Kubo formula and response functions
We want to see how the Kubo formula connects time dependent perturbations and their linear responses.
We can then apply a forcing field (electric, magnetic, mechanic) and determine susceptibilities in the linear response regime.
Susceptibility as a Green’s Function
When a weak external electric field \mathbf E(\mathbf r,t) is applied, the system responds linearly with an induced polarization \mathbf P(\mathbf r,t). In the linear response regime, the most general linear relation between cause (field) and effect (polarization) is
P_i(\mathbf r,t) = \int d\mathbf r' \int_{-\infty}^\infty dt' \, \chi_{ij}(\mathbf r,t;\mathbf r',t') \, E_j(\mathbf r',t'). \tag{1}
Here \chi is the susceptibility tensor, has certain properties by default or by imposition:
Because the system is weakly perturbed, the polarization is a linear functional of the applied field.
A perturbation at (\mathbf r',t') can influence the polarization at (\mathbf r,t). This makes \chi depend on two spacetime points.
A result cannot precede its cause. Therefore, \chi_{ij}(\mathbf r,t;\mathbf r',t') = 0 \quad \text{for } t < t' .
We can extract Equation 1 from the Kubo formula, which which expresses the linear response of an observable quantity due to a time-dependent perturbation.
From Kubo Formula to Response
We start with the Hamiltonian
\hat H(t) = H_0 - \int d\mathbf r'\,\hat{\mathbf P}(\mathbf r') \cdot \mathbf E(\mathbf r',t), \tag{2}
where \hat{\mathbf P}(\mathbf r) is the polarization operator.
From linear response theory (Kubo formula), the change in expectation value of an operator \hat A is \delta\langle \hat A(t)\rangle = \frac{i}{\hbar}\int_{-\infty}^t dt' \, \langle [\hat H_I(t'), \hat A(t)] \rangle_0 , \tag{3}
with \hat H_I(t') the interaction Hamiltonian.
For \hat A(t) = \hat P_i(\mathbf r,t), \delta \langle \hat P_i(\mathbf r,t)\rangle = \frac{i}{\hbar}\int_{-\infty}^t dt' \int d\mathbf r' \, \Big\langle \big[ -\hat P_j(\mathbf r',t') E_j(\mathbf r',t'), \, \hat P_i(\mathbf r,t) \big] \Big\rangle_0 .
Since E_j(\mathbf r',t') is a classical field (a c-number, not an operator), it commutes with operators and can be pulled out: = -\frac{i}{\hbar}\int_{-\infty}^t dt' \int d\mathbf r' \, E_j(\mathbf r',t') \, \langle [\hat P_j(\mathbf r',t'), \hat P_i(\mathbf r,t)] \rangle_0 .
Using [X,Y] = -[Y,X], this becomes \delta \langle \hat P_i(\mathbf r,t)\rangle = \frac{i}{\hbar}\int_{-\infty}^t dt' \int d\mathbf r' \, \langle [\hat P_i(\mathbf r,t), \hat P_j(\mathbf r',t')] \rangle_0 \, E_j(\mathbf r',t') .
Definition of the Retarded Susceptibility
Introduce the retarded susceptibility (response function): \chi_{ij}^R(\mathbf r,t;\mathbf r',t') = \frac{i}{\hbar} \, \Theta(t-t') \, \langle [\hat P_i(\mathbf r,t), \hat P_j(\mathbf r',t')] \rangle_0 .
Then the response takes the compact Green’s-function form: \delta \langle \hat P_i(\mathbf r,t)\rangle = \int d\mathbf r' \int_{-\infty}^\infty dt' \, \chi_{ij}^R(\mathbf r,t;\mathbf r',t') \, E_j(\mathbf r',t').
Why \chi is a Green’s Function
If the applied field is a \delta-pulse at (\mathbf r',t'), then \delta \langle \hat P_i(\mathbf r,t)\rangle = \chi_{ij}^R(\mathbf r,t;\mathbf r',t'), exactly the defining property of a Green’s function.
The step function \Theta(t-t') enforces causality: only past fields influence present responses.
In frequency space, causality implies that \chi_{ij}^R(\omega) is analytic in the upper half-plane, which leads to the Kramers–Kronig relations between the real and imaginary parts of the susceptibility.
The susceptibility \chi then is formally the retarded Green’s function of the polarization operator. It is introduced because: - Linearity \Rightarrow convolution with a kernel.
- Spatial/temporal memory \Rightarrow dependence on (\mathbf r,t;\mathbf r',t').
- Causality \Rightarrow retarded boundary condition.
- Kubo formula \Rightarrow explicit commutator expression, confirming its Green’s-function character.