Crossover of magnetoconductance autocorrelation for a ballistic chaotic quantum dot

The autocorrelation function $C_{\varphi,\eps}(\Delta\varphi,\,\Delta \eps)=$ $\langle \delta g(\varphi,\,\eps)\, \delta g(\varphi+\Delta\varphi,\,\eps+\Delta \eps)\rangle$ ($\varphi$ and $\eps$ are rescaled magnetic flux and energy) for the magnetoconductance of a ballistic chaotic quantum dot is calculated in the framework of the supersymmetric non-linear $\sigma$-model. The Hamiltonian of the quantum dot is modelled by a Gaussian random matrix. The particular form of the symmetry breaking matrix is found to be relevant for the autocorrelation function but not for the average conductance. Our results are valid for the complete crossover from orthogonal to unitary symmetry and their relation with semiclassical theory and an $S$-matrix Brownian motion ensemble is discussed.