Tensor Renormalization-Group study of the surface critical behavior of a frustrated two-layer Ising model

Two replicas of a 2D Ising model are coupled by frustrated spin-spin interactions. It is known that this inter-layer coupling is marginal and that the bulk critical behavior belongs to the Ashkin-Teller (AT) universality class, as the $J_1-J_2$ Ising model. In this work, the surface critical behavior is studied numerically by Tensor Renormalization-Group calculations. The Bond-Weight Tensor Renormalization Group algorithm is extended to tackle systems with boundaries. It is observed that the two-fold degeneracy of the surface magnetic scaling dimension of the AT model is lifted in the frustrated two-layer Ising model (F2LIM). The splitting is explained by the breaking of the ${\mathbb{Z}}_2$-symmetry under spin reversal of a single Ising replica in the F2LIM. The two distinct surface magnetic scaling dimensions $x_1^s$ and $x_2^s$ of the F2LIM satisfies a simple duality relation $x_1^s=1\mathrm{/}4x_2^s$.