  We address the quantum-classical correspondence for chaotic systems with a
crossover between symmetry classes. We consider the energy level statistics of
a classically chaotic system in a weak magnetic field. The generating function
of spectral correlations is calculated by using the semiclassical
periodic-orbit theory. An explicit calculation up to the second order,
including the non-oscillatory and oscillatory terms, agrees with the prediction
of random matrix theory. Formal expressions of the higher order terms are also
presented. The nonlinear sigma (NLS) model of random matrix theory, in the
variant of the Bosonic replica trick, is also analyzed for the crossover
between the Gaussian orthogonal ensemble and Gaussian unitary ensemble. The
diagrammatic expansion of the NLS model is interpreted in terms of the periodic
orbit theory.
