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

TL;DR

This study uses tensor renormalization-group methods to analyze the surface critical behavior of a frustrated two-layer Ising model, revealing symmetry breaking effects and duality relations in surface magnetic scaling dimensions.

Contribution

It extends the Bond-Weight Tensor Renormalization Group algorithm to systems with boundaries and investigates surface critical phenomena in a frustrated two-layer Ising model.

Findings

The degeneracy of surface magnetic scaling dimensions is lifted in the frustrated model.

Breaking of ${ m Z}_2$-symmetry explains the splitting of surface scaling dimensions.

Surface magnetic scaling dimensions satisfy a duality relation $x_1^s=1/4x_2^s$.

Abstract

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 $x_1^s=1/4x_2^s$.