9.5. XXZ model Consider the quantum spin chain of the XXZ model with Hamiltonian H = J1 N



k=1

(Sx kSx k+1

+Sy kSyk

+1)+J2

k Sz kSz k+1.

Assume periodic boundary condition. The spin operator  S is in the 1/2 representation and given by  S = 1/2σ, where σ is the set of Pauli matrices.

a. Show that the sign of J1 in H can be chosen freely without changing the physical properties of the model. Show that the same is not true for the sign of J2.

b. Discuss the symmetry of the system when J2/J1→0 and J2/J1→∞.

c. Use the fermionic representation of the operators S±

k

= Sx k

±iSx k and S3 k to write the Hamiltonian as H = J1 2

N



a=1 c†(a)c(a+1)+c†(a+1)c(a)

+J2 N



a=1

c†(a)c(a)− 1 2

c†(a+1)c(a+1)− 1 2

.

d. Consider the case J2 = 0, the so-called XY model. Use the Bogoliubov transformation to find in this case the spectrum of the fermionic form of the Hamiltonian.