Topological phases of matter are primarily studied in quantum many-body
systems with short-range interactions. Whether various topological phases can
survive in the presence of long-range interactions, however, is largely
unknown. Here we show that a paradigmatic example of a symmetry-protected
topological phase, the Haldane phase of an antiferromagnetic spin-1 chain,
surprisingly remains intact in the presence of arbitrarily slowly decaying
power-law interactions. The influence of long-range interactions on the
topological order is largely quantitative, and we expect similar results for
more general systems. Our conclusions are based on large-scale
matrix-product-state simulations and two complementary effective-field-theory
calculations. The striking agreement between the numerical and analytical
results rules out finite-size effects. The topological phase considered here
should be experimentally observable in a recently developed trapped-ion quantum
simulator.
