Debye Model of Steps on Vicinal Crystal Surfaces

The steps on a vicinal crystal surface can be mapped onto the world lines of spinless fermions, with the average direction of the steps (the $y$-direction) being mapped to time. If the interaction energy per unit length between neighboring steps is given by $V(L) \! = \! A/L^2$ (as is common), this resulting quantum system is integrable for only three values of $\tilde {A} \! \equiv \! \tilde{\beta}A/(k_{\rm B}T)^2$. For other values of $\tilde{A}$, the Pairwise Einstein Model gives an excellent approximation for the Terrace Width Distribution (TWD, the histogram of $x_{i+1}(y)-x_{i}(y)$) but is severely limited in describing $g_x(\Delta y) \! \equiv \! \langle [x_i (y+\Delta y) - x_i(y)]^2\rangle$, particularly for $\Delta y \! > \! \xi$, the correlation length. Here we show how the one- dimensional Debye model correctly gives $g_x(\Delta y)$ even for large $\Delta y$. The Pairwise Einstein Model also suggests a relationship between the compressibility of the steps and the tails of the TWD, a relationship we clarify using the Debye model.