A Toy Model of Boundary States with Spurious Topological Entanglement Entropy.

Topological entanglement entropy has been extensively used as an indicator of topologically ordered phases. However, it has been observed that there exist ground states in the topologically trivial phase that has nonzero "spurious" contribution to the topological entanglement entropy. In this work, we study conditions for two-dimensional topologically trivial states to exhibit such a spurious contribution. We introduce a tensor network model of the degrees of freedom along the boundary of a subregion. We then characterize the spurious contribution in the model using the theory of operator-algebra quantum error correction. We show that if the state at the boundary is a stabilizer state, then it has non-zero spurious contribution if, and only if, the state is in a non-trivial one-dimensional $G_1\times G_2$ symmetry-protected topological (SPT) phase. However, we provide a candidate of a boundary state that has a non-zero spurious contribution but does not belong to any of such SPT phases.