Classification of the sign of the critical Casimir force in two-dimensional systems at asymptotically large separations

We classify the sign of the critical Casimir force between two finite objects separated by a large distance in the two-dimensional systems that can be described by conformal field theory (CFT). Specifically, we show that, as long as the smallest scaling dimension present in the spectrum of the system is smaller than one, the sign of the force is independent of the shape of the objects and can be determined by the elements of the modular $S$ matrix of the CFT. The provided formula for the sign of the force indicates that the force is always attractive for equal boundary conditions, independent of the shape of the objects. However, different boundary conditions can lead to attractive or repulsive forces. Using the derived formula, we prove the known results regarding the Ising model and the free bosons. As new examples, we give detailed results regarding the $Q=3$ state Potts model and the compactified bosons. For example, for the latter model we show that the Dirichlet boundary condition does not always lead to an attractive force.

Figure 1

The conformal map $w(z)$ takes the whole plane minus the domains $D_1$ and $D_2$ to an annulus.