The transfer matrix for a pure phase in the two-dimensional Ising model

The problem of diagonalizing the transfer matrix for the two dimensional Ising model with all boundary spins equal to +1 is solved by use of the spinor method. This provides a simple proof that the spontaneous magnetization is actually given by the well known formula for the long range order with torodial boundary conditions, and this means that the critical temperature is precisely that temperature above which the state is unique and below which it is non unique. An expression for the magnetization at finite distance from the boundary is also given, and a simple derivation of the formula for the surface tension between two coexisting phases is presented. Finally the relation between the degeneracy of the spectrum and the phase transition is discussed.