Approximately Quantized Thermal Hall Effect of Chiral Liquids Coupled to Phonons

The recent observation of a half-integer quantized thermal Hall effect in $\alpha \text{−}{\mathrm{RuCl}}_3$ is interpreted as a unique signature of a chiral spin liquid with a Majorana edge mode. A similar quantized thermal Hall effect is expected in chiral topological superconductors. The unavoidable presence of gapless acoustic phonons, however, implies that, in contrast to the quantized electrical conductivity, the thermal Hall conductivity ${\kappa }_{xy}$ is never exactly quantized in real materials. Here, we investigate how phonons affect the quantization of the thermal conductivity, focusing on the edge theory. As an example, we consider a Kitaev spin liquid gapped by an external magnetic field coupled to acoustic phonons. The coupling to phonons destroys the ballistic thermal transport of the edge mode completely, as energy can leak into the bulk, thus drastically modifying the edge picture of the thermal Hall effect. Nevertheless, the thermal Hall conductivity remains approximately quantized, and we argue that the coupling to phonons to the edge mode is a necessary condition for the observation of the quantized thermal Hall effect. The strength of this edge coupling does, however, not affect the conductivity. We argue that for sufficiently clean systems the leading correction to the quantized thermal Hall effect, $\mathrm{\Delta }{\kappa }_{xy}/T\sim \mathrm{sign}(\mathrm{B})T^2$, arises from an intrinsic anomalous Hall effect of the acoustic phonons due to Berry phases imprinted by the chiral (spin) liquid in the bulk. This correction depends on the sign but not the amplitude of the external magnetic field.

Popular Summary

In the material ${\text{RuCl}}_3$, the thermal conductivity becomes quantized in the direction perpendicular to a gradient in the temperature. This recently discovered behavior—a new type of “quantum Hall effect”—indicates that the material can be described as a chiral quantum spin liquid, a much sought-after exotic state of matter. Quantum Hall effects are usually observed in insulators, but ${\text{RuCl}}_3$ can conduct heat very efficiently using lattice vibrations called phonons. Here, we explain why phonons do not destroy the thermal quantum Hall effect and why they are necessary to make it observable.

We mathematically show that when a temperature gradient is applied to the sample, the edge of the chiral spin liquid injects a quantized heat current into the phonon system. This injected phonon heat current enforces a second temperature gradient perpendicular to the first, which is then measured in experiments.

We also show that a tiny correction to the quantized thermal Hall effect arises because the coupling of the phonons to the chiral spin liquid imprints onto the acoustic phonons a Berry phase, a quantum mechanical phase that describes how a wave function of the phonon acquires an extra phase factor due to the coupling with the spin liquid. This results in a change to the phonons’ equations of motion, allowing their trajectories to bend and carry heat in the perpendicular direction.

Our work implies that quantized thermal Hall effects are very robust. Furthermore, we show how chiral spin liquids can imprint Berry phases on the phonons, which can be used to bend sound waves.

Figure 1

Sketch of heat currents (arrows) and local temperatures (color scale) in the presence of chiral edge channels. In the absence of phonons, panel (a), heat is transported ballistically by the edge channel, resulting in a quantization of ${\kappa }_{xy}/T$. Panel (b) shows a regime where the sample size is smaller than the phonon-edge scattering length, such that the edge states and phonons have different temperatures. No quantization of ${\kappa }_{xy}$ is expected in this case. In the opposite limit, panel (c), the edge channel and the phonons have the same local temperatures. The temperature gradient imprinted by the phonons onto the edge channel implies that a quantized heat current is injected from the edge mode into the bulk phonons, as shown in panel (d). This leads to quantization of ${\kappa }_{xy}$ with high precision.

Figure 2

Schematic plot of the $T$ dependence of ${\kappa }_{xy}$ per layer of a chiral (spin) liquid coupled to phonons. An approximately quantized Hall effect, $[({\kappa }_{xy})/T]=\{[(c_r-c_l)\pi k_B^2]/(6\hslash )\}+O(T^2)$, is observed for $T$ much smaller than the bulk gap ${\mathrm{\Delta }}_b$ of the chiral liquid under the condition that the phonon-phonon and the phonon-edge scattering length be smaller than the sample size, ${\ell }_{\mathrm{ph}}^{\mathrm{ph}}$, ${\ell }_{\mathrm{ph}}^e\ll L$. The leading correction for clean systems arises from an intrinsic anomalous Hall effect of phonons, Eq. (31). At lower $T$, spins and phonons decouple (mean-free path larger than system size) and one obtains a nonuniversal result (see Sec. 2b), which can be larger or smaller than the quantized value depending on contact properties. For $T\gg {\mathrm{\Delta }}_b$, the thermal Hall effects of both the spins and the phonons are expected to decay rapidly with increasing temperatures.

Figure 3

Feynman diagrams used in calculating the expectation values of the heat current to leading order. Here, straight continuous lines are the fermion propagators and dotted wiggly lines are phonon propagators. The “x” marks the space-time point at which the expectation value is evaluated. The different diagrams are (a) the ${\lambda }^2$ contribution to $⟨c_k^{†}{c}_k⟩$, (b) the ${\lambda }^2$ contribution to $⟨a_{\mathit{q}}^{†}{a}_{{\mathit{q}}^{\prime }}⟩$, (c) the $\lambda$ contribution to $⟨c_k^{†}{c}_{k^{\prime }}{a}_{\mathit{q}}⟩$, and (d) the ${\lambda }^0$ contribution to $⟨c_k^{†}{c}_{k^{\prime }}{a}_{\mathit{q}}^{†}{a}_{{\mathit{q}}^{\prime }}⟩$.

Figure 4

Dependence of the heat current flowing in the perpendicular direction on (a) the distance $d$ from the edge and (b) the temperature. Here, we plot the temperature derivative of the difference between the heat current for an infinite system and the heat current when considering only a slab of width $d$, for different temperatures in (a) and different distances in (b). The green line shows (a) $1/d^2$ behavior and (b) linear $T$ behavior.