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A general framework is proposed to solve the two-dimensional fully frustratedXY model for the Josephson junction arrays in a perpendicular magnetic field.The essential idea is to encode the ground-state local rules induced byfrustrations in the local tensors of the partition function. The partitionfunction is then expressed in terms of a product of one-dimensional transfermatrix operator, whose eigen-equation can be solved by an algorithm of matrixproduct states rigorously. The singularity of the entanglement entropy for theone-dimensional quantum analogue provides a stringent criterion to distinguishvarious phase transitions without identifying any order parameter a prior. Twovery close phase transitions are determined at $T_{c1}\approx 0.4459$ and$T_{c2}\approx 0.4532$, respectively. The former corresponding to aBerezinskii-Kosterlitz-Thouless phase transition describing the phase coherenceof XY spins, and the latter is an Ising-like continuous phase transition belowwhich a chirality order with spontaneously broken $Z_2$ symmetry isestablished.

Tensor network approach to the two-dimensional fully frustrated XY model and a chiral ordered phase