# How shift the system Hamiltonian change the interaction term?

I'm reading this paper about a model of a qubit coupled to an Ising spin bath. The interaction between the system qubit and the bath is described by the Ising Hamiltonian: $$H_{I}^{\prime}=\alpha \sigma^{z} \otimes \sum_{n=1}^{N} g_{n} \sigma_{n}^{z}$$ The system and bath Hamiltonians are $$H_{S}=\frac{1}{2} \omega_{0} \sigma^{2}$$ and $$H_{B}=\sum_{n=1}^{N} \frac{1}{2} \Omega_{n} \sigma_{n}^{z} .$$

My question is about the shift of the system Hamiltonian. The paper state in eq.(14) that, if we change $$H_{S} \mapsto H_{S}+\theta I$$ where $$\theta\equiv Tr(\sum_n g_n \sigma_n^z \rho_B)$$, then the interaction Hamiltonian is modified to

$$H_{I}^{\prime} \mapsto H_{I}=\alpha \sigma^{z} \otimes B$$ where $$B \equiv \sum_{n} g_{n} \sigma_{n}^{z}-\theta I_{B} .$$

I can't see how this change happened since I cannot add the new term up to get the old term.

• Importantly $\theta$ is not a constant. The definition is also given in eq 14. Commented Aug 14, 2022 at 12:25
• @MengCheng Yeah, I just think that why $H_I^\prime$ can be changed into form of $H_I$ does not have something to do with the actual definition of $\theta$. I added the definition of $\theta$ in the main post. Thanks! Commented Aug 14, 2022 at 12:45