TraSPI: A numerically exact method for nonequilibrium quantum transport through nanostructures

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The theoretical description of quantum transport through nanostructures is a challenging task. The complex interplay of nonequilibrium dynamics, Coulomb interaction, finite temperature and, if present, collective order does, in general, not allow for an exact analytical treatment. Therefore, a plethora of different theoretical methods have been developed to address various regimes of quantum transport in nanoscale devices. They all have their advantages and disadvantages. Some are restricted to the linear-response regime while others can cover strong nonequilibrium situations. In scenarios with a clear hierarchy of the involved parameters, a perturbation expansion in one of them can be used. However, in real experiments very often many of these parameters characterizing, e.g., temperature, tunnel coupling strength, Coulomb interaction as well as gate and bias voltage, are energetically of the same order of magnitude. Then, numerically exact methods are desirable. In a recent paper, we presented TraSPI ("Transfer-matrix Summation of Path Integrals") as such a numerically exact method [Mundinar et al., Phys. Rev. B 106, 165427 (2022)]. It is based on an iterative summation of path integrals, established as ISPI in the last decade. The virtue of ISPI (and thus also TraSPI) is that it naturally takes into account all orders in tunneling of electrons between quantum dot and leads, allows for arbitrary bias voltages that drive the system out of equilibrium, is not restricted to either low or high temperatures, and is able to include a finite Coulomb interaction. A major virtue of TraSPI is that the stationary limit is implemented by construction, and that certain derivatives can be calculated analytically. In addition, the use of transfer matrices allows for further improvements that enhance the efficiency and accuracy of the method. We will boost the performance of the TraSPI method by further development and through an efficient implementation, also on GPUs. Thereby, we will make use of similarities between the nonequilibrium TraSPI method and an effective long-range equilibrium Ising model. This will allow us to address several problems in quantum transport through nanostructures that are difficult or impossible to be tackled by other methods. This includes aspects of quantum-dot interferometry as well as the generation of superconducting correlations in quantum dots.