The Method of Effective Potentials, and its Applications.

This thesis presents a new approach, the method of effective potentials, for studying the ground states of one-dimensional systems with competing length scales, which can exhibit complex modulated phases and phase transitions. The systems have a Hamiltonian H which is a sum of terms V(u(,n)) + W(u(,n)-u(,n-1)), where the u(,n) are real variables and V is a periodic function. The method involves solving a non-linear eigenvalue equation whose solution is the "effective potential". It works for both convex and non -convex W, and yields the ground state energy and configuration, in distinction to metastable or unstable states. It also gives some information about "soliton" defects, which can be used to determine the locations of phase boundaries in case W is a parabola. The method can be generalized to H a sum of terms K(x(,n), x(,n-1)), where the arguments may be multi-dimensional. Numerical solutions of the eigenvalue problem are used to work out phase diagrams for various choices of W and V: (i) For W a parabola and V a cosine, the phase diagram is similar to that found by Aubry, with continous transitions and a devil's staircase structure. (ii) For W a parabola and V a cosine plus a small admixture of a second or third harmonic with the proper sign, the phase diagram exhibits additional complicated structures: a lot of horizontal bars corresponding to first-order transitions between states of the same winding number (omega) but different symmetry, and accumulation points of these bars. (iii) For W a parabola and V a particular piecewise parabolic function with continuous first derivative, the first-order transitions in case (ii) occur at pinching points, which are associated with sliding states (invariant circles) with rational (omega). (iv) For both W and V piecewise parabolic of the type used in (iii), the phase diagram can be divided into a convex region (in which only the convex part of W plays a role in determining the ground state) and a non-convex region (in which the non-convex part of W is involved). The latter possesses new phase structures which do not occur for a convex W, including first-order transitions between phases with different (omega)'s, tricritical points, and an accumulation point of bicritical points. A mathematical proof for the existence of a solution of the eigenvalue equation is given in the appendix.