The Kubo Formula is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation. Among the numerous applications of linear response formula, one can mention charge and spin susceptibilities of, for instance, electron systems due to external electric or magnetic fields. Responses to external mechanical forces or vibrations can also be calculated using the very same formula.
The general Kubo formula
Consider a quantum system described by the (time independent) Hamiltonian . The expectation value of a physical quantity, described by the operator , can be evaluated as:
where is the partition function. Suppose now that some time , an external perturbation is applied to the system, driving it out of equilibrium. The perturbation is described by an additional time dependent in the Hamiltonian: . We can find the time evolution of the density matrix to know the expectation value of
The time dependence of the states is governed by the Schrödinger equation Since is to be regarded as a small perturbation, it is convenient to utilize the interaction picture representation . The time dependence in this representation is given by
where by definition
To linear order in , we have . Thus one obtains the expectation value of up to linear order in the perturbation.
The brackets mean an equilibrium average with respect to he Hamiltonian . Here we have used an example, where the operators are bosonic operators, while for fermionic operators, the retarded functions are defined with anti-communtators instead of the usual (see Second quantization)