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Suppose I have some pertubative Hamiltonian on the Hubbard Hamiltonian and I want to calculate the change in current in linear response using the Kubo formalism. Now the kind of perturbative Hamiltonian I am using increases strength of hopping between two of the chosen lattice sites. Can you show how to proceed with the commutator that we get in the result of the Kubo formalism ?

Hubbard Hamiltonian is given here Due to perturbation, I increase the hopping strength only for a pair of lattice sites while the hopping strength for the rest of the lattice sites remain the same. I want to calculate the change in the current - any type-particle current or energy current or heat current - through the Kubo formalism which gives the result for all types of currents as follows: $ J_{\alpha} = = -i\int_{-\infty}^{t}dt^{\prime}<{[j_{\alpha}(\mathbf{r},t),H^{\prime}(t^{\prime})]}> $ where < > brackets denote the thermodynamic average or the expectation value.

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Needs a little more detail. – AJK Jul 11 '13 at 3:14
Hi, I have added important details. For more, please refer Mahan Many-Particle Physics Chapter 3. It has detailed account of Kubo formalism. – cleanplay Jul 11 '13 at 9:27
@cleanplay: You've asked four or five questions now on closely related topic in rapid succession. It's not a problem, but may I suggest you take a break and try to digest what you've learned? For your personal development you need to spend some time attacking these problems yourself. – BebopButUnsteady Jul 12 '13 at 16:01
@BebopButUnsteady Thanks. I understand the importance of your suggestion. Actually I worked the answers out partly to some of the questions but still wanted to confirm few things. Like this particular problem, the answer I found for calculating the commutator is to use what I suppose, is called the Leehman representation- to take the trace over the energy eigenstates and insert a resolution operator to get the time-evolution-exponentials of $j(r,t)$ out of the equation. – cleanplay Jul 14 '13 at 13:26

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