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I don't know if that is the case, but you may have been misguided by the formalism. For convenience, the language of second quantization is often utilized in non-relativistic many body physics. So you may have stumbled upon creation and annihilation operators in that context. Nevertheless, even if written in second quantization, the Hamiltonians of many ...


Note that this is a two-site problem, so the many-body Hilbert space is finite (and actually just four). Therefore one can simply express all the operators in terms of $4\times 4$ matrices. Let us order the many-body basis states as $|s_1\rangle=a_{1\uparrow}^\dagger a_{2\downarrow}^\dagger|\Omega\rangle$, $|s_2\rangle=a_{2\uparrow}^\dagger ...

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