It is not an obvious result. Technically, the claim is that the ground state and first few excited states of a local gapped Hamiltonian verify the area law. It is proved in some special cases, and is now well understood in 1 space dimension thanks to a proof by Matt Hastings "An area law for one-dimensional quantum systems" iopscience.iop.org/article/10.1088/1742-5468/2007/08/P08024/meta Intuively the proof goes something like: with a gapped Hamiltonian, correlations decay exponentially in time. If the theory were relativistic, this would be equivalent to exponential decay in space. For non-relativistic models on the lattice, locality of interactions gives weaker than the strict light cone of relativistic theories, but still allowing to prove decay of correlations in space. If correlation functions in space decay fast, it means there isn't much entanglement. Of course one needs to make it quantitative which Hastings does. For more general discussion about the area law, there is a big review by Eisert, Cramer and Plenio in Rev Mod Phys : journals.aps.org/rmp/abstract/10.1103/RevModPhys.82.277 . Although it is from 2008, so not super recent anymore, I don't think there has been huge results on area law since.
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Can you explain or maybe give a reference as to why the low energy eigen states verrify the area law ? (10:30)
It is not an obvious result. Technically, the claim is that the ground state and first few excited states of a local gapped Hamiltonian verify the area law. It is proved in some special cases, and is now well understood in 1 space dimension thanks to a proof by Matt Hastings "An area law for one-dimensional quantum systems" iopscience.iop.org/article/10.1088/1742-5468/2007/08/P08024/meta
Intuively the proof goes something like: with a gapped Hamiltonian, correlations decay exponentially in time. If the theory were relativistic, this would be equivalent to exponential decay in space. For non-relativistic models on the lattice, locality of interactions gives weaker than the strict light cone of relativistic theories, but still allowing to prove decay of correlations in space. If correlation functions in space decay fast, it means there isn't much entanglement. Of course one needs to make it quantitative which Hastings does.
For more general discussion about the area law, there is a big review by Eisert, Cramer and Plenio in Rev Mod Phys : journals.aps.org/rmp/abstract/10.1103/RevModPhys.82.277 . Although it is from 2008, so not super recent anymore, I don't think there has been huge results on area law since.