Doubling of Entanglement Spectrum in Tensor Renormalization Group

TL;DR

This paper reveals that the entanglement spectrum doubles under HOTRG tensor renormalization in 2D classical models, with detailed analysis at criticality and off-critical regions, supported by numerical calculations up to tensor dimension 64.

Contribution

It demonstrates the doubling of the entanglement spectrum in HOTRG for 2D classical models and explores its behavior at critical points, providing new insights into tensor RG properties.

Findings

Entanglement spectrum doubles in off-critical HOTRG analysis.

Doubling confirmed for the square-lattice Ising model up to D=64.

Non-trivial D-scaling behavior observed in entanglement entropy at criticality.

Abstract

We investigate the entanglement spectrum in HOTRG ---tensor renormalization group (RG) method combined with the higher order singular value decomposition--- for two-dimensional (2D) classical vertex models. In the off-critical region, it is explained that the entanglement spectrum associated with the RG transformation is described by `doubling' of the spectrum of a corner transfer matrix. We then demonstrate that the doubling actually occurs for the square-lattice Ising model by HOTRG calculations up to $D = 64$, where $D$ is the cut-off dimension of tensors. At the critical point, we also find that a non-trivial $D$ scaling behavior appears in the entanglement entropy. We mention about the HOTRG for the 1D quantum system as well.