9.2. Fermion identities Prove that c (a) c(a+1) = (a) + (a+1) c (a)

9.2. Fermion identities Prove that c

(a) c†(a+1) = −˜σ

−

(a) ˜σ

+

(a+1)

c†

(a) c†(a+1) = ˜σ

+

(a) ˜σ

+

(a+1)

c

(a) c(a+1) = −˜σ

−

(a) ˜σ

−

(a+1).

Moreover, show that the order operators of the Ising model can be expressed in terms of the fermion operators c

(a) and c†

(a) as

σ
3

(a) = 2c†

(a) c(a)−1, ˜σ
1

(a) ˜σ1(a+1) = c†(a)−c(a)
c†(a+1)−c(a+1)
.