Please wait a minute...
Frontiers of Physics

ISSN 2095-0462

ISSN 2095-0470(Online)

CN 11-5994/O4

Postal Subscription Code 80-965

2018 Impact Factor: 2.483

Front. Phys.    2018, Vol. 13 Issue (5) : 130507    https://doi.org/10.1007/s11467-018-0798-7
RESEARCH ARTICLE
Machine learning of frustrated classical spin models (II): Kernel principal component analysis
Ce Wang1, Hui Zhai1,2()
1. Institute for Advanced Study, Tsinghua University, Beijing 100084, China
2. Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
 Download: PDF(2044 KB)  
 Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks
Abstract

In this work, we apply a principal component analysis (PCA) method with a kernel trick to study the classification of phases and phase transitions in classical XY models of frustrated lattices. Compared to our previous work with the linear PCA method, the kernel PCA can capture nonlinear functions. In this case, the Z2 chiral order of the classical spins in these lattices is indeed a nonlinear function of the input spin configurations. In addition to the principal component revealed by the linear PCA, the kernel PCA can find two more principal components using the data generated by Monte Carlo simulation for various temperatures as the input. One of them is related to the strength of the U(1) order parameter, and the other directly manifests the chiral order parameter that characterizes the Z2 symmetry breaking. For a temperature-resolved study, the temperature dependence of the principal eigenvalue associated with the Z2 symmetry breaking clearly shows second-order phase transition behavior.

