Phase-locking in the multidimensional Frenkel-Kontorova model

We consider the multidimensional Frenkel-Kontorova model with one degree of freedom which is a variational problem for real functions on the lattice ${\bf Z}^n$. For every vector $\alpha\in{\bf R}^n$ there is a special class of minimal solutions $u_\alpha:{\bf Z}^n\to{\bf R}$ lying in finite distance to the linear function $x^{n+1}=\alpha x$ with $x\in{\bf Z}^n$. Due to periodicity properties of the variational problem $\alpha$ is called the rotation vector of these solutions. The average action $A(\alpha)$ of a minimal solution $u_\alpha$ is obtained by averaging the variational sum over ${\bf Z}^n$. One shows that this average action is the same for any minimal solution with finite distance to the linear function $x^{n+1}=\alpha x$ with rotation vector $\alpha$. Our main results concern the differentiability properties of $A(\alpha)$ as a function of the rotation vector: Typically, $A$ is not differentiable at $\alpha\in{\bf Q}^n$. This will be interpreted in a dual form as phase-locking. The phase $\alpha(\mu)$ of $\mu\in{\bf R}^n$ is defined by the unique vector in ${\bf R}^n$ for which $A(\alpha)-\alpha\cdot\mu$ is minimal. If one perturbs the variational principle by changing the parameter $\mu$, the non-differentiability of $A$ at $\alpha\in{\bf Q}^n$ forces the phase to be locked onto the rational value $\alpha(\mu)$.