Momentum space Hamiltonian of a disordered system of tight binding model

Question

I have introduced onsite disorder in Graphene. How will I write the momentum space Hamiltonian for such a system?(System size is 25 x 25 unit cells) If I don't add onsite disorder then we can write the momentum space Hamiltonian(k space) easily. The tight binding Hamiltonian in terms of creation and annihilation operator reads as, $$H = -t \sum_{<i,j>} c_i^{\dagger} c_j + \sum_{i} w_i c_i^{\dagger} c_i$$ where,the first quantity on r.h.s is the nearest neighbor hopping term, t is the nearest neighbor hopping strength, $c_i^\dagger$ is the creation operator, $c_i$ is the annihilation operator. The second quantity is for onsite disorder where, $w_i$ is the randomly generated onsite energy at site $i$.

The important thing here is that the second quantity on r.h.s breaks the translation symmetry.

Answer 1

In momentum-space, the hopping term will be diagonal in momenta and the on-site disorder term will mix/scatter different momenta. (In other words, momentum is no longer conserved when disorder is present.) With periodic boundary conditions, one natural way of expressing the disorder term is $$ \sum_i w_i c_i^\dagger c_i = \sum_{k,k'} W_{k,k'} c_k^\dagger c_{k'}, $$ where the scattering matrix is $$ W_{k,k'} = \frac{1}{N} \sum_{j=1}^N w_j e^{i(2\pi/N)j(k-k')}. $$