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The quantum harmonic oscillator is the fundamental building block to computethermal properties of virtually any dielectric crystal at low temperatures interms of phonons, extended further to cases with anharmonic couplings, or evendisordered solids. In general, Path Integral Monte Carlo (PIMC) or MolecularDynamics (PIMD) methods are powerful tools to determine stochasticallythermodynamic quantities without systematic bias, not relying on perturbativeschemes. Addressing transport properties, for instance calculating thermalconductivity from PIMC, however, is substantially more difficult. Althoughcorrelation functions of current operators can be determined by PIMC fromanalytic continuation on the imaginary-time axis, Bayesian methods are usuallyemployed for the numerical inversion back to real-time response functions. Thistask not only strongly relies on the accuracy of the PIMC data, but alsointroduces noticeable dependence on the model used for the inversion. Here, weaddress both difficulties with care. In particular, we first devise improvedestimators for current correlations which substantially reduce the variance ofthe PIMC data. Next, we provide a neat statistical approach to the inversionproblem, blending into a fresh workflow the classical stochastic maximumentropy method together with recent notions borrowed from statistical learningtheory. We test our ideas on a single harmonic oscillator and a collection ofoscillators with a continuous distribution of frequencies, and provideindications of the performance of our method in the case of a particle in adouble well potential. This work establishes solid grounds for an unbiased,fully quantum mechanical calculation of transport properties in solids.

Time correlation functions for quantum systems: validating Bayesian approaches for harmonic oscillators and beyond