Relationship among Exactly Soluble Models of Critical Phenomena. I

By relating a transfer matrix associated with a classical system to the Hamiltonian of its corresponding quantum mechanical system, it is proved that the two-dimensional Ising model in the absence of a magnetic field is equivalent to the ground state of the linear XY-model in the presence of a magnetic field, under appropriate relations among coupling parameters appearing in the two Hamiltonians. Simple relations are established for spin correlations in both systems. Consequently, the Ising system and the ground state of the XY-model are shown to exhibit similar singularities with respect to temperature and a magnetic field, respectively. It is also proved that the two-dimensional dimer problem is equivalent to the ground state of a generalized XY-model, which is solved exactly in general.