Entanglement Entropy From Tensor Network States for Stabilizer Codes

TL;DR

This paper uses the TNS formalism to obtain the entanglement spectrum and entropy of these ground-states for some special cuts of 3D stabilizer codes, and conjecture that the negative linear correction to the area law is a signature of extensive 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.

Figures (19)

• Figure 1: Examples of TNS lattice wave functions in 1D and 2D. Each node is a tensor whose indices are the lines connecting to it. The physical indices - of the quantum Hilbert space - are the lines with arrows, while the lines without any arrows are the virtual indices. Connected lines means the corresponding indices are contracted. Panel (a) is an MPS for 1D systems. Panel (b) is a PEPS on a 2D square lattice.
• Figure 2: An illustration of the TNS gauge in MPS. (a) A part of an MPS. $A_1$ and $A_2$ are two local tensors contracted together. (b) We insert the identity operator $\mathbb{I}=UU^{-1}$ at the virtual level - it acts on the virtual bonds. The tensor contraction of $A_1$ and $A_2$ does not change. (c) We further multiply $U$ with $A_1$ and $U^{-1}$ with $A_2$, resulting in $\tilde{A}_1$ and $\tilde{A}_2$ respectively in Panel (d). The tensor contraction of $A_1$ and $A_2$ is the same as the tensor contraction of $\tilde{A}_1$ and $\tilde{A}_2$. The TNS does not change as well. Similar TNS gauges also appear in other TNS such as PEPS.
• Figure 3: (a) A plane of TNS on a cubic lattice. (b) TNS on a cube. The lines with arrows are the physical indices. The connected lines are the contracted virtual indices, while the open lines are not contracted. On each vertex, there lives a $T$ tensor, and on each bond, we have a projector $g$ tensor.
• Figure 4: Transfer matrix (red dashed square) of a 1D MPS. The connected lines are the contracted virtual indices. The connected arrow lines are the contracted physical indices. The MPS norm (or any other quantities) can be built using the transfer matrix. Higher dimensional transfer matrices are similarly defined for TNS on a cylinder or a torus, by contracting in all directions except one. This leads to a 1D MPS with a bond dimension exponentially larger than the TNS one.
• Figure 5: The Hamiltonian terms of the 3D toric code model. Panel (a) is $A_{v}$ which is a product of 6 $Z$ operators, and Panel (b) is $B_{p}$ which is a product of 4 $X$ operators. The circled $X$ and $Z$ represent the Pauli matrices acting on the spin-$1/2$'s. The toric code Hamiltonian includes $A_v$ terms on all vertices $v$ and $B_p$ terms on all plaquettes $p$.