# Boundary Tensor Renormalization Group

## TL;DR

This paper introduces a boundary tensor renormalization group algorithm that enables the study of both bulk and boundary properties in statistical systems, revealing boundary fixed points and conformal structures at criticality.

## Contribution

It extends the tensor renormalization group method to include open boundaries, capturing boundary phenomena and conformal boundary conditions.

## Key findings

- Boundary tensors exhibit fixed point structures similar to bulk tensors.
- At criticality, boundary fixed point tensors encode conformal tower information.
- The method allows analysis of surface magnetization and boundary critical phenomena.

## Abstract

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.