Exact description of the boundary theory of the Kitaev toric code with open boundary conditions

Abstract

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