When the external field 
is applied to the system
then the zero field spin-flip symmetry is broken. The operator
was chosen as one of the complete commuting set in section 8.5 to label the energy eigenvectors.
Thus those eigenvectors are also eigenvectors of
which is the Hamiltonian of equation 8.11 with an external field applied only in the z direction. The eigenvalues in the energy eigenvector basis are
now given by 
. The probabilities of each eigenstate are calculated from the Boltzmann distribution using these energies, and as before the probability distribution of the 
eigenstates, the information correlation functions, moments, correlations, and cumulants may be calculated.