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Equivalence between the two-dimensional Ising model and the quantum XY chain with randomness and with open boundary

The equivalence between the two-dimensional Ising model and the one-dimensional quantum XY model is generalized to the cases with alternating/random interactions and with periodic/free boundary conditions. It is proved that the eigenstate for the maximum eigenvalue of the transfer matrix of the two-dimensional Ising model corresponds to the ground state of the Hamiltonian of the one-dimensional XY model, which certifies direct relations between the physical quantities of the two models.