in whichEk¼2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2gξ
~
S
2
ð1γ
k
Þ½2gð1þγ
kÞþD
q
are the eigenvalues of
the energy operator, 〈…〉0 means a thermal average calculated
through the quadratic Hamiltonian and the integral is evaluated over
thefirst Brillouin zone. Theξparameter is given by