Direct computation of entanglement entropy from Rindler Hamiltonian

Is there any references in which entanglement entropy (EE) for half space is directly computed from the modular Hamiltonian?

For half space $A$ specified by ${x^1>0}$, the modular Hamiltonian $K_A=-log rho_A$ is given by
$$K_A=2pi int_{x^1>0} d^d x x_1 T^00 (x).$$
While from the definition of EE, EE equals to
$$S_A=mathrm{tr} (rho K_A) +log Z.$$

I have tried to compute $S_A$ following above equations using the Hamiltonian of free scalar but the obtained answer was not consistent with known results (e.g. using replica method, $mathbb{Z}_N$ orbifold).