Nonlinear lattice dynamics in two dimensions

In general, dynamical and quantum-mechanical properties of anharmonic lattices are best expressed in terms of bond variables. A transformation from atomic (site) to bond coordinates is generated in two dimensions, with the help of a number of subsidiary conditions expressed on plaquettes. It is shown that these commute with the Hamiltonian and yield a number of zero-energy modes. With the use of this trick, two-dimensional problems are now only slightly more difficult to study than the one-dimensional analogs. We discuss some exact results.