  We have developed a tensor network approach to the two-dimensional fully
frustrated classical XY spin model on the kagome lattice, and clarified the
nature of the possible phase transitions of various topological excitations.We
find that the standard tensor network representation for the partition function
does not work due to the strong frustrations in the low temperature limit. To
avoid the direct truncation of the Boltzmann weight, based on the duality
transformation, we introduce a new representation to build the tensor network
with local tensors lying on the centers of the elementary triangles of the
kagome lattice. Then the partition function is expressed as a product of
one-dimensional transfer matrix operators, whose eigen-equation can be solved
by the variational uniform matrix product state algorithm accurately. The
singularity of the entanglement entropy for the one-dimensional quantum
operator provides a stringent criterion for the possible phase transitions.
Through a systematic numerical analysis of thermodynamic properties and
correlation functions in the thermodynamic limit, we prove that the model
exhibits a single Berezinskii-Kosterlitz-Thouless phase transition only, which
is driven by the unbinding of $1/3$ fractional vortex-antivortex pairs
determined at $T_{c}\simeq 0.075J_{1}$ accurately. The absence of long-range
order of chirality or quasi-long range order of integer vortices has been
verified in the whole finite temperature range. Thus the long-standing
controversy about the phase transitions in this fully frustrated XY model on
the kagome lattice is solved rigorously, which provides a plausible way to
understand the charge-6e superconducting phase observed experimentally in the
two-dimensional kagome superconductors.
