# 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.

## General Kubo formula

Consider a quantum system described by the (time independent) Hamiltonian ${\displaystyle H_{0}}$. The expectation value of a physical quantity, described by the operator ${\displaystyle {\hat {A}}}$, can be evaluated as:

${\displaystyle \langle {\hat {A}}\rangle ={1 \over Z_{0}}\operatorname {Tr} \,[{\hat {\rho _{0}}}{\hat {A}}]={1 \over Z_{0}}\sum _{n}\langle n|{\hat {A}}|n\rangle e^{-\beta E_{n}}}$
${\displaystyle {\hat {\rho _{0}}}=e^{-\beta {\hat {H}}_{0}}=\sum _{n}|n\rangle \langle n|e^{-\beta E_{n}}}$

where ${\displaystyle Z_{0}=\operatorname {Tr} \,[{\hat {\rho }}_{0}]}$ is the partition function. Suppose now that just above some time ${\displaystyle t=t_{0}}$ an external perturbation is applied to the system. The perturbation is described by an additional time dependence in the Hamiltonian: ${\displaystyle {\hat {H}}(t)={\hat {H}}_{0}+{\hat {V}}(t)\theta (t-t_{0}),}$ where ${\displaystyle \theta (t)}$ is the Heaviside function ( = 1 for positive times, =0 otherwise) and ${\displaystyle {\hat {V}}(t)}$ is hermitian and defined for all t, so that ${\displaystyle {\hat {H}}(t)}$ has for positive ${\displaystyle t-t_{0}}$ again a complete set of real eigenvalues ${\displaystyle E_{n}(t).}$ But these eigenvalues may change with time.

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

The time dependence of the states ${\displaystyle |n(t)\rangle }$ is governed by the Schrödinger equation ${\displaystyle i\partial _{t}|n(t)\rangle ={\hat {H}}(t)|n(t)\rangle ,}$ which thus determines everything, corresponding of course to the Schrödinger picture. But since ${\displaystyle {\hat {V}}(t)}$ is to be regarded as a small perturbation, it is convenient to now use instead the interaction picture representation, ${\displaystyle |{\hat {n}}(t)\rangle ,}$ in lowest nontrivial order. The time dependence in this representation is given by ${\displaystyle |n(t)\rangle =e^{-i{\hat {H}}_{0}t}|{\hat {n}}(t)\rangle =e^{-i{\hat {H}}_{0}t}{\hat {U}}(t,t_{0})|{\hat {n}}(t_{0})\rangle ,}$ where by definition for all t and ${\displaystyle t_{0}}$ it is: ${\displaystyle |{\hat {n}}(t_{0})\rangle =e^{i{\hat {H}}_{0}t_{0}}|n(t_{0})\rangle }$

To linear order in ${\displaystyle {\hat {V}}(t)}$, we have ${\displaystyle {\hat {U}}(t,t_{0})=1-i\int _{t_{0}}^{t}dt'{\hat {V}}(t')}$. Thus one obtains the expectation value of ${\displaystyle {\hat {A}}(t)}$ up to linear order in the perturbation.

${\displaystyle {\begin{array}{rcl}\langle {\hat {A}}(t)\rangle &=&\langle {\hat {A}}\rangle _{0}-i\int _{t_{0}}^{t}dt'{1 \over Z_{0}}\sum _{n}e^{-\beta E_{n}}\langle n(t_{0})|{\hat {A}}(t){\hat {V}}(t')-{\hat {V}}(t'){\hat {A}}(t)|n(t_{0})\rangle \\&=&\langle {\hat {A}}\rangle _{0}-i\int _{t_{0}}^{t}dt'\langle [{\hat {A}}(t),{\hat {V}}(t')]\rangle _{0}\end{array}}}$

The brackets ${\displaystyle \langle \rangle _{0}}$ mean an equilibrium average with respect to the Hamiltonian ${\displaystyle H_{0}.}$ 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 ${\displaystyle t>t_{0}}$.

The above expression is true for any kind of operators. (see also Second quantization)[1]