The Bethe Ansatz

We can start our considerations by looking at a system represented by a Hamiltonian \(H_0\), without having to specify in how many dimensions our system is, whether it is on the lattice or in the continuum, or what its operator content is. Let us imagine that we perturb our system by applying some external forces on it. We represent this by adding to the Hamiltonian a time-dependent term

\[ H (t) = H_0 + F (t) \hat{P} \]

in which \(\hat{P}\) is the (Hermitian) perturbing operator with which we connect to our system and \(F\) are the (real-valued) time-dependent parameters setting the scale of this change (this equation is in the Schrödinger picture, so \(\hat{P}\) is time-independent).

What we are interested in is the effect of the presence of this perturbation on the expectation value which generic observables \(\hat{O}\) take, that is we would like to calculate \(\bar{O} (t) = \langle \psi (t) | \hat{O} | \psi (t) \rangle\). Generically, these expectation values will be complicated functionals of the applied perturbations.

Let us try to compute these expectation values using a minimal set of assumptions. Written in the interaction representation, such an expectation value becomes

\[ \bar{O} (t) = \langle \psi^I(t) | \hat{O}^I (t) | \psi^I(t) \rangle = \langle \psi^I(t_0) | \left(U^I(t, t_0)\right)^{-1} \hat{O}^I (t) U^I (t, t_0) |\psi^I (t_0) \rangle \]

in which the propagator is (we put \(\hbar = 1\) from now on)

\[ U^I (t, t_0) = T_t \left[ e^{-i \int_{t_0}^t dt' F(t') \hat{P}^I (t')} \right]. \]

Let us assume that the system starts in an eigenstate \(|\psi_0 \rangle\) of the unperturbed system at \(t = t_0 = -\infty\), and that the perturbation parametrized by \(F(t)\) is very small (the precise definition of "very small" is actually quite complicated; for our purposes it suffices to say that it does not lead to modifications of order of one in the state occupation probability distribution). Expanding the propagator in powers of \(F\) allows to write

\[ \bar{O}(t) = \bar{O}_{\psi_0} - i \int_{-\infty}^t dt' \langle \psi_0 | [ \hat{O}^I (t), \hat{P}^I (t') ] | \psi_0 \rangle F(t') + O(F^2) \]

in which \(\bar{O}_{\psi_0} = \langle \psi_0 | \hat{O} | \psi_0 \rangle\) is the original expectation value in the unperturbed system. We now define the retarded correlation function (in eigenstate \(|\psi \rangle\) of \(H_0\)) linking \(\hat{P}\) and \(\hat{O}\) as

\[ {\cal C}^{\hat{O}, \hat{P}}_{ret, \psi} (t- t') \equiv -i \theta (t - t') \langle \psi | [ \hat{O}^I (t), \hat{P}^I (t') ] | \psi \rangle \tag{Cret}\label{Cret} \]

in which the operators are in the interaction representation (e.g. \(\hat{O}^I (t) = e^{i H_0 t} \hat{O} e^{-i H_0 t}\) ). Note that this in only a function of \(t-t'\) (and not of the individual times) in view of our assumption that \(|\psi \rangle\) is an eigenstate of \(H_0\).

In terms of this, we find that in the presence of the perturbation \(F(t) \hat{P}\), the expectation value of \(\hat{O}\) obtains a linear correction as compared to its original, unperturbed value:

\[ \bar{O} (t) = \bar{O}_{\psi_0} + \int_{-\infty}^\infty dt' {\cal C}^{\hat{O}, \hat{P}}_{ret, \psi_0} (t- t') F(t') + O(F^2). \tag{Kubo}\label{Kubo} \]

This known as the Kubo formula and is the fundamental equation of linear response theory. Physically, the retarded function \({\cal C}^{\hat{O}, \hat{P}}_{ret, \psi_0} (t-t')\) thus connects a perturbation enforced by \(\hat{P}\) acting at time \(t'\) to the modification of the value of \(\bar{O}\) at time \(t\). Since the retarded function vanishes for \(t < t'\), only past perturbations can influence an expectation value, in other words the retarded function reflects the principle of causality.