Ground state entanglement in one-dimensional translationally invariant quantum systems

We examine whether it is possible for one-dimensional translationally invariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians {H-n} for the infinite chain. The spectral gap of H-n is Omega(1/poly(n)). Moreover, for any state in the ground space of H-n and any m, there are regions of size m with entanglement entropy Omega(min{m, n}). A similar construction yields translationally invariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings [An area law for one dimensional quantum systems, J. Stat. Mech.: Theory Exp. 2007 (08024)] gives a constant upper bound on the entanglement entropy for one-dimensional ground states that is independent of the size of the region but exponentially dependent on 1/Delta, where Delta is the spectral gap. This paper provides a lower bound, showing a family of Hamiltonians for which the entanglement entropy scales polynomially with 1/Delta. Previously, the best known such bound was logarithmic in 1/Delta. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3254321]