Introduction to the Mathematics of the XY -Spin Chain

In the following we present an introduction to the mathematical theory of the XY spin chain. The importance of this model lies in the fact, first understood by Lieb, Schultz and Mattis in [4], that the XY spin chain is one of very few “exactly solvable” models in the theory of quantum many-body systems. Lieb, Schultz and Mattis considered the constant coefficient case. In the variable coefficient case considered here, “exactly solvable” should be understood as “reducible to an effective one-particle Hamiltonian”. The key method behind this is the Jordan-Wigner transform, which allows to map the XY chain to a free Fermion system. 1 The isotropic XY -spin chain For a positive integer n, consider the Hamiltonian