  The experimentally measured phase diagram of cuprate superconductors in the
temperature-applied magnetic field plane illuminates key issues in
understanding the physics of these materials. At low temperature, the
superconducting state gives way to a long-range charge order with increasing
magnetic field; both the orders coexist in a small intermediate region. The
charge order transition is strikingly insensitive to temperature, and quickly
reaches a transition temperature close to the zero-field superconducting $T_c$.
We argue that such a transition along with the presence of the coexisting phase
cannot be described simply by a competing orders formalism. We demonstrate that
for some range of parameters there is an enlarged symmetry of the strongly
coupled charge and superconducting orders in the system depending on their
relative masses and the coupling strength of the two orders. We establish that
this sharp switch from the superconducting phase to the charge order phase can
be understood in the framework of a composite SU(2) order parameter comprising
the charge and superconducting orders. Finally, we illustrate that there is a
possibility of the coexisting phase of the competing charge and superconducting
orders only when the SU(2) symmetry between them is weakly broken due to
biquadratic terms in the free energy. The relation of this sharp transition to
the proximity to the pseudogap quantum critical doping is also discussed.
