Emergence of magnetic long-range order in kagome quantum antiferromagnets

The existence of a spin-liquid ground state of the s = 1/2 Heisenberg kagome antiferromagnet (KAFM) is well established. Meanwhile, also for the s = 1 Heisenberg KAFM evidence for the absence of magnetic long-range order (LRO) was found. Magnetic LRO in Heisenberg KAFMs can emerge by increasing the spin quantum number s to s > 1 and for s = 1 by an easy-plane anisotropy. In the present paper we discuss the route to magnetic order in s = 1/2 KAFMs by including an isotropic interlayer coupling (ILC) J_⊥ as well as an easy-plane anisotropy in the kagome layers by using the coupled-cluster method to high orders of approximation. We consider ferro- as well as antiferromagnetic J_⊥. To discuss the general question for the crossover from a purely two-dimensional (2D) to a quasi-2D and finally to a three-dimensional system we consider the simplest model of stacked (unshifted) kagome layers. Although the ILC of real kagome compounds is often more sophisticated, such a geometry of the ILC can be relevant for barlowite. We find that the spin-liquid ground state present for the strictly 2D s = 1/2 X X Z KAFM survives a finite ILC, where the spin-liquid region shrinks monotonously with increasing anisotropy. If the ILC becomes large enough (about 15% of intralayer coupling for the isotropic Heisenberg case and about 4% for the XY limit) magnetic LRO can be established, where the q = 0 symmetry is favorable if J_⊥ is of moderate strength. If the strength of the ILC further increases, $\sqrt{3}\times \sqrt{3}$ LRO can become favorable against q = 0 LRO. (paper)