Ground states of one-dimensional systems using effective potentials

A nonlinear eigenvalue equation whose solution is an ‘‘effective potential’’ is used to study the ground states of one-dimensional systems (such as the Frenkel-Kontorova model) whose Hamiltonian H is a sum of terms V($u_n$)+W($u_{n+1\mathrm{}}$-$u_n$), where the $u_n$ are real and V is periodic. The procedure is not limited to convex W, and it yields the ground-state energy and orbit, in contrast to metastable or unstable states, and some information about ‘‘soliton’’ defects. It can be generalized to H a sum of K($x_{n+1\mathrm{}}$,$x_n$), where the arguments may be multidimensional. Numerical solutions of the eigenvalue problem are used to work out phase diagrams for W a parabola, and various choices of V. With V a cosine plus a small admixture of a second or third harmonic with the proper sign, we find first-order transitions between states of the same winding number ω but different symmetry. A piecewise parabolic V with continuous first derivative can yield sliding states (invariant circles) with rational ω.