Separability and entanglement of resonating valence-bond states

TL;DR

This paper analyzes the entanglement properties of Rokhsar-Kivelson and resonating valence-bond states, demonstrating conditions under which these states are separable or entangled across various lattice configurations.

Contribution

It provides exact and asymptotic results on the separability and entanglement measures of RK and RVB states on arbitrary lattices, including explicit formulas for logarithmic negativity.

Findings

Reduced density matrices of certain RK states are exactly separable.

Logarithmic negativity vanishes or is exponentially suppressed with subsystem distance.

Separable states include quantum spin liquids and critical systems.

Abstract

We investigate separability and entanglement of Rokhsar-Kivelson (RK) states and resonating valence-bond (RVB) states. These states play a prominent role in condensed matter physics, as they can describe quantum spin liquids and quantum critical states of matter, depending on their underlying lattices. For dimer RK states on arbitrary tileable graphs, we prove the exact separability of the reduced density matrix of $k$ disconnected subsystems, implying the absence of bipartite and multipartite entanglement between the subsystems. For more general RK states with local constraints, we argue separability in the thermodynamic limit, and show that any local RK state has zero logarithmic negativity, even if the density matrix is not exactly separable. In the case of adjacent subsystems, we find an exact expression for the logarithmic negativity in terms of partition functions of the underlying statistical model. For RVB states, we show separability for disconnected subsystems up to exponentially small terms in the distance $d$ between the subsystems, and that the logarithmic negativity is exponentially suppressed with $d$. We argue that separability does hold in the scaling limit, even for arbitrarily small ratio $d/L$, where $L$ is the characteristic size of the subsystems. Our results hold for arbitrary lattices, and encompass a large class of RK and RVB states, which include certain gapped quantum spin liquids and gapless quantum critical systems.