Semiclassical theory of magnetization for a two-dimensional non-interacting electron gas

We compute the semiclassical magnetization and susceptibility of non-interacting electrons, confined by a smooth two-dimensional potential and subjected to a uniform perpendicular magnetic field, in the general case when their classical motion is chaotic. It is demonstrated that the magnetization per particle m(B) is directly related to the staircase function N(E), which counts the single-particle levels up to energy E. Using Gutzwiller's trace formula for N, we derive a semiclassical expression for m. Our results show that the magnetization has a non-zero average, which arises from quantum corrections to the leading-order Weyl approximation to the mean staircase and which is independent of whether the classical motion is chaotic or not. Fluctuations about the average are due to classical periodic orbits and do represent a signature of chaos. This behaviour is confirmed by numerical computations for a specific system.