We present a general quantum kinetic theory that accounts for the interplay between a temperature gradient, momentum-space Berry curvatures of Bloch electrons, and Bloch-state scattering. Using a theory that incorporates the presence of a temperature gradient by introducing a "thermal vector potential", we derive a quantum kinetic equation for Bloch electrons in the presence of disorder and a temperature gradient. In contrast to the semiclassical Boltzmann formalism in which a temperature gradient is introduced by setting $\dot{𝐫}\cdot \frac{\partial }{\partial 𝐫}\to \dot{𝐫}\cdot \frac{\partial T}{\partial 𝐫}\frac{\partial }{\partial T}$ in the Boltzmann equation, the presence of a temperature gradient in our formalism is described as a driving force just as in the case of an electric field (i.e., comes from $\dot{𝐤}$ in the language of the semiclassical Boltzmann formalism). Taking also into account the presence of electric and magnetic fields, the quantum kinetic equation we derive makes it possible to compute transport coefficients at arbitrary orders of electric-field $𝐄$, magnetic-field $𝐁$, and temperature-gradient $\nabla T$ strengths $|𝐄{|}^a|𝐁{|}^b|\nabla T{|}^c$. Our theory enables a systematic calculation of magnetothermoelectric and magnetothermal conductivities of systems with momentum-space Berry curvatures. As an illustration, we derive from a general microscopic electron model a general expression for the rate of pumping of electrons between valleys in parallel temperature gradient and magnetic field. From this expression we find a relation, which is analogous to the Mott relation, between the rate of pumping due to a temperature gradient and that due to an electric field. We also apply our theory to a two-band model for Weyl semimetals to study thermoelectric and thermal transport in a magnetic field. We show that the Mott relation is satisfied in the chiral-anomaly induced thermoelectric conductivity, and that the Wiedemann-Franz law is violated in the chiral-anomaly induced thermal conductivity, which are both consistent with the results obtained by invoking semiclassical wave-packet dynamics.

中文翻译：

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磁场中热电和热输运的量子动力学理论

我们提出了一个通用的量子动力学理论，该理论解释了温度梯度，布洛赫电子的动量空间贝里曲率和布洛赫状态散射之间的相互作用。使用一种通过引入“热矢量势”而引入温度梯度的理论，我们得出了在存在无序和温度梯度的情况下布洛赫电子的量子动力学方程。与半经典的玻尔兹曼形式主义不同，后者通过设置温度梯度来引入温度梯度。$\dot{𝐫}\cdot \frac{\partial }{\partial 𝐫}\to \dot{𝐫}\cdot \frac{\partial Ť}{\partial 𝐫}\frac{\partial }{\partial Ť}$ 在玻耳兹曼方程中，形式上存在温度梯度被描述为驱动力，就像在电场的情况下（即，来自 $\dot{𝐤}$在半古典的玻尔兹曼形式主义的语言中）。还考虑到电场和磁场的存在，我们得出的量子动力学方程使得可以计算任意阶次电场下的传输系数$𝐄$磁场 $𝐁$和温度梯度 $\nabla Ť$ 长处 $|𝐄{|}^{\mathrm{一种}}|𝐁{|}^b|\nabla Ť{|}^C$。我们的理论可以系统地计算具有动量空间贝里曲率的系统的磁热电和磁热导率。作为说明，我们从一般的微观电子模型中得出了在平行温度梯度和磁场中谷之间电子泵送速率的一般表达式。从该表达式中，我们发现由于温度梯度引起的抽速与由于电场引起的抽速之间的关系类似于莫特关系。我们还将我们的理论应用于Weyl半金属的两波段模型，以研究磁场中的热电和热传输。我们证明在手征异常诱导的热导率中满足了Mott关系，并且在手征异常诱导的热导率中违反了Wiedemann-Franz定律，