# How to write reduced density matrix of a single site

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.

• This isn’t a general solution. However in this case the Hamiltonian can be diagonalized. Perhaps do it and use the equilibrium expression $\rho=e^{-\beta H}$? – Yair M Oct 13 '18 at 13:48
• Note that it's 'reduced' density matrix. – Emilio Pisanty Oct 13 '18 at 17:05