The Hamiltonian of a theory describes its dynamics. Symmetry is "spontaneously broken" when a certain state of the quantum theory doesn't have the same symmetry as the Hamiltonian (dynamics). The standard example is a theory with a quartic potential. In field theory, say we have a potential $V (\phi) \sim \lambda{(\phi^2 - a^2)}^2$ in the lagrangian/hamiltonian, for some real valued scalar field $\phi$. This potential has minima at $\phi_{\pm}=\pm a$.

The theory (Hamiltonian) has a $\phi \rightarrow - \phi$ symmetry, but note that each of those *vacua don't have the same symmetry* i.e. vacuum states are not invariant under the symmetry. The solutions $\phi_{\pm}$ transform into each other under the transformation. So this is an example where the symmetry of the theory is *broken* by the vacuum state, "spontaneously" (roughly by itself, as the theory settles into the vacuum state).

So, to find whether a symmetry will be spontaneously broken (given a Hamiltonian), you have to check the symmetry of the states (typically vacua) of the theory and compare them to the symmetry of the Hamiltonian.

This post imported from StackExchange Physics at 2014-04-24 02:31 (UCT), posted by SE-user Siva