Phonon Thermal Hall Conductivity from Scattering with Collective Fluctuations

Because electrons and ions form a coupled system, it is a priori clear that the dynamics of the lattice should reflect symmetry breaking within the electronic degrees of freedom. Recently, this has been clearly evidenced for the case of time-reversal and mirror symmetry breaking by observations of a large phononic thermal Hall effect in many strongly correlated electronic materials. However, the mechanism by which time-reversal breaking and chirality is communicated to the lattice is far from evident. In this paper, we discuss how this occurs via many-body scattering of phonons by collective modes: a consequence of non-Gaussian correlations of the latter modes. We derive fundamental new results for such skew (i.e., chiral) scattering and the consequent thermal Hall conductivity. We emphasize that these results apply to any collective variables in any phase of matter: electronic, magnetic, or neither; highly fluctuating and correlated, or not. As a proof of principle, we compute general formulas for the above quantities for ordered antiferromagnets. From the latter, we obtain the scaling behavior of the phonon thermal Hall effect in clean antiferromagnets. The calculations show several different regimes and give quantitative estimates of similar order to that seen in recent experiments.

Popular Summary

When a temperature gradient is applied to any material, a heat current arises. In most solids, heat is carried by phonons, which are quantized elastic waves in the atomic lattice. The effectiveness of heat transport is typically controlled by phonon scattering. In this paper, we show that phonons scattering off collective excitations of electronic origin can induce a fascinating phenomenon, the thermal Hall conductivity, which appears when phonons scatter more clockwise than anticlockwise (or vice versa).

We provide a very general formulation to calculate the scattering of phonons with an arbitrary quantum field (i.e., a collective excitation). From there, we derive the consequences on the phonons’ thermal conductivity tensor, which characterizes the proportionality between a temperature gradient and a heat current. This should apply to any material—conductor, insulator, or magnet—for any amount of correlation between its degrees of freedom at different times, provided only that the material is not too disordered. We find that the thermal Hall effect is proportional to a specific correlation function of the quantum field, taken at four different time points. The latter is a highly nontrivial property of the quantum field, which can thus be probed through the phonon Hall conductivity.

As an illustration of this general formulation, we apply our results to the case where the fluctuating field arises from magnons, which are quantized spin-wave excitations, in an ordered antiferromagnet. For a reasonable set of parameters, we find a large thermal Hall conductivity (relative to the longitudinal one), comparable to that found in experiments. We also obtain theoretical predictions for both the longitudinal and Hall thermal conductivity as a function of temperature.

This work will help identify the contribution of phonons to thermal transport in magnets. More broadly, the general results we have obtained can now be applied to many systems where phonons are coupled to a fluctuating quantum field.

Figure 1

Illustration of a scattering mechanism responsible for a Hall effect. Only scattering processes that involve at least two (virtual) collisions with collective fluctuations can contribute to a Hall effect.

Figure 2

Calculated skew-scattering rate ${\mathfrak{W}}_{n\mathbf{k}{n}^{\prime }{\mathbf{k}}^{\prime }}^{\ominus ,-+}/{\gamma }_0$ [see Eqs. (10) and (19)] to transfer a phonon in mode $n^{\prime }{\mathbf{k}}^{\prime }$ to mode $n\mathbf{k}$, induced by coupling the lattice to a two-dimensional antiferromagnet. The density plot shows the angular dependence as a function of $\theta ({\mathbf{k}}^{\prime })\in [0,\pi /2]$ (horizontal axis) and $\phi (\mathbf{k},{\mathbf{k}}^{\prime })=\varphi ({\mathbf{k}}^{\prime })-\varphi (\mathbf{k})$ (vertical axis) for fixed $|{\mathbf{k}}^{\prime }|=0.8/\mathfrak{a}$, $k_x=0.2/\mathfrak{a}$, $k_y=0$, $k_z=0.1/\mathfrak{a}$, ${\mathit{m}}_0=0.05\widehat{\mathit{z}}$, and temperature $T=0.5T_0$. Here, $\mathfrak{a}$ is the in-layer lattice spacing, and $\varphi ({\mathbf{k}}^{({}^{\prime })})$ and $\theta ({\mathbf{k}}^{({}^{\prime })})$ are the azimuthal and polar angles of ${\mathbf{k}}^{({}^{\prime })}$, defined in the usual way. Note that the color bar is not scaled linearly.

Figure 3

Longitudinal thermal conductivity ${\kappa }_L$ with respect to temperature $T$, for four different values of ${\gamma }_{\mathrm{ext}}$. (a) Results on an order-one temperature scale, with ${\gamma }_{\mathrm{ext}}=1\times {10}^{-z}(v_{\mathrm{ph}}/\mathfrak{a}),z\in ⟦4,7⟧$, from darker ($z=4$) to lighter ($z=7$) shade. A crossover occurs between two scaling regimes with $x=1$ and $x=-1$ [see Eq. (23)]. Inset: log-log plot. The scaling behaviors are consistent with the analysis presented in the text. (b) Results on a smaller temperature scale, with ${\gamma }_{\mathrm{ext}}=1\times {10}^{-z}(v_{\mathrm{ph}}/\mathfrak{a}),z\in ⟦6,9⟧$, from darker ($z=6$) to lighter ($z=9$) shade. The peaks are features related to the magnon gaps.

Figure 4

Thermal Hall resistivity ${\varrho }_H^{xy}$ and ${\varrho }_H^{zx}$ (in units of ${\varrho }_0={\kappa }_0^{-1}$) with respect to temperature $T$. The transverse magnetization values $(m_0^y,m_0^z)$ used for computing ${\varrho }_H^{xy}$ and ${\varrho }_H^{xz}$ are (0.0,0.05) and (0.05,0.0), respectively. Inset: log-log plot. The scaling behavior is consistent with the analysis presented in the text.