The XY Model

Abstract

The XY Model is another linear chain of spin-1/2 atoms but with a different type of exchange interaction in which only the x-components and y-components of the spins are involved, and with unequal weights. This system has interesting and unusual features and it is exactly soluble i.e. integrable. However the mathematical techniques involved here are not based on the Bethe Ansatz, but instead use a Jordan-Wigner transformation. The first step is to introduce new operators which are fermion operators, unlike the spin operators considered previously. The Jordan-Wigner transformation then involves combining these fermion operators into new ‘quasiparticle’ operators that are still fermions. The ground state is then a state in which all eigenstates of these operators are occupied up to the ‘Fermi surface’. It is then possible to calculate the ground state energy of this system.