Controversy on the Quantized Thermal Hall Effect in α-RuCl3

 [Introduction]

    In recent years, research and development of quantum computers has been actively pursued as the next generation of computational methods. Various quantum computer methods have been proposed, including superconducting, semiconductors, and optical quantum computers [1].
    One such method is called a topological quantum computer using Majorana particles (or excitation) [2]. In order to realize this method, it is necessary to manipulate Majorana particles (Majorana fermions) in a solid.
    Materials in a state (phase) called a quantum spin liquid have been proposed as a stage for the realization of Majorana particles. While various candidates for quantum spin liquid materials have been explored, the existence of Majorana particles is strongly suggested by the magnetic field-induced quantum spin liquid phase of α-RuCl3 [3]. The half-integer quantized thermal Hall effect observed in this magnetic field-induced quantum spin liquid phase is considered to be strong evidence for the existence of Majorana particles. However, the quantized thermal Hall effect has been controversial, with some saying that it has been observed and others saying that it has not.
    The current status of the recent controversy is summarized in a post by Prof. P. A. Lee [4] published in the Journal Club for Condensed Matter Physics. In this article, I summarize the current status of the controversy based on this post.

[Controversy history and current status]

1, Starting point

    The controversy was sparked by a Nature paper [5] reported in 2018, led by a group from Kyoto University. In this paper, the authors measured the quantized thermal Hall effect in a magnetic field-induced quantum spin liquid phase and observed a quantized thermal Hall effect with half the value predicted from fermionic excitations, arguing for the existence of Majorana fermionic excitations. 
    In this first paper, a magnetic field was applied both perpendicular and horizontal to the honeycomb lattice plane of α-RuCl3 (meaning 45° diagonally), and heat flow was applied in the in-plane a-axis direction (perpendicular to the Ru-Ru bond). In a follow-up paper published in Science [6], it was reported that the quantized thermal Hall effect can be observed by simply applying a magnetic field in the a-axis direction in the same plane as the heat flow direction (this is why the observed thermal Hall effect is also called the planar thermal Hall effect). 
    In the Nature paper, the quantization is observed when a magnetic field equivalent to 6.5-8T is applied in the plane, while in the Science paper it is observed at 10-11T. Prof. Lee points out that the former is in the magnetic field range corresponding to the field-induced quantum spin liquid phase, while the latter is outside that range.
Figure, Direction of applied magnetic field and thermal Hall effect. The thermal Hall effect can be seen only when the magnetic field is applied in the a-axis direction.

2, Follow up the experiment

　In response to these surprising discoveries, four groups have attempted to follow up on them.

2-1, The group claiming quantization

　First, a group from the University of Tokyo made a follow-up attempt and reported that the quantized values could be observed [7]. However, they pointed out the necessity of using a sample with high Phonon thermal conductivity for quantization.
Figure, Magnetic field and temperature dependence of the thermal Hall effect, quantized at a higher field than in the Nature paper.

　A group at the Max Planck Institute in Germany also conducted a follow-up test and reported that the quantized thermal Hall effect can be observed [8]. They reported that quantization occurs below 6.5 K and above 10 T, but the value is 20% smaller than the quantized value. However, the observed magnetic field region is outside the 7T-10T range where the quantum spin liquid phase is observed. It is also observed that the thermal Hall conductivity becomes smaller at lower temperatures.
Figure, Thermal Hall conductivity values observed for magnetic field and temperature. The white area is the quantized region.

2-2, The group claiming that quantization cannot be observed

　A group at Princeton University in the US reported that the quantized thermal Hall effect cannot be reproduced [9]. They reported only 60% of the quantized value at 5K and 10T, the same conditions as in the Science paper (on the other hand, they observed quantum oscillations of heat conduction in the magnetic field-induced quantum spin liquid phase, and proposed the existence of a mysterious neutral particle).
Figure, Quantum oscillations in thermal conductivity observed in the quantum spin liquid phase. The existence of quantum oscillations suggests the possibility of the existence of a spinon Fermi surface.

　Later, Princeton University submitted a second follow-up paper [10]. In that paper, they measured the thermal Hall effect over a wider temperature and magnetic field range than in the previous paper, and reported that the quantized value was still unobservable. Furthermore, it is argued that the thermal Hall conductivity decreases at lower temperatures, which is contrary to the convergence to a constant value expected in the presence of Majorana particles. This leads them to speculate that the cause of the thermal Hall effect may be the mysterious Bose particle-like excitation.
Figure, Temperature dependence of thermal Hall conductivity for different magnetic fields. It tends to zero at low temperatures, which should converge to a constant value if the phenomenon is due to Majorana fermion excitation.

    Sherbrooke University in Canada has also measured the thermal Hall effect [11]. They argue that since thermal conductivity is proportional to the magnitude of the thermal Hall effect, the contribution of phonon excitations should be carefully considered when discussing the thermal Hall effect. However, strictly speaking, they have not measured the quantized thermal Hall effect, so I will not include it here as a direct follow-up (the value of thermal Hall conductivity observed at an in-plane field of 7 T is about 1/10 of the quantized value).
Figure, Temperature dependence of thermal conductivity and thermal Hall conductivity on magnetic field in the in-plane direction.

3, Possible causes of controversy

　In response to these conflicting results, Prof. Lee, based on his personal correspondence with Prof. Takagi of the Max Planck Institute, pointed out the possibility that this was due to the different methods of making the samples used in the measurements [4]. The samples used by Kyoto University, University of Tokyo, and Max Planck Institute, where the quantized values were observed, were made using the Bridgman method [12] provided by Tokyo Institute of Technology. On the other hand, the Princeton (and Sherbrooke) groups, which were unable to observe the quantization, used samples made by the Chemical Vapor Transport (CVT) method [12] provided by the Oak Ridge National Laboratory (ORNL) for their measurements.
　Despite the above remarks, the actual cause has not yet been clarified. It is hoped that further research will be conducted in the future.

[Comparison of experimental results]

　The following is a side-by-side comparison of the magnetic field dependence of the thermal Hall effect for the papers mentioned above. It is true that there is a clear difference between the results with and without quantization.
There is a difference, isn't there?
Figure, Measured results of thermal Hall effect for each group.

[Summary]

　This is a summary of the controversy over the quantized thermal Hall effect in α-RuCl3. It's interesting.
Because it is an exciting result, it has generated a lot of discussion among experimentalists and theorists alike. This is a field that I would like to continue to pay attention to.

[References]

[5] Kasahara, Y., Ohnishi, T., Mizukami, Y. et al. Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid. Nature 559, 227–231 (2018).
[6] Yokoi, T., S. Ma, Y. Kasahara, S. Kasahara, T. Shibauchi, N. Kurita, H. Tanaka et al. ”Half-integer quantized anomalous thermal Hall effect in the Kitaev material candidate α − RuCl3.” Science 373, no. 6554 (2021): 568-572.
[9] Czajka, P., Gao, T., Hirschberger, M. et al. Oscillations of the thermal conductivity in the spin-liquid state of α-RuCl3. Nat. Phys. 17, 915–919 (2021).
[10] Peter Czajka et al., The planar thermal Hall conductivity in the Kitaev magnet α-RuCl3, arXiv:2201.07873
[12] 平林 良次、単結晶作成とその装置

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