  Quantum spin liquids (QSLs) have become a key area of research in magnetism
due to their remarkable properties, such as long-range entanglement, fractional
excitations, pinch-point singularities, and topologically protected phenomena.
In recent years, the search for QSLs has expanded into the three-dimensional
world, where promising features have been found in materials that form
pyrochlore and hyper-kagome lattices, despite the suppression of quantum
fluctuations due to high dimensionality. One such material is the $S = 1$
K$_2$Ni$_2$(SO$_4$)$_3$ compound, which belongs to the langbeinite family
consisting of two interconnected trillium lattices. Although magnetically
ordered, K$_2$Ni$_2$(SO$_4$)$_3$ has been found to exhibit a highly dynamical
and correlated state which can be driven into a pure quantum spin liquid under
magnetic fields of only $B \simeq 4$~T. In this article, we combine inelastic
neutron scattering measurements with pseudo-fermion functional renormalization
group (PFFRG) and classical Monte Carlo (cMC) calculations to study the
magnetic properties of K$_2$Ni$_2$(SO$_4$)$_3$, revealing a high level of
agreement between the experiment and theory. We further reveal the origin of
the dynamical state in K$_2$Ni$_2$(SO$_4$)$_3$ by studying a larger set of
exchange parameters, uncovering an `island of liquidity' around a focal point
given by a magnetic network composed of tetrahedra on a trillium lattice.

Strongly Correlated Electrons;Materials Science