Entanglement area law in thermodynamically gapped spin systems

Bei Zeng, Xie Chen, D. L. Zhou +1 more

Anurag Anshu, Aram W. Harrow, Mehdi Soleimanifar

Jaeyoon Cho

We consider general locally interacting arbitrary-dimensional lattice spin systems that are gapped for any system size. We show under reasonable conditions that nondegenerate ground states of such systems obey the entanglement area law. In so doing, we offer an intuitive picture on how a spectral gap restricts the correlations that a ground state can accommodate and leads to such a special feature.

Jaeyoon Cho

We introduce the boundary effect on the ground state as an attribute of general local spin systems that restricts the correlations in the ground state. To this end, we introduce what we call a boundary effect function, which characterizes not only the boundary effect, but also the thermodynamic limit of the ground state. We prove various aspects of the boundary effect function to unfold its relationship to other attributes of the system such as a finite spectral gap above the ground state, two-point correlation functions, and entanglement entropies. In particular, it is proven that an exponentially decaying boundary effect function implies the exponential clustering of two-point correlation functions in arbitrary spatial dimension, the entanglement area law in one-dimension, and the logarithmically corrected area law in higher dimension. It is also proven that gapped local spin systems with nondegenerate ground states ordinarily fall into that class. In one-dimension, the area law can also result from a moderately decaying boundary effect function, in which case the system is thermodynamically gapless.

Jaeyoon Cho

We study a class of many-body states enlarged by a series of approximately local unitary transformations, which obey the exponential clustering in any spatial dimension, the area law for entanglement entropy in one dimension, and the logarithmically-corrected area law in higher dimensions. Such states appear as the ground states of local spin Hamiltonians if the system converges sufficiently fast to a thermodynamic limit as the system is enlarged. We prove that nondegenerate ground states of gapped frustration-free Hamiltonians fall into that class and hence exhibit all such correlations.

Hosho Katsura, Takaaki Hirano, Yasuhiro Hatsugai