On the computation of imaginary-time correlation functions beyond the ℏβ limit

The prevailing formalism for the computation of exact time correlation functions is imaginary time path integration followed by analytic continuation. This formalism is constrained to the fundamental imaginary-time interval [0, ℏβ] which vanishes in the classical limit. While going beyond the ℏβ limit may substantially increase the accuracy of the analytic continuation, it is beyond the reach of straightforward path integration, and, furthermore, it is impossible for systems and operators for which imaginary-time correlation functions do not exist for arbitrary times. To circumvent this problem, we propose evaluating the correlation functions of auxiliary operators whose existence is guaranteed for all imaginary times and from which the spectral densities of the operators of interest can be retrieved. To this end, we formulate a novel path integration method that treats exponentials of the squared Liouville operator. Initial numerical results, which give mixed indications regarding the feasibility of the method, are provided and discussed.