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In this work we consider the Kitaev Toric Code with specific open boundaryconditions. Such a physical system has a highly degenerate ground statedetermined by the degrees of freedom localised at the boundaries. We can writedown an explicit expression for the ground state of this model. Based on this,the entanglement properties of the model are studied for two types ofbipartition: one, where the subsystem A is completely contained in B; and thesecond, where the boundary of the system is shared between A and B. In theformer configuration, the entanglement entropy is the same as for the periodicboundary condition case, which means that the bulk is completely decoupled fromthe boundary on distances larger than the correlation length. In the latter,deviations from the torus configuration appear due to the edge states and leadto an increase of the entropy. We then determine an effective theory for theboundary of the system. In the case where we apply a small magnetic field as aperturbation the degrees of freedom on the boundary acquire a dispersionrelation. The system can there be described by a Hamiltonian of the Ising typewith a generic spin-exchange term.

Exact description of the boundary theory of the Kitaev Toric Code with open boundary conditions