In recent years, the role of predictive computational modeling has become a cornerstone for the study of fundamental electronic, optical, and thermal properties in complex forms of condensed matter, including Dirac and topological materials. The simulation of quantum transport in realistic materials calls for the development of linear scaling, or order-$N$, numerical methods, which then become enabling tools for guiding experimental research and for supporting the interpretation of measurements. In this review, we describe and compare different order-$N$ computational methods that have been developed during the past twenty years, and which have been intensively used to explore quantum transport phenomena in disordered media. We place particular focus on the electrical conductivities derived within the Kubo-Greenwood and Kubo-Streda formalisms, and illustrate the capabilities of these methods to tackle the quasi-ballistic, diffusive, and localization regimes of quantum transport. The fundamental issue of computational cost versus accuracy of various proposed numerical schemes is addressed in depth. We then extend the review to the study of spin dynamics and topological transport, for which efficient approaches of inspecting charge, spin, and valley Hall conductivities are outlined. The usefulness of these methods is illustrated by various examples of transport in disordered materials, such as polycrystalline and defected graphene models, 3D metals and Dirac semimetals, carbon nanotubes, and organic semiconductors.

中文翻译：

---

线性标度量子传输方法

近年来，预测计算建模的作用已成为研究复杂形式的凝聚态物质（包括狄拉克和拓扑材料）的基本电子、光学和热特性的基石。现实材料中量子传输的模拟需要开发线性标度或阶数 N$ 数值方法，然后这些方法成为指导实验研究和支持测量解释的有利工具。在这篇综述中，我们描述和比较了过去 20 年中发展起来的不同阶 $N$ 计算方法，这些方法已被广泛用于探索无序介质中的量子传输现象。我们特别关注源自 Kubo-Greenwood 和 Kubo-Streda 形式主义的电导率，并说明这些方法处理量子传输的准弹道、扩散和局域化机制的能力。深入解决了各种提议的数值方案的计算成本与准确性的基本问题。然后，我们将回顾扩展到自旋动力学和拓扑输运的研究，概述了检查电荷、自旋和谷霍尔电导率的有效方法。这些方法的有用性通过无序材料中的各种传输示例来说明，例如多晶和有缺陷的石墨烯模型、3D 金属和狄拉克半金属、碳纳米管和有机半导体。深入解决了各种提议的数值方案的计算成本与准确性的基本问题。然后，我们将回顾扩展到自旋动力学和拓扑输运的研究，概述了检查电荷、自旋和谷霍尔电导率的有效方法。这些方法的有用性通过无序材料中的各种传输示例来说明，例如多晶和有缺陷的石墨烯模型、3D 金属和狄拉克半金属、碳纳米管和有机半导体。深入解决了各种提议的数值方案的计算成本与准确性的基本问题。然后，我们将回顾扩展到自旋动力学和拓扑输运的研究，概述了检查电荷、自旋和谷霍尔电导率的有效方法。这些方法的有用性通过无序材料中的各种传输示例来说明，例如多晶和有缺陷的石墨烯模型、3D 金属和狄拉克半金属、碳纳米管和有机半导体。