Spin Squeezing with Short-Range Spin-Exchange Interactions

We investigate many-body spin squeezing dynamics in an XXZ model with interactions that fall off with distance r as 1/r(alpha) in D = 2 and 3 spatial dimensions. In stark contrast to the Ising model, we find a broad parameter regime where spin squeezing comparable to the infinite-range alpha = 0 limit is achievable even when interactions are short ranged, alpha > D. A region of collective behavior in which optimal squeezing grows with system size extends all the way to the alpha -> infinity limit of nearest-neighbor interactions. Our predictions, made using the discrete truncated Wigner approximation, are testable in a variety of experimental cold atomic, molecular, and optical platforms.