Because of this property, automatically \( \hat{a}^\dagger_i \hat{a}^\dagger_i = 0 \), enforcing the Pauli exclusion principle. Thus when writing a Slater determinant using creation operators, $$ \hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k \ldots |0 \rangle $$ each index \( i,j,k, \ldots \) must be unique.

For some relevant exercises with solutions see chapter 8 of Lecture Notes in Physics, volume 936.