Proof of Gauge-Invariance of Magnetic Susceptibility of Topologically Trivial Insulators in the Independent Particle Approximation

We have derived an expression for the magnetic susceptibility of topologically trivial insulators, however an important consideration for any response tensor is whether it is gauge-invariant. By this we refer to the gauge-freedom in choosing the Bloch functions one uses in a computation since multiplication by an arbitrary $\mathbf{\text{k}}$-dependent complex phase produces equally valid Bloch functions. Additionally Wannier functions, which can be constructed by a unitary mixing of the Bloch functions, are a useful basis for computation due to their localized properties, but are strongly non-unique. Therefore, we show that the theoretical expression we derived is independent of the choice of Bloch functions or Wannier functions used, and thus is gauge-invariant in this sense.