Kubo formula

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 $H_0$. The expectation value of a physical quantity, described by the operator $\hat{A}$, can be evaluated as:

$\langle \hat{A}\rangle={1\over Z_0}Tr[\hat{\rho_0}\hat{A}]={1\over Z_0}\sum_n \langle n | \hat{A} |n \rangle e^{-\beta E_n}$

$\hat{\rho_0}=e^{-\beta \hat{H}_0}=\sum_n |n \rangle\langle n |e^{-\beta E_n}$

where $Z_0=Tr[\hat\rho_0]$ 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, we can again find the time evolution of the density matrix $\hat{\rho}(t)$ rsp. of the partition function $Z(t)=Tr [\hat\rho (t)],$ to evaluate the expectation value of $\langle\hat A\rangle = Tr [\hat \rho (t)\,\hat A]/Tr [\hat\rho (t)].$

 The time dependence of the states $|n(t) \rangle$ is governed by the Schrödinger equation $i\partial_t|n(t) \rangle=\hat{H}(t)|n(t) \rangle ,$ 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, $|\hat n(t)\rangle ,$ in lowest nontrivial order. The time dependence in this representation is given by $|n(t) \rangle=e^{-i\hat H_0t}|\hat{n}(t) \rangle=e^{-i\hat H_0t}\hat{U}(t,t_0)|\hat{n}(t_0) \rangle ,$ where by definition for all t and $t_0$ it is: $|\hat{n}(t_0) \rangle=e^{i\hat H_0t_0}|n(t_0) \rangle$

To linear order in $\hat{V}(t)$, we have $\hat {U}(t,t_0)=1-i\int_{t_0}^t dt'\hat V(t')$. Thus one obtains the expectation value of $\hat{A}(t)$ up to linear order in the perturbation.

$\begin{array}{rcl} \langle \hat{A}(t)\rangle &=& \langle \hat{A}\rangle_0-i\int_{t_0}^t dt'{1\over Z_0}\sum_n e^{-\beta E_n} \langle n (t_0)| \hat{A}(t)\hat{V}(t')- \hat{V}(t')\hat{A}(t) |n(t_0) \rangle\\ &=& \langle \hat{A}\rangle_0-i\int_{t_0}^t dt'\langle [\hat{A}(t),\hat{V}(t')]\rangle_0 \end{array}$

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 $t > t_0$.

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)^[1]

References

1. Mahan, GD (1981). many particle physics. New York: springer. ISBN 0306463385.