4.2. Low-temperature series in the presence of a magnetic field Consider the two-dimensional Ising model on a square lattice with equal coupling along the horizontal and vertical links and in presence of an external magnetic field B.

Generalize the discussion on the series expansion of the free energy in the lowtemperature phase and show that ZN can be written as ZN = exp[2NK +NβB]

∞



r,s=1 n(r, s) exp[−2Kr] exp[−sβB]

where K = βJ and n(r, s) is the number of closed graphs made of r links on the dual lattice, having in their internal region s points of the original lattice.