Exact eigenvalues and thermodynamic signatures of Heisenberg-Kitaev interactions in spin-$\frac{1}{2}$ quantum clusters

We investigate the finite-size effects, energy spectra, and thermodynamics of a spin-$\frac{1}{2}$ trimer, a three-sided star tetramer, and tetrahedron clusters with Heisenberg and bond-dependent Kitaev exchange. Using exact diagonalization, we compare the isotropic-exchange case with the corresponding Heisenberg-Kitaev clusters in an applied magnetic field. The Kitaev term lifts degeneracies, hybridizes states with different total-spin character, and produces nonlinear field-dependent eigenvalues associated with avoided crossings. These spectral changes generate field-driven cluster transitions, including first- and second-order ground-state transitions associated with level crossings and continuous ground-state evolution. They also redistribute finite-level Schottky weight in the magnetic heat capacity and can produce additional low-temperature heat-capacity features when the ratio $K/J$ is large. The results provide analytic cluster benchmarks for identifying how bond-dependent symm