Spurious long-range entanglement and replica correlation length

Topological entanglement entropy has been regarded as a smoking-gun signature of topological order in two dimensions, capturing the total quantum dimension of the topological particle content. An extrapolation method on cylinders has been used frequently to measure the topological entanglement entropy. Here, we show that a class of short-range entangled 2D states, when put on an infinite cylinder of circumference $L$, exhibits the entanglement Rényi entropy of any integer index $\alpha \ge 2$ that obeys $S_{\alpha }=aL-\gamma$, where $a,\gamma >0$. Under the extrapolation method, the subleading term $\gamma$ would be identified as the topological entanglement entropy, which is spurious. A nonzero $\gamma$ is always present if the 2D state reduces to a certain symmetry-protected topological 1D state, upon disentangling spins that are far from the entanglement cut. The internal symmetry that stabilizes $\gamma >0$ is not necessarily a symmetry of the 2D state, but should be present after the disentangling reduction. If the symmetry is absent, $\gamma$ decays exponentially in $L$ with a characteristic length, termed as a replica correlation length, which can be arbitrarily large compared to the two-point correlation length of the 2D state. We propose a simple numerical procedure to measure the replica correlation length through replica correlation functions. We also calculate the replica correlation functions for representative wave functions of Abelian discrete gauge theories and the double semion theory in 2D, to show that they decay abruptly to zero. This supports a conjecture that the replica correlation length being small implies that the subleading term from the extrapolation method determines the total quantum dimension.

Figure 1

Two methods of extracting TEE. (Left) Kitaev-Preskill prescription divides the system into four parts and extract TEE by using (2). (Right) DMRG calculations put the system on an infinite cylinder and divide the system into two parts, then calculate the entanglement entropy $S\left(L\right)$ between the two parts for different circumferences $L$ of the cylinder. Fitting the results into (1), TEE is identified as $-S(L=0)$.

Figure 2

A seven-spin interaction in the summation of Eq. (8), which is a product of the ${\sigma }^x$ operator on the center site and the ${\sigma }^z$ operators on the sites that surround it.

Figure 3

Triangular lattice with an entanglement cut parallel to one of the sides of a triangle. The upper region is $A$ and the lower is $B$.

Figure 4

The reduced 1D chain of the 2D cluster state. The red circles represent qubits in region A and the green circles represent qubits in region B.

Figure 5

Bravyi's example. There is a qubit on each vertex of the zigzag chain, and the dashed line represents an entanglement cut that divides the chain into two halves. If the chain is in the cluster state of Eq. (10), the entanglement entropy is $\frac{1}{2}b-1$ in the units of $ln2,\mathrm{where}b$ is the number of bond cuts.

Figure 6

(a) $\rho$ obtained by tracing out one physical qudits for every site. (b) ${\rho }^2$. Connected bonds are contracted. Wiggling lines represent complex conjugation. The horizontal virtual bonds are contracted due to the periodic boundary condition, which is not drawn. $Tr\left(\rho \right)$ and $Tr\left({\rho }^2\right)$ are computed by contracting the upper vertical wiggling bonds with the lower straight ones. For this purpose, it is enough and more efficient to consider the transfer matrix designated by the dotted rectangle [see Eq. (21)].

Figure 7

Transfer matrix and its symmetry. The left-most diagram of (b) represents the transfer matrix ${\mathbb{T}}_3$ for ${\rho }^3$. The ensuing equalities are direct consequences of the symmetry lifted to the virtual level. This implies that the transfer matrix ${\mathbb{T}}_{\alpha }$ has degeneracy $q^{\alpha -1}$ for some integer $q>1$.

Figure 8

Replica correlation function calculation. One prepares two copies of the state, and apply the swap operator on the shaded region; insert observables in the circles, and compute the overlap with the original, unswapped state. The overlap is generally exponentially small in the boundary length of the shaded region, but after normalization this reveals the replica correlation length.

Figure 9

Measuring replica correlation functions. Observables are inserted in the circles. Even if the global state ${\rho }_{AB}$ has a short correlation length, the positive semidefinite operator ${\rho }_A^2$, treated as a normalized state ${\rho }_A^2/Tr\left({\rho }_A^2\right)$, may have much longer correlation length. The latter length scale, which we call as the replica correlation length, can be simply measured in numerical calculations, and is the relevant length scale for the subleading term in the entanglement entropy.

Figure 10

Double-semion model defined in Eq. (58). (a) shows the lattice configuration and the location of the degrees of freedom. Each edge of the hexagon accommodates two spins, denoted by the blue dots on the edge. The dashed line is the entanglement cut. (b)–(d) pictorially show the three terms in the Hamiltonian for a plaquette, vertex, and edge. The symbol near a spin denotes the operator acting on this spin, where $X$ means ${\sigma }_x,Z$ means ${\sigma }_z,\sqrt{Z}=\mathrm{diag}(1,i)$, and $\mathrm{Id}$ is the identity matrix. The total Hamiltonian is the summation over all plaquettes, vertices and edges, with appropriate sign factors defined in Eq. (58).

Figure 11

Deforming the state while respecting the lattice reflection or inversion symmetry. From the original 1D chain, which consists of red $\left(a_i\right)$ and green $\left(b_i\right)$ qudits, one can insert an auxiliary yellow $\left(c_i\right)$ qudit. Then one can apply the swap operator that exchanges the red qudits and yellow qudits circled by the dashed ellipses. This swap operation can be implemented continuously without breaking the lattice reflection symmetry. The numbers in each qudits label the positions of the corresponding qudits before the swap operation. It is understood that we have a 1D chain of qudits, although only a few sites are shown here.

Figure 12

Deforming state while respecting symmetry. In addition to the red $\left(a_i\right)$ and green $\left(b_i\right)$ qudits, a pair of time reversal-invariant spin singlets are inserted in each site (the yellow qudits are $a_i^{\prime }$ and blue qudits are $b_i^{\prime })$. The swap unitary is applied to the qudits circled by the dashed ellipses, so that the entanglement across the cut solely arises from the inserted singlet. The swap can be implemented continuously during which the time-reversal symmetry is unbroken. It is understood we have a 1D chain of qudits, although only one site is shown here.