Keywords machine learning      classical XY model      kernel PCA      frustrated lattice     
Corresponding Author(s): Hui Zhai   
Issue Date: 13 June 2018
 Cite this article:   
Ce Wang,Hui Zhai. Machine learning of frustrated classical spin models (II): Kernel principal component analysis[J]. Front. Phys. , 2018, 13(5): 130507.
 URL:  
https://academic.hep.com.cn/fop/EN/10.1007/s11467-018-0798-7
https://academic.hep.com.cn/fop/EN/Y2018/V13/I5/130507
1 C. Wang and H. Zhai, Machine learning of frustrated classical spin models (I): Principal component analysis, Phys. Rev. B 96(14), 144432 (2017)
https://doi.org/10.1103/PhysRevB.96.144432
2 L. Wang, Discovering phase transitions with unsupervised learning, Phys. Rev. B 94(19), 195105 (2016)
https://doi.org/10.1103/PhysRevB.94.195105
3 J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nat. Phys. 13(5), 431 (2017)
4 E. P. L. van Nieuwenburg, Y. H. Liu, and S. D. Huber, Learning phase transitions by confusion, Nat. Phys. 13(5), 435 (2017)
5 G. Torlai and R. G. Melko, Learning thermodyamics with Boltzmann machines, Phys. Rev. B 94(16), 165134 (2016)
https://doi.org/10.1103/PhysRevB.94.165134
6 S. Wetzel, Unsupervised learning of phase transitions: From principal component analysis to variational autoencoders, Phys. Rev. E 96(2), 022140 (2017)
https://doi.org/10.1103/PhysRevE.96.022140
7 P. Ponte and R. G. Melko, Kernel methods for interpretable machine learning of order parameters, Phys. Rev. B 96(20), 205146 (2017)
https://doi.org/10.1103/PhysRevB.96.205146
8 W. J. Hu, R. Singh, and R. Scalettar, Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination, Phys. Rev. E 95(6), 062122 (2017)
https://doi.org/10.1103/PhysRevE.95.062122
9 K. Ch’ng, N. Vazquez, and E. Khatami, Unsupervised machine learning account of magnetic transitions in the Hubbard model, Phys. Rev. E 97(1), 013306 (2018)
https://doi.org/10.1103/PhysRevE.97.013306
10 N. C. Costa, W. J. Hu, Z. J. Bai, R. Scalettar, and R. Singh, Principal component analysis for fermionic critical points, Phys. Rev. B 96(19), 195138 (2017)
https://doi.org/10.1103/PhysRevB.96.195138
11 S. Wetzel and M. Scherzer, Machine learning of explicit order parameters: From the Ising model to SU(2) lattice gauge theory, Phys. Rev. B 96(18), 184410 (2017)
https://doi.org/10.1103/PhysRevB.96.184410
12 K. Ch’ng, J. Carrasquilla, R. G. Melko, and E. Khatami, Machine learning phases of strongly correlated fermions, Phys. Rev. X 7(3), 031038 (2017)
https://doi.org/10.1103/PhysRevX.7.031038
13 P. Broecker, J. Carrasquilla, R. G. Melko, and S. Trebst, Machine learning quantum phases of matter beyond the fermion sign problem, Sci. Rep. 7(1), 8823 (2017)
https://doi.org/10.1038/s41598-017-09098-0
14 P. Broecker, F. F. Assaad, and S. Trebst, Quantum phase recognition via unsupervised machine learning, arXiv: 1707.00663 (2017)
15 M. Beach, A. Golubeva, and R. G. Melko, Machine learning vortices at the Kosterlitz–Thouless transition, Phys. Rev. B 97(4), 045207 (2018)
https://doi.org/10.1103/PhysRevB.97.045207
16 Y. Zhang and E. Kim, Quantum loop topography for machine learning, Phys. Rev. Lett. 118(21), 216401 (2017)
https://doi.org/10.1103/PhysRevLett.118.216401
17 Y. Zhang, R. G. Melko, and E. Kim, Machine learning Z2 quantum spin liquids with quasiparticle statistics, Phys. Rev. B 96(24), 245119 (2017)
https://doi.org/10.1103/PhysRevB.96.245119
18 P. Zhang, H. Shen, and H. Zhai, Machine learning topological invariants with neural networks, Phys. Rev. Lett. 120(6), 066401 (2018)
https://doi.org/10.1103/PhysRevLett.120.066401
19 J. Villain, Spin glass with non-random interactions, J. Phys. Chem. 10, 1717 (1977)
https://doi.org/10.1088/0022-3719/10/10/014
20 J. Villain, Two-level systems in spin-glass model (I): General formalism and two-dimensional model, J. Phys. Chem. 10, 4793 (1977)
https://doi.org/10.1088/0022-3719/10/23/013
21 D. H. Lee, J. D. Joannopoulos, J. W. Negele, and D. P. Landau, Discrete-symmetry breaking and novel critical phenomena in an antiferromagnetic planar (XY) model in two dimensions, Phys. Rev. Lett. 52(6), 433 (1984)
https://doi.org/10.1103/PhysRevLett.52.433
22 S. Miyashita and H. Shiba, Nature of phase transition of the two-dimensional antiferromagnetic plane rotator model on the triangular lattice, J. Phys. Soc. Jpn. 53(3), 1145 (1984)
https://doi.org/10.1143/JPSJ.53.1145
23 S. Lee and K. C. Lee, Phase transitions in the fully frustrated XY model studied with use of the microcanonical Monte Carlo technique, Phys. Rev. B 49(21), 15184 (1994)
https://doi.org/10.1103/PhysRevB.49.15184
24 S. Korshunov, Kink pairs unbinding on domain walls and the sequence of phase transitions in fully frustrated XYmodels, Phys. Rev. Lett. 88(16), 167007 (2002)
https://doi.org/10.1103/PhysRevLett.88.167007
25 M. Hasenbusch, A. Pelissetto, and E. Vicari, Transitions and crossover phenomena in fully frustrated XYsystems, Phys. Rev. B 72(18), 184502 (2005)
https://doi.org/10.1103/PhysRevB.72.184502
26 T. Obuchi and H. Kawamura, Spin and chiral orderings of the antiferromagnetic XYmodel on the triangular lattice and their critical properties, J. Phys. Soc. Jpn. 81(5), 054003 (2012)
https://doi.org/10.1143/JPSJ.81.054003
27 J. P. Lv, T. M. Garoni, and Y. J. Deng, Phase transitions in XYantiferromagnets on plane triangulations, Phys. Rev. B 87(2), 024108 (2013)
https://doi.org/10.1103/PhysRevB.87.024108
28 P. Olsson, Monte Carlo analysis of the two-dimensional XYmodel (II): Comparison with the Kosterlitz renormalization-group equations, Phys. Rev. B 52(6), 4526 (1995)
https://doi.org/10.1103/PhysRevB.52.4526
29 T. Ohta and D. Jasnow, XYmodel and the superfluid density in two dimensions, Phys. Rev. B 20(1), 139 (1979)
https://doi.org/10.1103/PhysRevB.20.139
30 H. Weber and P. Minnhagen, Monte Carlo determination of the critical temperature for the two-dimensional XYmodel, Phys. Rev. B 37(10), 5986 (1988)
https://doi.org/10.1103/PhysRevB.37.5986
31 C. M. Bishop, Pattern Recognition and Machine Learning, Springer, 2007
[1] Wen Tong, Qun Wei, Hai-Yan Yan, Mei-Guang Zhang, Xuan-Min Zhu. Accelerating inverse crystal structure prediction by machine learning: A case study of carbon allotropes[J]. Front. Phys. , 2020, 15(6): 63501-.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed