Two-peak structure in temperature dependence of the specific heat in spin-S Heisenberg-Kitaev models on a honeycomb lattice

An interesting phase called as the Kitaev's spin liquid (KSL) phase exists in the S$=$1/2 Heisenberg-Kitaev (HK) model on a honeycomb lattice. The low-energy excitations in the KSL state is characterized by Majorana fermions resulting from fractionalization of quantum spins. In the Kitaev model [PRL 113, 197205 (2014)], the fractionalization is observed as a two peak structure in the temperature dependence of the specific heat $C(T)$. This two peak structure of $C(T)$ \quad survives in the magnetic ordered phase, if the system is located in the vicinity of the KSL phase, and this offers criteria for measuring the closeness to the KSL phase [PRB 93, 174425 (2016)]. Similarly, it has been studied that $C(T)$ of the HK model at the large S limit (classical HK model) shows a two peak structure [PRL 109, 187201 (2012)]. In this study, we have calculated $C(T)$ for spin-S HK model and found that the origin of two peaks is different between the quantum and classical cases. The difference is evident in the higher temperature peaks in $C(T)$: For the quantum spin S$=$1/2, the higher temperature peak shrinks and the two-peak structure disappears for 3/2 \textless S. At the classical limit, an additional tiny peak seems to appear, independently of the two-peak structures in the quantum case.