Boundary tensor renormalization group

We develop the tensor renormalization group (TRG) algorithm for statistical systems with open boundaries, which allows us to investigate not only the bulk but also the boundary property, such as the surface magnetization. We demonstrate that the tensors representing the boundary in our algorithm exhibit the fixed-point structures, just as bulk tensors in previous TRG algorithms. At criticality, the scale-invariant boundary fixed-point tensors have the information of the conformal tower, which is described by the underlying boundary conformal field theory.