  We introduce a generic scheme to perform non-perturbative linked cluster
expansions in long-range ordered quantum phases. Clusters are considered to be
surrounded by an ordered reference state leading to effective edge-fields in
the exact diagonalization on clusters which break the associated symmetry of
the ordered phase. Two approaches, based either on a self-consistent solution
of the order parameter or on minimal sensitivity with respect to the
ground-state energy per site, are formulated to find the optimal edge-field in
each NLCE order. Furthermore, we investigate the scaling behavior of the NLCE
data sequences towards the infinite-order limit. We apply our scheme to gapped
and gapless ordered phases of XXZ Heisenberg models on various lattices and for
spins 1/2 and 1 using several types of cluster expansions ranging from a
full-graph decomposition, rectangular clusters, up to more symmetric square
clusters. It is found that the inclusion of edge-fields allows to regularize
non-perturbative linked-cluster expansions in ordered phases yielding
convergent data sequences. This includes the long-range spin-ordered ground
state of the spin-1/2 and spin-1 Heisenberg model on the square and triangular
lattice as well as the trimerized valence bond crystal of the spin-1 Heisenberg
model on the kagome lattice.
