# 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 $Z_0=Tr[\rho_0]$ is the partition function. Suppose now that some time $t=t_0$, 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: $\hat{H}(t)=\hat{H}_0+\hat{V}(t) \theta (t-t_0)$. We can find the time evolution of the density matrix $\hat{\rho}(t)$ to know the expectation value of $\hat{A}(t)$

$\langle \hat{A}(t)\rangle={1\over Z_0}Tr[\hat{\rho}(t)\hat{A}]={1\over Z_0}\sum_n \langle n(t) | \hat{A} |n(t) \rangle e^{-\beta E_n}$
$\hat{\rho(t)}=\sum_n |n(t) \rangle\langle n(t) |e^{-\beta E_n}$.

The time dependence of the states $|n(t) \rangle$ is governed by the Schrödinger equation $i\partial_t|n(t) \rangle=\hat{H}(t)|n(t) \rangle$ Since $\hat{V}(t)$ is to be regarded as a small perturbation, it is convenient to utilize the interaction picture representation $|\hat{n}(t) \rangle$. 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 $|\hat{n}(t_0) \rangle=e^{i\hat H_0t_0}|n(t_0) \rangle$

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 he Hamiltonian $H_0$. 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.