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I want to study a system coupled to a bath, however I do not fully understand how to implement/think of the Hamiltonian. For simplicity say the bath is given by a spin chain (PBC), e.g. Ising-like $$H_B=\sum \sigma^j_z \sigma_z^{j+1}$$ and I want to couple it to a single spin $H_S=\sigma^S_z$ at, say site 1 via coupling $H_I=\sigma_x^S\sigma_x^1$.

Now my question is how to set up the overall setting. Given that the bath $H_B$ is set up, how does $H_S$ precisely look like? Simply $\sigma_z\otimes1\otimes1\otimes\dots$? And is it correct that $H_I=\sigma_X^2\otimes 1 \otimes 1\otimes \dots$?

Many thanks in advance!

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Recall that by definition, you assume the relations (with $j=0,...,N$, $j=0$ being the isolated spin and $j=1,...,N$ being the spins of the bath): $$ [\sigma_i^j,\sigma_{i'}^{j'}] = i\delta_{jj'}\epsilon_{ii'k}\sigma_k^j $$ which you can represent by tensor products of $N+1$ factors, with $N-1$ identity matrices and a Pauli matrix at position $j$: $$ \sigma_i^j = 1^{\otimes j}\otimes\sigma_i\otimes 1^{\otimes {N-j}} $$ You were correct for $H_S$, but not for $H_I$ which is rather (you need to multiply factor by factor in the tensor product): $$ H_I = \sigma_x\otimes \sigma_x \otimes 1^{\otimes N-1} $$

Hope this helps.

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