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The Berezinskii-Kosterlitz-Thouless (BKT) transition in magnetic system is anintriguing phenomena and an accurate estimation of the BKT transitiontemperature has been a long-standing problem. In this work we explore theanisotropic classical Heisenberg XY and XXZ models with ferromagnetic exchangeon a square lattice and antiferromagnetic exchange on a triangular latticeusing an unsupervised machine learning approach called principal componentanalysis (PCA). In earlier studies of the BKT transition, spin configurationsand vorticities calculated from Monte Carlo method are used to determine thetransition temperature $T_{BKT}$, but those methods fail to give any conclusiveresults by analyzing the principal components in the PCA approach. In this workvorticities are used as initial input to the PCA and curve of the firstprincipal component with temperature is fitted with a function to determine anaccurate value of $T_{BKT}$. This procedure works well for anisotropicclassical Heisenberg with ferromagnetic exchange on square lattice as well asfor frustrated antiferromagnetic exchange on a triangular lattice. Theclassical anisotropic Heisenberg antiferromagnetic model on the triangularlattice has two close transitions; the BKT at $T_{BKT}$ and Ising like phasetransition for chirality at $T_c$ and it is difficult to separate thesetransition points. It is also noted that using the PCA method and manipulationof their first principal component, not only separation of transition pointsare possible but also transition temperature can be determined accurately.

Study of the Berezinskii-Kosterlitz-Thouless transition: An unsupervised machine learning approach