitá degli Studi di Trieste, via Alfonso Valerio 12/1, 34127 Trieste, Italy
| | - Marcello Dalmonte
- The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
- SISSA-International School of Advanced Studies, via Bonomea 265, 34136 Trieste, Italy
| | - Ginestra Bianconi
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
- The Alan Turing Institute, 96 Euston Road, London NW1 2DB, United Kingdom
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Vitale V, Mendes-Santos T, Rodriguez A, Dalmonte M. Topological Kolmogorov complexity and the Berezinskii-Kosterlitz-Thouless mechanism. Phys Rev E 2024; 109:034102. [PMID: 38632805 DOI: 10.1103/physreve.109.034102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2023] [Accepted: 02/08/2024] [Indexed: 04/19/2024]
Abstract
Topology plays a fundamental role in our understanding of many-body physics, from vortices and solitons in classical field theory to phases and excitations in quantum matter. Topological phenomena are intimately connected to the distribution of information content that, differently from ordinary matter, is now governed by nonlocal degrees of freedom. However, a precise characterization of how topological effects govern the complexity of a many-body state, i.e., its partition function, is presently unclear. In this paper, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. We concretely present a theory that shows how the Kolmogorov complexity of a classical partition function sampling carries unique, distinctive features depending on the presence of topological excitations in the system. We confront two-dimensional Ising, Heisenberg, and XY models on several topologies and study the corresponding samplings as high-dimensional manifolds in configuration space, quantifying their complexity via the intrinsic dimension. While for the Ising and Heisenberg models the intrinsic dimension is independent of the real-space topology, for the XY model it depends crucially on temperature: across the Berezkinskii-Kosterlitz-Thouless (BKT) transition, complexity becomes topology dependent. In the BKT phase, it displays a characteristic dependence on the homology of the real-space manifold, and, for g-torii, it follows a scaling that is solely genus dependent. We argue that this behavior is intimately connected to the emergence of an order parameter in data space, the conditional connectivity, which displays scaling behavior. Our approach paves the way for an understanding of topological phenomena emergent from many-body interactions from the perspective of Kolmogorov complexity.
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Affiliation(s)
- Vittorio Vitale
- International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
- SISSA, via Bonomea 265, 34136 Trieste, Italy
- Université Grenoble Alpes, CNRS, Laboratoire de Physique et Modélisation des Milieux Condensés (LPMMC), Grenoble 38000, France
| | - Tiago Mendes-Santos
- Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany
| | - Alex Rodriguez
- International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
- Dipartimento di Matematica e Geoscienze, Universitá degli Studi di Trieste, via Alfonso Valerio 12/1, 34127, Trieste, Italy
| | - Marcello Dalmonte
- International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
- SISSA, via Bonomea 265, 34136 Trieste, Italy
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Leykam D, Rondón I, Angelakis DG. Dark soliton detection using persistent homology. CHAOS (WOODBURY, N.Y.) 2022; 32:073133. [PMID: 35907713 DOI: 10.1063/5.0097053] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2022] [Accepted: 07/01/2022] [Indexed: 06/15/2023]
Abstract
Classifying images often requires manual identification of qualitative features. Machine learning approaches including convolutional neural networks can achieve accuracy comparable to human classifiers but require extensive data and computational resources to train. We show how a topological data analysis technique, persistent homology, can be used to rapidly and reliably identify qualitative features in experimental image data. The identified features can be used as inputs to simple supervised machine learning models, such as logistic regression models, which are easier to train. As an example, we consider the identification of dark solitons using a dataset of 6257 labeled atomic Bose-Einstein condensate density images.
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Affiliation(s)
- Daniel Leykam
- Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
| | - Irving Rondón
- School of Computational Sciences, Korea Institute for Advanced Study, 85 Hoegi-ro, Seoul 02455, Republic of Korea
| | - Dimitris G Angelakis
- Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
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Sale N, Giansiracusa J, Lucini B. Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology. Phys Rev E 2022; 105:024121. [PMID: 35291098 DOI: 10.1103/physreve.105.024121] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2021] [Accepted: 02/01/2022] [Indexed: 06/14/2023]
Abstract
We use persistent homology and persistence images as an observable of three variants of the two-dimensional XY model to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and k-nearest neighbor models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behavior and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.
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Affiliation(s)
- Nicholas Sale
- Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, Swansea, Wales, United Kingdom
| | - Jeffrey Giansiracusa
- Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, Swansea, Wales, United Kingdom
| | - Biagio Lucini
- Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, Swansea, Wales, United Kingdom
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Masoomy H, Askari B, Najafi MN, Movahed SMS. Persistent homology of fractional Gaussian noise. Phys Rev E 2021; 104:034116. [PMID: 34654089 DOI: 10.1103/physreve.104.034116] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2021] [Accepted: 08/18/2021] [Indexed: 11/07/2022]
Abstract
In this paper, we employ the persistent homology (PH) technique to examine the topological properties of fractional Gaussian noise (fGn). We develop the weighted natural visibility graph algorithm, and the associated simplicial complexes through the filtration process are quantified by PH. The evolution of the homology group dimension represented by Betti numbers demonstrates a strong dependency on the Hurst exponent (H). The coefficients of the birth and death curves of the k-dimensional topological holes (k-holes) at a given threshold depend on H which is almost not affected by finite sample size. We show that the distribution function of a lifetime for k-holes decays exponentially and the corresponding slope is an increasing function versus H and, more interestingly, the sample size effect completely disappears in this quantity. The persistence entropy logarithmically grows with the size of the visibility graph of a system with almost H-dependent prefactors. On the contrary, the local statistical features are not able to determine the corresponding Hurst exponent of fGn data, while the moments of eigenvalue distribution (M_{n}) for n≥1 reveal a dependency on H, containing the sample size effect. Finally, the PH shows the correlated behavior of electroencephalography for both healthy and schizophrenic samples.
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Affiliation(s)
- H Masoomy
- Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran
| | - B Askari
- Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran
| | - M N Najafi
- Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran
| | - S M S Movahed
- Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran
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