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arXiv:1003.4401 · Journal of Physics A Mathematical and Theoretical

Entanglement in resonating valence bond states: ladder versus isotropic lattices

Himadri Shekhar Dhar, Aditi Sen

Published Oct 26, 2011• 23 citations
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Abstract

We study the behavior of bipartite, as well as genuine multipartite, entanglement of a resonating valence bond state on a ladder. Although there is negligible bipartite entanglement (BE) present in the rails, as is the case for isotropic lattices in two and higher dimensions, we show that the system possesses significant amounts of BE in the steps of the ladder. However, we find that the genuine multipartite entanglement in the system is substantially low. Therefore, both the properties of bipartite as well as multipartite entanglement in the ladder are contrary to the behavior of entanglement in isotropic lattices, indicating that the geometry of the lattice can play a significant role in quantum information processing tasks. Our results also indicate that numerical and analytical investigations obtained on ladders cannot infer the same on two-dimensional lattices.

DOI
10.1088/1751-8113/44/46/465302

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Kaszlikowskiet al.Reply:

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We study the behavior of bipartite, as well as genuine multipartite, entanglement of a resonating valence bond state on a ladder. Although there is negligible bipartite entanglement (BE) present in the rails, as is the case for isotropic lattices in two and higher dimensions, we show that the system possesses significant amounts of BE in the steps of the ladder. However, we find that the genuine multipartite entanglement in the system is substantially low. Therefore, both the properties of bipartite as well as multipartite entanglement in the ladder are contrary to the behavior of entanglement in isotropic lattices, indicating that the geometry of the lattice can play a significant role in quantum information processing tasks. Our results also indicate that numerical and analytical investigations obtained on ladders cannot infer the same on two-dimensional lattices.
We investigate the trends of genuine multipartite entanglement of high-temperature superconducting states, usually described by doping the Mott insulator. We begin by analyzing the ground states of a Hubbard model with large onsite interactions obtained by exact diagonalization in regimes where the superconducting phase is observed for relatively high doping. The ground states are observed to be genuine multipartite entangled, with the maximum value achieved close to the superconducting phase boundary. Subsequently, for finite hole doping, the short-range resonating valence bond (RVB) state is considered to be the ground state of the Hubbard model. We prove that doped RVB ladder states are always genuine multipartite entangled. We then formulate an analytical recursion method for the wave function, which allows us to efficiently estimate the entanglement as well as other physical quantities in large doped RVB ladders. The maximum genuine multipartite entanglement in these RVB states is observed at doping corresponding to the superconducting phase of the Hubbard model.
In Phys. Rev. Lett. 99, 170502 (2007) (arXiv:quant-ph/0703227), we prove that resonating-valence-bond (RVB) states on arbitrary lattices possess genuine multiparty entanglement, and provide bounds on two-particle entanglement (and correlators). We also conjecture on the entanglement in large and small portions of the whole lattice. The proof, bound, and conjecture are not considered in the Comment in Phys. Rev. Lett. 101, 248901 (2008) (arXiv:0812.1932v1 [quant-ph]). Next we take two examples: "RVB gas" and "RVB liquid". RVB liquid for arbitrary lattice size is not considered in the Comment. Two-party entanglement of RVB gas for relatively large lattice size is derived by an independent method in the Comment, and it agrees with our result. Multi-party entanglement of RVB gas is not considered in the Comment. Lastly, we consider an RVB liquid on a 4x4 square lattice and there was an error in the calculation of two-party entanglement in the second decimal place, as noted in the Comment. Note that this calculation was nowhere used in our Letter to derive other results. Moreover, the conclusion of "no two-particle entanglement" is still true for the RVB liquid on a 4x4 square lattice.
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 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 d between the subsystems, and that the logarithmic negativity is exponentially suppressed with d d . We argue that separability does hold in the scaling limit, even for arbitrarily small ratio d/L d / L , where L 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.
We investigated the scaling behaviors of entanglements on the two-leg ladder lattice structure implemented with Affleck–Kennedy–Lieb–Tasaki (AKLT) model on six types of topological surfaces, namely, torus, sphere, real projective plane, Klein bottle, cylinder and Möbius strip. All six topological surfaces on which our lattices were embedded are classified according to whether their boundaries are closed or open. Lattice structures on these topological surfaces are formed either by a trivalent or four-valent site to fulfill a complete spin-32 or spin-2 system, respectively. Their ground states, which are the valence-bond-solid states, are derived by the tensor-product method. The von Neumann entropies for entanglement of the bipartite region on all six topological surfaces are calculated and all results show that the entanglement entropies per area are bounded from above or below when approaching the thermodynamic limit. For the spin-2 and the spin-32 systems on all surfaces, except on the sphere, numerical results between even and odd cases of the number of lattice sites should be considered separately for scrutinizing their scaling behaviors. However, their entanglement entropies still satisfy the area law, but with a correction term.
We present a direct comparison of the recently proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin-1/2 Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy; hence, it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entanglement entropy has a multiplicative logarithmic correction.