Non-parametric learning critical behavior in Ising partition functions: PCA entropy and intrinsic dimension
The Abdus Salam International Centre for Theoretical Physics (ICTP)（阿卜杜斯·萨拉姆国际理论物理中心）；SISSA — International School of Advanced Studies（SISSA高等研究院）；INFN Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Dipartimento di Matematica e Geoscienze, Universitá degli Studi di Trieste（的里雅斯特大学数学与地球科学系）；School of Mathematical Sciences, Queen Mary University of London（伦敦大学皇家玛丽学院数理科学学院）；Nordita, KTH Royal Institute of Technology and Stockholm University（北欧理论天体物理学研究所，瑞典皇家理工学院和斯德哥尔摩大学）；The Alan Turing Institute（艾伦·图灵研究所）
Rajat K. Panda, Roberto Verdel, Alex Rodriguez, Hanlin Sun, Ginestra Bianconi, Marcello Dalmonte
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20 pages, 10 figures, Submission to SciPost. Comments are welcome
We provide and critically analyze a framework to learn critical behavior in classical partition functions through the application of non-parametric methods to data sets of thermal configurations. We illustrate our approach in phase transitions in 2D and 3D Ising models. First, we extend previous studies on the intrinsic dimension of 2D partition function data sets, by exploring the effect of volume in 3D Ising data. We find that as opposed to 2D systems for which this quantity has been successfully used in unsupervised characterizations of critical phenomena, in the 3D case its estimation is far more challenging. To circumvent this limitation, we then use the principal component analysis (PCA) entropy, a "Shannon entropy" of the normalized spectrum of the covariance matrix. We find a striking qualitative similarity to the thermodynamic entropy, which the PCA entropy approaches asymptotically. The latter allows us to extract -- through a conventional finite-size scaling analysis with modest lattice sizes -- the critical temperature with less than $1\%$ error for both 2D and 3D models while being computationally efficient. The PCA entropy can readily be applied to characterize correlations and critical phenomena in a huge variety of many-body problems and suggests a (direct) link between easy-to-compute quantities and entropies.
A Microcanonical Inflection Point Analysis via Parametric Curves and its Relation to the Zeros of the Partition Function
Universidade Federal de Ouro Preto - UFOP（欧罗普雷托联邦大学）；Universidade Federal de Juiz de Fora - UFJF（茹伊斯德福拉联邦大学）；Universidade Federal de Minas Gerais - UFMG（米纳斯吉拉斯联邦大学）
