Spurious long-range entanglement and replica correlation length

Abstract

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 R\'enyi entropy of any integer index $\ensuremath{\alpha}\ensuremath{\ge}2$ that obeys ${S}_{\ensuremath{\alpha}}=aL\ensuremath{-}\ensuremath{\gamma}$, where $a,\ensuremath{\gamma}>0$. Under the extrapolation method, the subleading term $\ensuremath{\gamma}$ would be identified as the topological entanglement entropy, which is spurious. A nonzero $\ensuremath{\gamma}$ 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 $\ensuremath{\gamma}>0$ is not necessarily a symmetry of the 2D state, but should be present after the disentangling reduction. If the symmetry is absent, $\ensuremath{\gamma}$ 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.