We consider the effect of coupling between phonons and a chiral Majorana edge in a gapped chiral spin liquid with Ising anyons (e.g., Kitaev's non-abelian spin liquid on the honeycomb lattice). This is especially important in the regime in which the longitudinal bulk heat conductivity $\kappa_{xx}$ due to phonons is much larger than the expected quantized thermal Hall conductance $\kappa_{xy}^{\rm q}=\frac{\pi T}{12} \frac{k_B^2}{\hbar}$ of the ideal isolated edge mode, so that the thermal Hall angle, i.e., the angle between the thermal current and the temperature gradient, is small. By modeling the interaction between a Majorana edge and bulk phonons, we show that the exchange of energy between the two subsystems leads to a transverse component of the bulk current and thereby an {\em effective} Hall conductivity. Remarkably, the latter is equal to the quantized value when the edge and bulk can thermalize, which occurs for a Hall bar of length $L \gg \ell$, where $\ell$ is a thermalization length. We obtain $\ell \sim T^{-5}$ for a model of the Majorana-phonon coupling. We also find that the quality of the quantization depends on the means of measuring the temperature and, surprisingly, a more robust quantization is obtained when the lattice, not the spin, temperature is measured. We present general hydrodynamic equations for the system, detailed results for the temperature and current profiles, and an estimate for the coupling strength and its temperature dependence based on a microscopic model Hamiltonian. Our results may explain recent experiments observing a quantized thermal Hall conductivity in the regime of small Hall angle, $\kappa_{xy}/\kappa_{xx} \sim 10^{-3}$, in $\alpha$-RuCl$_3$.

中文翻译：

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小霍尔角下的霍尔热导率的量化

我们考虑了在具有Ising anyons的带间隙手性旋转液体中声子与手性Majorana边缘之间的耦合效应（例如，蜂窝晶格上的Kitaev非阿贝尔旋转液体）。这在由于声子引起的纵向整体热导率$ \ kappa_ {xx} $比预期的量化热霍尔电导率$ \ kappa_ {xy} ^ {\ rm q} = \ frac {理想隔离边缘模式的\ pi T} {12} \ frac {k_B ^ 2} {\ hbar} $，因此热霍尔角（即热电流与温度梯度之间的角度）很小。通过对马约拉纳边与体声子之间的相互作用进行建模，我们表明两个子系统之间的能量交换导致体电流的横向分量，从而产生{\ em有效}的霍尔电导率。值得注意的是 后者等于边和块可以热化时的量化值，这发生在长度为$ L \ gg \ ell $的霍尔棒上，其中$ \ ell $是热化长度。对于Majorana-声子耦合模型，我们获得\ ell \ sim T ^ {-5} $。我们还发现，量化的质量取决于测量温度的方式，而且令人惊讶的是，当测量晶格温度而不是自旋温度时，可以获得更可靠的量化。我们介绍了系统的一般流体动力学方程式，温度和电流曲线的详细结果，以及基于微观模型哈密顿量的耦合强度及其对温度的估计。我们的结果可能解释了最近的实验，即在小霍尔角状态下观察到量化的霍尔热导率，