**Abstract**

Solving the complete eigenvalue problem of a matrix of size N, on the memory consumption scale, is on the order of N^2, and on the time scale is N^3. On the other hand, the kernel polynomial method is linearly scalable with the size of a matrix. The fact comes from the usage of the polynomial, in our case Chebyshev, expansion, as an efficient way to calculate spectral quantities of large sparse matrices, using only computational-resources inexpensive vector–matrix multiplication.

Starting from the expansion of the delta and Green's functions, we can derive expressions for other spectral quantities such as the density of states or the conductivity tensor based on the linear response Kubo formalism. In this way, we can efficiently perform calculations of systems that include the presence of disorder and magnetic fields, and apply the formalism to disordered monolayer, and (twisted) bilayer graphene. We will also focus on the numerical difficulties that can arise in the implementation.

Misa Andelkovic, Condensed Matter Theory, UAntwerpen