Entanglement Entropy From Tensor Network States for Stabilizer Codes

TL;DR

This paper constructs tensor network states for 3D stabilizer code ground states, analyzes their entanglement properties, and reveals unique corrections and degeneracy signatures, especially in fracton models like the X-cube and Haah codes.

Contribution

It introduces tensor network representations for 3D stabilizer codes and computes their entanglement spectra and entropy, highlighting corrections and degeneracy features.

Findings

Entanglement entropy shows constant and linear corrections in fracton models.

Entanglement spectra are flat for the models considered.

Transfer matrices are projectors with eigenvalues 0 or 1, related to ground state degeneracy.

Abstract

In this paper, we present the construction of tensor network states (TNS) for some of the degenerate ground states of 3D stabilizer codes. We then use the TNS formalism to obtain the entanglement spectrum and entropy of these ground-states for some special cuts. In particular, we work out the examples of the 3D toric code, the X-cube model and the Haah code. The latter two models belong to the category of "fracton" models proposed recently, while the first one belongs to the conventional topological phases. We mention the cases for which the entanglement entropy and spectrum can be calculated exactly: for these, the constructed TNS is the singular value decomposition (SVD) of the ground states with respect to particular entanglement cuts. Apart from the area law, the entanglement entropies also have constant and linear corrections for the fracton models, while the entanglement entropies for the toric code models only have constant corrections. For the cuts we consider, the entanglement spectra of these three models are completely flat. We also conjecture that the negative linear correction to the area law is a signature of extensive ground state degeneracy. Moreover, the transfer matrices of these TNS can be constructed. We show that the transfer matrices are projectors whose eigenvalues are either 1 or 0. The number of nonzero eigenvalues is tightly related to the ground state degeneracy.