Modular Hamiltonian ($K=-\log \rho$)

The modular Hamiltonian of a density operator is $K=-\log \rho$. For a Gibbs state, $K=\beta H+\log Z$.

Linearizing the von Neumann entropy around a reference state gives $\delta S=\mathrm{Tr}(\delta \rho K)$. Around a thermal state this becomes $\delta S=\beta \delta ⟨H⟩$.