Spin Correlations in the Ground State

For the sake of simplicity we show the calculation of the spin correlation function in the symmetric case x = —1/4, when the Hamiltonian (2) takes the form 

Since in this case there is one spin in each elementary cell, the singlet ground state wave function T0 can be written in a more simple and symmetric form  

One can check that wave functions (10) is the singlet ground-state wave function with zero energy for Hamiltonian (9). Therefore, the equivalence of wave functions (4) and (10) follows from the non-degeneracy of the ground state in the 5 = 0 sector (though functions T in Eqs.(10) and (4) are different). 

Now we calculate the norm and correlation function of the wave function To (10). The norm of the singlet wave function To is 

It is easy to check that the function T has Sz = 0. Then the projector P0 in Eq.(6) takes the form [11] 

---