Frustration is a ubiquitous phenomenon in many-body physics that influences
the nature of the system in a profound way with exotic emergent behavior.
Despite its long research history, the analytical or numerical investigations
on frustrated spin models remain a formidable challenge due to their extensive
ground state degeneracy. In this work, we propose a unified tensor network
theory to numerically solve the frustrated classical spin models on various
two-dimensional (2D) lattice geometry with high efficiency. We show that the
appropriate encoding of emergent degrees of freedom in each local tensor is of
crucial importance in the construction of the infinite tensor network
representation of the partition function. The frustrations are thus relieved
through the effective interactions between emergent local degrees of freedom.
Then the partition function is written as a product of a one-dimensional (1D)
transfer operator, whose eigen-equation can be solved by the standard algorithm
of matrix product states rigorously, and various phase transitions can be
accurately determined from the singularities of the entanglement entropy of the
1D quantum correspondence. We demonstrated the power of our unified theory by
numerically solving 2D fully frustrated XY spin models on the kagome, square
and triangular lattices, giving rise to a variety of thermal phase transitions
from infinite-order Brezinskii-Kosterlitz-Thouless transitions, second-order
transitions, to first-order phase transitions. Our approach holds the potential
application to other types of frustrated classical systems like Heisenberg spin
antiferromagnets.Comment: 20 pages, 19 figure