 $ [{\cal{A}}_{\mathbf{k}}, {\cal{A}}_{{\mathbf{k^{\prime}}}}^{\dagger }]= \delta_{{\mathbf{k}}, {\mathbf{k^{\prime}}}}  $ and $ [{\cal{B}}_{\mathbf{k}}, {\cal{B}}_{{\mathbf{k^{\prime}}}}^{\dagger }]= \delta_{{\mathbf{k}}, {\mathbf{k^{\prime}}}}  $