Thermodynamic measure of the magnetoelectric coupling in a three-dimensional topological insulator

We show that the magnetoelectric coupling in three-dimensional (strong) topological insulators is related to a second derivative of the bulk magnetization. The formula we derive is the nonlinear-response analog of the Streda formula for Hall conductivity [Streda, J. Phys. C 15, L717 (1982)], which relates the Hall conductivity to the derivative of the magnetization with respect to the chemical potential. Our finding allows one to extract the magnetoelectric coefficient by measuring the magnetization, while varying the chemical potential and one more perturbing field. The relation we find also makes transparent the effect of disorder on the magnetoelectric response, which occurs only through the density of states, and has no effect when the system is gapped