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

TL;DR

The paper develops a boundary-capable Bond-Weight Tensor Renormalization Group approach to study surface critical behavior in a frustrated two-layer Ising model (F2LIM) that shares bulk Ashkin–Teller universality with the $J_1$-$J_2$ Ising model ($c=1$). It reveals a splitting of the surface magnetic scaling dimensions, producing two nondegenerate exponents $x_1^s$ and $x_2^s$ related by $x_2^s\approx 1/(4x_1^s)$ due to boundary symmetry breaking, and embeds these findings in AT-like AT-surface spectra along the bulk-critical line. The critical line in the $(J_1,J_2)$ plane is determined from domain-wall free-energy differences with high precision, and the near-Ising and near-4-state-Potts limits are used to benchmark the AT expectations, while the tricritical point at $J_1\approx 0.425$ marks the onset of first-order behavior. Overall, the work provides a robust tensor-network framework for boundary conformal field theory analyses of coupled-layer spin systems and clarifies how boundary conditions influence surface criticality in marginally coupled models.

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

Figures (10)

• Figure 1: The different steps of the Tensor-Renormalization Group algorithm for a finite strip. The black dots correspond to the tensors. To each line is attached an index which is one of the indices of the tensors at the two edges of the line. A black dot at the crossing of four lines is therefore a rank-4 tensor. A summation over the index of each line is implicit. The pink dots correspond to the initial tensors. After the two steps of the TRG algorithm, one can see that the number of tensors has decreased by a factor of 4.
• Figure 2: Estimates of the bulk (left) and surface (right) critical dimensions of the Ashkin-Teller model versus the inverse of the number of iterations of the BTRG algorithm at the point $K=0.04894351$ (close to the Ising point) of the critical line. The dashed lines are the exact values.
• Figure 3: Estimates of the bulk (left) and surface (right) critical dimensions of the Ashkin-Teller model versus the inverse of the number of iterations of the BTRG algorithm at the point $K=0.2724481$ (close to the 4-state Potts point) of the critical line. The dashed lines are the exact values.
• Figure 4: Estimates of the bulk (left) and surface (right) critical dimensions of the Ashkin-Teller model versus the coupling $J_1$. The dashed lines are the exact values.
• Figure 5: Estimates of surface critical dimensions of the Ashkin-Teller model with Fixed Boundary Conditions (Identical on the left subplot and Mixed on the right one) versus the coupling $J_1$. The dashed lines are the exact values.