Answer Happy

3. Massive Dirac Model Consider a 2D tight binding model for fermions on a square lattice which in the single particle sector looks like: II=​2i1​(x,y)∈Z2∑​(∣x+1,y⟩⟨x,y∣−∣x,y⟩⟨x+1,y∣)⊗σx​+2i1​(x,y)∈Z2∑​(∣x,y+1⟩⟨x,y∣−∣x,y⟩⟨x,y+1∣)⊗σy​+21​(x,y)∈Z2∑​(∣x+1,y⟩⟨x,y∣+∣x,y⟩⟨x+1,y∣)⊗σz​+21​(x,y)∈Z2∑​(∣x,y+1⟩⟨x,y∣+∣x,y⟩⟨x,y+1∣)⊗σz​+MI⊗σz​​ where we have chosen units where the lattice constant is a=1. This is yet another example of a two-dimensional Chern insulator, as you are about to understand. 3.1. Show that the above Hamiltonian can be written in momentum space as H(k)=μ=1∑2​sin(kμ​)σμ​+(M−μ=1∑2​cos(kμ​))σz​