QUANTUM TRANSPORT WITH BERRYCURVATURE IN TOPOLOGICAL MATTERS

输运测量是探索固体材料性质最有力和最直接的方法之一。测量对象主要包 括电流和热流，对应的驱动场是电场和温度梯度场。根据测量对象和驱动场的不 同，可将载流子输运可分为三种，即电输运、热电输运和热输运。第一种在科学 文献和公众传播中更加广为人知，现代物理学中的许多重大发现都与其相关，如 超导体、量子霍尔效应、巨磁阻效应等。相比于电输运后两者虽然没有那么普遍， 但它们仍然是不可或缺的，尤其是在应用科学和工程领域。可见输运测量作为一 种基本的研究方法，在凝聚态物理，材料科学以及工程应用中起着至关重要的作 用。另一方面，贝里相位的发现为凝聚态物理的研究开辟了一条崭新的视角。在过 去，相比于体系的本征能量相位因子并没有得到太多的关注，原因是人们总是可 以增加一个全局的相位因子，而不会影响任何可观测量。然而，自从发现贝里相 位以来，这一认识发生了显著的变化。所谓贝里相位描述的是当一个量子态经过 绝热演化后，除了动力学相位之外，还会累积一个额外的相位。此后的相关发展 证明，这一发现的重要程度不亚于任何一次诺奖级的成果，特别是对凝聚态物理、 量子光学、原子物理和量子信息等邻域产生了极其深远的影响，同时贝利曲率的 发现也让我们对量子力学的本质有了更深刻的理解和认识。近些年来，以拓扑材 料为代表的凝聚态物理的巨大进展很大程度上归功于对量子态几何结构的深入理 解，后者则直接与贝里曲率相关。 狄拉克材料或拓扑材料的主要特征是在费米能级附近具有线性的色散关系， 这意味着电子的低能激发行为需要用相对论性的狄拉克方程来描述，此时被激发 的准粒子也被称为狄拉克费米子。由于这些准粒子是相对论性的，这导致了诸多 与普通费米子相异的特性。一般来说，拓扑物质包括拓扑绝缘体、拓扑半金属和 拓扑超导体。奇异的能带结构以及拓扑保护的新颖表面态使得拓扑物质具有各种 独特的输运性质。本文中我们主要关注拓扑绝缘体和拓扑半金属。拓扑绝缘体的 体态是有能隙的绝缘态，而边界态是无能隙的金属态，这种特殊的无能隙表面态 由于受拓扑保护所以是无耗散的，这一特性突显了拓扑绝缘体在未来电子工业应 用中的巨大潜力。拓扑半金属可以看作是三维的石墨烯，根据对称性，可以将其 简单地分为狄拉克半金属、外尔半金属和节线半金属。其中外尔半金属是最为出 名的，由于其能带的交叉点（外尔点）没有对称性的要求，即不受微扰的影响。外 尔半金属的体态存在成对的贝里曲率单极子，且在某些特殊表面存在受拓扑保护的费米弧表面态。所有这些使得拓扑物质成为探索奇特输运性质的极好平台。 本文中我们首先回顾了一些必要的概念，包括输运模型、贝里曲率和有效哈 密顿量。然后介绍了玻尔兹曼动理学，这是本文进行理论分析的主要方法。基于半 经典输运理论我们给出了一个统一的理论框架来描述各种输运现象，包括电、热 电和热输运。我们证明了这种方法可以从相同的角度出发来处理不同的狄拉克材 料 (拓扑半金属和拓扑绝缘体)。此外，数值结果表明，轨道磁矩可以显著地促进磁 场下输运效应。同时，利用该理论我们对最近的磁阻和磁塞贝克实给出了定量解 释。 此外，我们还探讨了反常输运中的贝里曲率效应。本文主要研究了反常霍尔 效应、反常能斯特效应和反常热霍尔效应中贝曲率的可观测效应，这些输运现象 为探索各种奇异现象提供了极好的平台。威德曼-弗朗兹定律是固体物理中关于热 导率和电导率之比的一个普适关系。本文中我们通过解析和数值分析，发现在高 温下热导率和电导率之比与经典结果有很大的偏离。以 2 维狄拉克模型为出发点， 我们导出了一个对威德曼-弗朗兹定律进行修正的解析公式。该公式的结果是相当 普适的，不依赖于具体的模型细节，只决定于于费米能级的相对位置。我们进一 步指出该解析表达式可以推广到 3 维，并且与相关实验测量 (例如笼目磁体) 吻合 的极好。这些结果将进一步促进对物质的拓扑态及其非平庸相位的深入理解。

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