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To clear my concepts about density matrices, I am trying to solve the following system as an example.

The density matrix is defined as $$\rho=|\psi\rangle \langle \psi |$$ where $|\psi\rangle$ is ground state of the system. Let we have a chain of fermions with hopping strength between two nearest neighbor sites $t$ i.e. $$--\bullet--\bullet--\bullet--\bullet--\bullet--$$ The Hamiltonian can be written as $$H=t\sum_n c_n^\dagger c_{n+1} + h.c.$$

I want to write reduce density matrix for single site ($\rho_1$) and two sites ($\rho_2$).

I do not understand how can I write reduced matrices for both cases.

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  • $\begingroup$ This isn’t a general solution. However in this case the Hamiltonian can be diagonalized. Perhaps do it and use the equilibrium expression $\rho=e^{-\beta H}$? $\endgroup$
    – Yair M
    Commented Oct 13, 2018 at 13:48
  • $\begingroup$ Note that it's 'reduced' density matrix. $\endgroup$
    – Emilio Pisanty
    Commented Oct 13, 2018 at 17:05

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