Orbital magnetism of mesoscopic integrable system

In the mesoscopic regime, magnetic properties (such as orbital magnetism) are sensitive to whether the corresponding classical dynamics is chaotic or integrable. Non-interacting electron gas in a rectangular box is proposed as a ``generic'' model to study orbital magnetism of integrable system. We derived the exact energy level correlation function for this system, including the perturbation by magnetic field. Combining the exact correlation function and Imry's formalism, we calculated the orbital magnetic susceptibility and discussed the field dependence at$T\to 0$ and temperature dependence at $B\to 0$. As a result, the susceptibility $\chi \sim \left| {\chi _L } \right|\sqrt {k_F L} \left\{ {\begin{array}{l} \log \left( {{\phi _0 } \mathord{\left/ {\vphantom {{\phi _0 } \phi }} \right. \kern-\nulldelimiterspace} \phi } \right)\mbox{ }(T\to 0,\phi <<\phi _0 ) \\ \log \left( {{\sqrt {\Delta E_F } } \mathord{\left/ {\vphantom {{\sqrt {\Delta E_F } } T}} \right. \kern-\nulldelimiterspace} T} \right)\mbox{ }(B\to 0,T<<\sqrt {\Delta E_F } ) \\ \end{array}} \right.$, where \textit{$\chi $}$_{L}$ is the Landau susceptibility, $k_{F}$ the is Fermi vector, $L$ is the rectangle's side, \textit{$\phi $}$_{0}$ is the magnetic flux quantum, $\Delta $ is the mean level spacing, and $E_{F}$ is the Fermi energy. For high temperature and large field, the mesoscopic part of susceptibility exponentially vanishes and only the bulk Landau diamagnetism is left. The logarithmic divergence at zero field and zero temperature is consistent with previous numerical calculations and is a manifestation of pronounced non-self-averaging properties of integrable systems.