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arXiv:1103.3994 · Physical Review B

Entanglement and SU( n ) symmetry in one-dimensional valence-bond solid states

Román Orús, Hong-Hao Tu

Published May 23, 2011• 21 citations
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Abstract

Here we evaluate the many-body entanglement properties of a generalized SU($n$) valence-bond solid state on a chain. Our results follow from a derivation of the transfer matrix of the system which, in combination with symmetry properties, allows for an elegant and straightforward evaluation of different entanglement measures. In particular, the geometric entanglement per block, correlation length, von Neumann and R\'enyi entropies of a block, localizable entanglement, and entanglement length are obtained in a very simple way. All our results are in agreement with previous derivations for the SU($2$) case.

DOI
10.1103/physrevb.83.201101

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We investigate the entanglement properties of the valence-bond-solid states with generic integer spin SSS. Using the Schwinger boson representation of the valence-bond-solid states, the entanglement entropy, the von Neumann entropy of a subsystem, is obtained exactly and its relationship with the usual correlation function is clarified. The saturation value of the entanglement entropy, 2log2(S+1)2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}(S+1)2log2​(S+1), is derived explicitly and is interpreted in terms of the edge-state picture. The validity of our analytical results and the edge-state picture is numerically confirmed. We also propose an application of the edge state as a qubit for quantum computation.
In this paper we provide the analytical derivation of the global geometric entanglement per block for the valence-bond solid ground state of the spin-1 Affleck-Kennedy-Lieb-Tasaki chain. In particular, we show that this quantity saturates exponentially fast to a constant when the sizes of the blocks are sufficiently large. Our result provides an example of an analytical calculation of the geometric entanglement for a gapped quantum many-body system in one dimension and far away from a quantum critical point.
We investigate entanglement properties of N initially disentangled spins, embedded in a ring or d-dimensional lattice. The spins interact via some long-range Ising-type interaction. A description in terms of generalized valence bond solids allows us, for arbitrarily large N, to efficiently calculate reduced density operators of up to ten particles. From these density operators we can compute many quantities, for example the bipartite entanglement S_L between a block of L neighboring spins and the remaining system, lower and upper bounds on the localizable entanglement, and higher order correlation functions. We vary the distance dependence of the interaction and investigate how this variation affects the scaling of the entropy S_L as L grows. We provide a sufficient condition when S_L saturates. In other cases, we find that the entanglement length diverges. For special configurations, we determine S_L analytically for all L in the limit N to infinity.