honeycomb lattice is straightforward.[@Zhong2012b; @ZhangLan2015]) Next, we use fermionic representation to rewrite the degree of freedom of f-electron local spins as $S_{i}^{f}=\frac{1}{2}\sum_{\sigma\sigma'}f_{i\sigma}^{\dag}\tau_{\sigma\sigma'}f_{i\sigma'}$ with $\tau$ being the standard Pauli matrices. Moreover, to exclude nonphysical charge fluctuation of auxiliary fermion $f_{\sigma}$, it is crucial to enforce the local constraint $\sum_{\sigma}f_{i\sigma}^{\dag}f_{i\sigma}=1$ at each site.

Physically, this model describes two competing tendencies: The first is the Kondo screening, which leads to the formation of collective spin-singlet state among local moments and conduction electrons. The second is the short-ranged antiferromagnetic fluctuation introduced by Heisenberg interaction between local moments. It is believed that the observed complicated phenomena in diverse heavy electron systems can be captured by these two active factors. Usually, to get qualitatively correct information in the paramagnetic heavy fermion liquid state, the fermionic large-N or slave-boson mean-field theory is widely utilized.[@Read1983; @Coleman1987] Here, we use the former one to get an effective mean-field Hamiltonian.

Mean-field model in paramagnetic state
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After performing the standard large-N mean-field approximation,[@Lee2006; @Senthil2004] we get the following Hamiltonian $$\begin{aligned}
H&&=\sum_{k\sigma}[\varepsilon_{k}c_{k\sigma}^{\dag}c_{k\sigma}+\chi_{k}f_{k\sigma}^{\dag}f_{k\sigma}+\frac{J_{K}V}{2}(c_{k\sigma}^{\dag}f_{k\sigma}+f_{k\sigma}^{\dag}c_{k\sigma})]\nonumber\\
&&+E_{0}.\label{eq1}\end{aligned}$$ Here, Kondo screening effect is encoded by an effective hybridization between conduction electron and local spins via $V=-\sum_{\sigma}\langle c_{i\sigma}^{\dag}f_{i\sigma}\rangle$. Meanwhile, local spins acquire dissipation $\chi_{k}=J_{H}\chi \eta_k + \lambda$ 