I've only recently, and still only haphazardly and rather poorly, begun to understand Ising models with local interactions. I'm interested in particular in the simple one-dimensional Ising model with nearest and next-nearest neighbor interactions, which have been referred to in the literature as ANNNI Hamiltonians, or anisotropic, next-nearest neighbor interactions (with none, either, or both a transverse and longitudinal external magnetic field).
Depending on the strengths of the nearest-neighbor interactions relative to the next-nearest neighbor interactions (and also to the external magnetic fields) there could be very lovely and dynamic frustration going on - the nearest neighbor might favor parallel spins $\mid\uparrow\uparrow\rangle$ or $\mid\downarrow\downarrow\rangle$ but the next-nearest neighbor interactions might force antiparallel spins $\mid\downarrow\uparrow\rangle$ or $\mid\uparrow\downarrow\rangle$, or vice-versa.
- Can we say anything about if and when frustration leads to an entangled ground state?
The answer might be related to so-called area laws, which consider the amount of entanglement relative to the dimension of the chain. For a one-dimensional chain an area law suggests that there may be little entanglement, but is there any additional entanglement borne out of frustration?
I gathered from her talk that the Kagome lattice puts some stress on naive implementations of classical many-body algorithms like DMRG, but nonetheless from clever DMRG simulations there's some evidence that the lattice does indeed have a uniquely quantum (non-degenerate and highly entangled) ground state.