  At strong magnetic fields double-layer two-dimensional-electron-gas systems
can form an unusual broken symmetry state with spontaneous inter-layer phase
coherence. In this paper we explore the rich variety of quantum and
finite-temperature phase transitions associated with this broken symmetry. We
describe the system using a pseudospin language in which the layer
degree-of-freedom is mapped to a fictional spin 1/2 degree-of-freedom. With
this mapping the spontaneous symmetry breaking is equivalent to that of a spin
1/2 easy-plane ferromagnet. In this language spin-textures can carry a charge.
In particular, vortices carry e/2 electrical charge and vortex-antivortex pairs
can be neutral or carry charge e. We derive an effective low-energy action and
use it to discuss the charged and collective neutral excitations of the system.
We have obtained the parameters of the Landau-Ginzburg functional from
first-principles estimates and from finite-size exact diagonalization studies.
We use these results to estimate the dependence of the critical temperature for
the Kosterlitz-Thouless phase transition on layer separation.
