Phase Separation in a Wedge: Exact Results

The exact theory of phase separation in a two-dimensional wedge is derived from the properties of the order parameter and boundary condition changing operators in field theory. For a shallow wedge we determine the passage probability for an interface with endpoints on the boundary. For generic opening angles we exhibit the fundamental origin of the filling transition condition and of the property known as wedge covariance.

Figure 1

Wedge geometry with boundary condition changing points at (0, $\pm R/2$) and an interface running between them.

Figure 2

Passage probability density $P(x,y)/m$ for the interface, with $P(x,y)$ given by Eq. (11), $mR=25$ and $\alpha =0.04$. The leftmost contour line corresponds to $P(x,y)=0$, and then to the wedge. The required divergence of the passage probability density as the pinning points (0, $\pm R/2$) are approached is also clearly visible.

Figure 3

Illustration of Eq. (16). The kink emission amplitude exhibits a bound state pole corresponding to adsorption of phase $b$ on the boundary.