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Systems as diverse as binary mixtures and inclusions in biological membranes,and many more, can be described effectively by interacting spins. When thecritical fluctuations in these systems are constrained by boundary conditions,critical Casimir forces (CCF) emerge. Here we analyze CCF between boundarieswith alternating boundary conditions in two dimensions, employing conformalfield theory (CFT). After presenting the concept of boundary changingoperators, we specifically consider two different boundary configurations for astrip of critical Ising spins: (I) alternating equi-sized domains of up anddown spins on both sides of the strip, with a possible lateral shift, and (II)alternating domains of up and down spins of different size on one side andhomogeneously fixed spins on the other side of the strip. Asymptotic resultsfor the CCF at small and large distances are derived. We introduce a novelmodified Szeg\"o formula for determinants of real antisymmetric block Toeplitzmatrices to obtain the exact CCF and the corresponding scaling functions at alldistances. We demonstrate the existence of a surface Renormalization Group flowbetween universal force amplitudes of different magnitude and sign. The Casimirforce can vanish at a stable equilibrium position that can be controlled byparameters of the boundary conditions. Lateral Casimir forces assume auniversal simple cosine form at large separations.

Conformal field theory of critical Casimir forces between surfaces with alternating boundary conditions in two dimensions