Kubo formula

The Kubo formula, named for Ryogo Kubo who first presented the formula in 1957, is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation.

Among numerous applications of the Kubo formula, one can calculate the charge and spin susceptibilities of systems of electrons in response to applied electric and magnetic fields. Responses to external mechanical forces and vibrations can be calculated as well.

General Kubo formula
Consider a quantum system described by the (time independent) Hamiltonian $$H_0$$. The expectation value of a physical quantity at equilibrium temperature $$T$$, described by the operator $$\hat{A}$$, can be evaluated as:
 * $$\left\langle\hat{A}\right\rangle = {1 \over Z_0}\operatorname{Tr}\,\left[\hat{\rho_0}\hat{A}\right] = {1 \over Z_0}\sum_n \left\langle n \left| \hat{A} \right| n \right\rangle e^{-\beta E_n}$$,

where $$\beta=1/k_{\rm B}T$$ is the thermodynamic beta, $$\hat{\rho}_0  $$ is density operator, given by
 * $$\hat{\rho_0} = e^{-\beta \hat{H}_0} = \sum_n | n \rangle\langle n | e^{-\beta E_n}$$

and $$Z_0 = \operatorname{Tr}\,\left[\hat\rho_0\right]$$ is the partition function.

Suppose now that just above some time $$t = t_0$$ an external perturbation is applied to the system. The perturbation is described by an additional time dependence in the Hamiltonian:
 * $$\hat{H}(t) = \hat{H}_0 + \hat{V}(t) \theta (t - t_0),$$

where $$\theta (t) $$ is the Heaviside function (1 for positive times, 0 otherwise) and $$\hat V(t)$$ is hermitian and defined for all t, so that $$\hat H(t)$$ has for positive $$t - t_0$$ again a complete set of real eigenvalues $$ E_n(t).$$ But these eigenvalues  may change with time.

However, one can again find the time evolution of the density matrix $$\hat{\rho}(t)$$ rsp. of the partition function $$Z(t) = \operatorname{Tr}\, \left[\hat\rho (t)\right],$$ to evaluate the expectation value of
 * $$\left\langle\hat A\right\rangle = \frac{\operatorname{Tr}\,\left[\hat \rho (t)\,\hat A\right]}{\operatorname{Tr}\,\left[\hat\rho (t)\right]}.$$

The time dependence of the states $$|n(t) \rangle$$ is governed by the Schrödinger equation
 * $$i\hbar\frac{\partial}{\partial t} | n(t) \rangle = \hat{H}(t)| n(t) \rangle ,$$

which thus determines everything, corresponding of course to the Schrödinger picture. But since $$\hat{V}(t)$$ is to be regarded as a small perturbation, it is convenient to now use instead the interaction picture representation, $$\left|\hat n(t) \right\rangle ,$$ in lowest nontrivial order. The time dependence in this representation is given by $$| n(t) \rangle = e^{-i\hat H_0t/\hbar} \left| \hat{n}(t) \right\rangle = e^{-i\hat H_0t/\hbar}\hat{U}(t, t_0) \left| \hat{n}(t_0) \right\rangle ,$$ where by definition for all t and $$t_0$$ it is: $$\left| \hat{n}(t_0) \right\rangle = e^{i\hat H_0t_0/\hbar} | n(t_0) \rangle$$

To linear order in $$\hat{V}(t)$$, we have
 * $$\hat{U}(t, t_0) = 1 - \frac{i}{\hbar}\int_{t_0}^t dt' \hat{V}\mathord\left(t'\right)$$.

Thus one obtains the expectation value of $$\hat{A}(t)$$ up to linear order in the perturbation:
 * $$\left\langle\hat{A}(t)\right\rangle= \left\langle \hat{A} \right\rangle_0 - \frac{i}{\hbar}\int_{t_0}^t dt' {1 \over Z_0}\sum_n e^{-\beta E_n} \left\langle n (t_0) \left| \hat{A}(t) \hat{V}\mathord\left(t'\right) - \hat{V}\mathord\left(t'\right)\hat{A}(t) \right| n(t_0) \right\rangle$$,

thus

The brackets $$\langle \rangle_0$$ mean an equilibrium average with respect to the Hamiltonian $$H_0 .$$ Therefore, although the result is of first order in the perturbation, it involves only the zeroth-order eigenfunctions, which is usually the case in perturbation theory and moves away all complications which otherwise might arise for $$t > t_0$$.

The above expression is true for any kind of operators. (see also Second quantization)