A general analytic continuation response formalism for large quantum systems

We develop a general nth-order analytic continuation formalism for evaluating the nth-order response of a many-body system which is subject to an arbitrary time-dependent mechanical perturbation K(t) and which is initially in a state of thermal equilibrium (i.e. its initial density matrix is e-βH/Z). The formalism is a generalization of the linear analytic continuation response formalism as presented by Luttinger and Nozieres and similarly entails the analytic continuation of certain suitably defined correlation functions which may, in principle, be evaluated exactly by the techniques of temperature perturbation theory. In this way the time development of the system is, in principle, exactly prescribed. Finally, as an example of the use of the generalized formalism, we evaluate the Hall effect in a metal with impurity scattering by means of the second-order formalism and the result justifies the usual expression.