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

纳米结构中量子输运的理论描述是一项具有挑战性的任务。非平衡动力学、库仑相互作用、有限温度和集体有序(如果存在)的复杂相互作用，通常不允许进行精确的分析处理。因此，人们发展了大量不同的理论方法来解决纳米器件中不同的量子输运问题。它们都有各自的优点和缺点。其中一些被限制在线性响应区域，而另一些则可以涵盖强烈的非平衡情况。在所涉及参数的层次结构清晰的情况下，可以使用其中一个参数的微扰展开。然而，在实际实验中，许多表征这些参数的参数，例如温度、隧道耦合强度、库仑相互作用以及栅极和偏置电压，在能量上是相同的数量级。然后，数值精确的方法是可取的。在最近的一篇论文中，我们提出了TraSPI(“路径积分的传递矩阵求和”)作为这样一种数值精确方法[Mundina等人，Phys.修订本B 106,165427(2022年)]。它基于路径积分的迭代求和，在过去十年中被建立为ISPI。ISPI(因此也包括TraSPI)的优点是它自然地考虑了量子点和引线之间电子隧穿的所有顺序，允许任意的偏置电压导致系统失去平衡，不受低温或高温的限制，并且能够包括有限的库仑相互作用。TraSPI的一个主要优点是静态极限是通过构造来实现的，并且某些导数可以解析地计算。此外，传递矩阵的使用允许进一步改进，从而提高方法的效率和准确性。我们将通过进一步的开发和高效的实现来提高TraSPI方法的性能，也是在GPU上。因此，我们将利用非平衡TraSPI方法和有效的长期平衡伊辛模型之间的相似之处。这将使我们能够解决通过纳米结构进行量子传输的几个问题，这些问题是其他方法难以或不可能解决的。这包括量子点干涉测量的各个方面，以及在量子点中产生超导关联。

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.