Measuring Renyi Entanglement Entropy with Quantum Monte Carlo

TL;DR

A quantum Monte Carlo procedure is developed to measure the Renyi entanglement entropy of a many-body ground state as the expectation value of a unitary Swap operator acting on two copies of the system.

Abstract

We develop a quantum Monte Carlo procedure, in the valence bond basis, to measure the Renyi entanglement entropy of a many-body ground state as the expectation value of a unitary {\it Swap} operator acting on two copies of the system. An improved estimator involving the ratio of {\it Swap} operators for different subregions enables simulations to converge the entropy in a time polynomial in the system size. We demonstrate convergence of the Renyi entropy to exact results for a Heisenberg chain. Finally, we calculate the scaling of the Renyi entropy in the two-dimensional Heisenberg model and confirm that the Néel groundstate obeys the expected area law for systems up to linear size L=28.

Figures (4)

• Figure 1: (color online) A six-site chain, with two non-interacting copies (top and bottom) before (a) and after (b) the $Swap_A$ operation. The region $A$ consists of three light-colored sites on the left; the complement region $B$ of three dark sites on the right. The curved lines denote singlets in the state $|V_{r}\rangle$, which is a product of two different valence bond states, one per copy. The ground state of the entire system is a linear combination of similar $|V_{r}\rangle$.
• Figure 2: (color online) The Renyi entropy $S_2$ as a function of site index $i \in A$, for a 100-site Heisenberg chain with open boundaries, calculated with DMRG and QMC. Data labeled "Swap" was calculated with Eq. \ref{['Swap']} with one QMC simulation, while data labeled $j_{\rm max}=5$ was calculated with Eq. \ref{['Ratio']} using 20 separate QMC simulations with a range of $j \in [1,5]$. The inset shows the convergence of $S_2$ to the exact value (dashed line) for $i=6$ with up to $m=4000$.
• Figure 3: (color online) The Renyi entropy divided by the linear dimension $\ell$ of the entangled region $A$, for 2D lattices with $L=8$ and $L=16$. The data labeled "Swap" is from a single simulation calculating Eq. \ref{['Swap']} directly. The other data sets are derived from the improved ratio estimator, Eq. \ref{['Ratio']}, with different ranges of $r \in [1,r_{\rm max}]$ (see text). The inset is a periodic $L=8$ lattice with region $A$ consisting of the 16 central (dark) sites labeled by $\ell = 4$.
• Figure 4: (color online) Scaling of the Renyi entropy divided by the linear dimension $\ell$ of the entangled region $A$ for different systems sizes in 2D. All data is calculated with the improved ratio estimator, Eq. \ref{['Ratio']}, with $r_{\rm max}=1$.