  We give a self-contained exposition of the combinatorial solution of quantum
mechanical systems of coupled spins on a one-dimensional lattice. Using Trotter
formula, we write the partition function as a generating function of a spanning
subgraph of a two-dimensional lattice and solve the combinatorial problem by
the method of Pfaffians provided the weights satisfy the so-called free fermion
condition. The free energy and the ground state energy as a function of the
inverse temperature, couplings J and magnetic fields h, for the XY model in a
transverse field with period p=1 and 2, is then obtained.
