Magnetic responses in the presence of a partial magnetic field on the rectangular Ising lattice

We study the two-dimensional Ising model for the square lattice in the presence of a magnetic field which applies on a part of spins. Specifically, we consider the situation in which only the spins on a row or a diagonal line can interact with the magnetic field. The low-temperature series expansions of the reduced partition functions for the square lattice with a magnetic field on a row are presented for both ferromagnetic and antiferromagnetic spin-spin couplings. The low- and high-temperature series expansions of the zero-field susceptibility for the above lattice are performed, and the critical amplitudes are estimated. The critical exponent γ′_r of the susceptibility for a ferromagnet is confirmed to be 3/4. The spontaneous magnetization per field-applied site of the same system is found to be the same as the ordinary spontaneous magnetization per site with an uniform magnetic field. The exact partition functions are also obtained for the super-exchange model applying both on a boundary row and on the central row of the rectangular Ising lattice.