Phase transitions for the one-dimensional Landau-Ginzburg system and their relation to the theory of polymers and of electrons in a random potential

A study is made of the phase transitions for the one-dimensional Landau-Ginzburg system in the limit of zero degrees of freedom found by Balian and Toulouse (see abstr. A30922 of 1974). It is shown that they correspond to a rather simple property of a polymer in one dimension. The phase transitions are exhibited for a discrete model as well as for the Landau-Ginzburg model. The properties of an electron in a one-dimensional random potential can be found by an analytic continuation of this theory, but the phase transitions lie away from the physical region of this system. An explicit expression is derived for the localization length for the electron in a white-noise potential, and it is indicated how the conductivity could be calculated.