R\'enyi entanglement entropies in quantum dimer models : from criticality to topological order

TL;DR

This paper investigates the Re9nyi entanglement entropies and spectrum of quantum dimer models on a triangular lattice, revealing topological order and criticality transitions through Pfaffian techniques and various geometries.

Contribution

It provides high-precision numerical analysis of entanglement properties in quantum dimer models, demonstrating the behavior of topological entanglement entropy across critical and topological phases.

Findings

In the topological phase, the entanglement entropy is a7- ln 2, independent of parameters.

At the critical point, the entropy exhibits a distinct, parameter-dependent form.

The study confirms the robustness of topological entanglement entropy in the a7 a7 phase.

Abstract

Thanks to Pfaffian techniques, we study the R\'enyi entanglement entropies and the entanglement spectrum of large subsystems for two-dimensional Rokhsar-Kivelson wave functions constructed from a dimer model on the triangular lattice. By including a fugacity $t$ on some suitable bonds, one interpolates between the triangular lattice (t=1) and the square lattice (t=0). The wave function is known to be a massive $\mathbb Z_2$ topological liquid for $t>0$ whereas it is a gapless critical state at t=0. We mainly consider two geometries for the subsystem: that of a semi-infinite cylinder, and the disk-like setup proposed by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)]. In the cylinder case, the entropies contain an extensive term -- proportional to the length of the boundary -- and a universal sub-leading constant $s_n(t)$. Fitting these cylinder data (up to a perimeter of L=32 sites) provides $s_n$ with a very high numerical accuracy ($10^{-9}$ at t=1 and $10^{-6}$ at $t=0.5$). In the topological $\mathbb{Z}_2$ liquid phase we find $s_n(t>0)=-\ln 2$, independent of the fugacity $t$ and the R\'enyi parameter $n$. At t=0 we recover a previously known result, $s_n(t=0)=-(1/2)\ln(n)/(n-1)$ for $n<1$ and $s_n(t=0)=-\ln(2)/(n-1)$ for $n>1$. In the disk-like geometry -- designed to get rid of the boundary contributions -- we find an entropy $s^{\rm KP}_n(t>0)=-\ln 2$ in the whole massive phase whatever $n>0$, in agreement with the result of Flammia {\it et al.} [Phys. Rev. Lett. 103, 261601 (2009)]. Some results for the gapless limit $R^{\rm KP}_n(t\to 0)$ are discussed.