  We reinvestigate the phase transitions of the Ising model on the Kagome
lattice with antiferromagnetic nearest-neighbor and ferromagnetic
next-nearest-neighbor interactions, which has a six-state-clock spin ice ground
state and two consecutive Berezinskii-Kosterlitz-Thouless (BKT) phase
transitions. Employing the classical Monte Carlo (MC) simulations, the phases
are characterized by the magnetic order parameter, and the critical
temperatures are obtained by the finite-size scaling of related physical
quantities. Moreover, we attempt to gain general information on the phase
transitions from the MC process instead of MC results and successfully extract
the correct transition points with surprisingly high accuracy. Specifically, we
focus on the selected data set of uncorrelated MC configurations and quantify
the MC process using the distribution of two-configuration Hamming distances in
this small data collection. This distribution is more than a quantity that
features different behaviors in different phases but also nicely supports the
same BKT scaling form as the order parameter, from which we successfully
determine the two BKT transition points with surprisingly high accuracy. We
also discuss the connection between the phase transitions and the intrinsic
dimension extracted from the Hamming distances, which is widely used in the
growing field of machine learning and is reported to be able to detect critical
points. Our findings provide a new understanding of the spin ice transitions in
the Kagome lattice and can hopefully be used similarly to identify transitions
in the quantum system on the same lattice with strong frustrations.
