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3 Synthesis and Discussion Ashvin Vishwanath, Harvard University, built upon the plenaries with a presentation synthesizing recent work in thermal transport and energy conversion. He then moderated a discussion among speakers and other workshop participants to draw out and expand upon key insights, themes, and next steps. FRONTIERS IN THERMAL TRANSPORT AND ENERGY CONVERSION Vishwanath summarized key themes from the workshop plenaries, detailed an example of thermal transport from recent research, and explored the pairing of semimetals and thermoelectrics. The workshop talks explored different aspects of thermal probes to examine new states of matter; new regimes for thermal transport, including with non-quasiparticles; and approaches for new applications based on this science. Regarding this last point, Vishwanath noted the significant gap that currently exists between fundamental science and engineering in this field. He emphasized that closing this gap is essential to making progress, but requires not-yet-achieved milestones such as raising the figure of merit (ZT). Thermal Transport Vishwanath expressed his belief that thermal transport, specifically using the quantum Hall effect to probe non-Abelian quantum states, is the best means to probe exotic states of matter. These exotic phases are the âdark matterâ of condensed matter physics: their presence is strongly suspected, and there are indirect signatures that can be probed, but they canât yet truly be seen. Vishwanath posited that the quantum Hall effect is the best probing method, especially for topologically ordered states, because it was the first route by which these phenomena were discovered, and it is the only place currently where these exotic states are being realized. This method also holds promise for building a robust quantum computer, he noted. There is still work to be done before such applications are possible, however, including understanding the difference between integer and fractional results, which have different implications.1,2 When the thermal Hall effect result is a fraction, it is an especially interesting state because there are anyons, which are non-Abelian quasiparticle excitations. These states, amazingly, have been realized in nature, and there is even a quantum Hall state with an even denominator, 5/2.3 If this 5/2 state could be measured, it would be considered a âsmoking gunâ signature of a non-Abelian quantum Hall phase, Vishwanath said. Experiments in this realm are extremely difficult, but success with non-Abelian excitations would have a large payoff: the implementation of quantum computing. This particular implementation would be 1 K. v.Klitzing, G. Dorda, and M. Pepper, 1980, New method for high-accuracy determination of the fine- structure constant based on quantized Hall resistance, Physical Review Letters (45) 494, https://link.aps.org/doi/10.1103/PhysRevLett.45.494. 2 D.C. Tsui, H.L. Stormer, and A.C. Gossard, 1982, Two-dimensional magnetotransport in extreme quantum limit, Physical Review Letters (48) 1559, https://link.aps.org/doi/10.1103/PhysRevLett.48.1559. 3 W. Pan et al., 1999, Exact quantization of the even-denominator fractional quantum Hall state at Î½=5/2 Landau level filling factor, Physical Review Letters (83) 3530, https://link.aps.org/doi/10.1103/PhysRevLett.83.3530. PREPUBLICATION COPYâSUBJECT TO FURTHER EDITORIAL CORRECTION 21

robust to errors because it has topological protections from these nonlocal excitations and nonlocal information encoding. The environment, which usually destroys quantum coherence, in this case cannot destroy the excitationsâ coherence. A recent experiment was able to measure thermal Hall conductance, and the difference between the predicted integers and the resulting fractions showed some non-Abelian quantum Hall phases.4 Another experiment found non-Abelian quantum Hall phases in the 5/2 state.5 Both experiments required very low temperatures, which added to their difficulty, but the results are part of the mounting, if indirect, evidence that this is either the Pfaffian state, which has these non-Abelian quasiparticle excitations, or the anti-Pfaffian state. However, 5/2 also implies that there is some version of the Pfaffian state that is particle-hole symmetric. If this state were realized in an experiment, it would have the exact edge state of the thermal Hall contribution, but finding the regime that creates these results is still out of reach. Semimetals and Thermoelectrics Vishwanath posited that semimetals in a strong magnetic field should be examined to find better thermoelectrics. The thermoelectric properties of composite fermions have been studied, but there are still many unanswered questions, and there is great potential to use their thermoelectric properties to probe exotic states of matter, Vishwanath said.6,7 Nodal semimetals, which have a zero density of states at a particular value of chemical potential, appear to have interesting thermoelectric properties, including for power generation and refrigeration, but alone they are unlikely to have good thermoelectric properties. However, if they are placed in a strong magnetic field, approaching the quantum limit, finite thermal power is possible. Converting this energy results in a very large ZT, creating the potential for new applications, Vishwanath suggested. In response to questions, Vishwanath clarified that the only thing that contributes to thermopower is the charged modes at the edge. Also, the non-Abelian state has a robust result of Â½ because it depends on which non-Abelian state is present. Other states will have other results. For example, 12/5 is suspected to result in Fibonacci anyons. Realizing that phase would be very exciting because, unlike the Pfaffian state or Majorana fermions, it could implement all the gates needed for universal quantum computing. First, however, the thermal Hall effect must be proven to be powerful, accurate, and able to distinguish between integers and fractions. Ong added that his group is studying thermopower of fractional quantum Hall states at very low temperatures in an effort to obtain results at 5/2, but this experiment is not finished. Finally, in response to a question about quasicrystals, Vishwanath agreed that there appears to be a connection, but it is not yet clear if it is beneficial. Quasicrystals also have symmetry, but it is unlikely that they would be useful in these experiments, he added. 4 M. Banerjee et al., 2018, Observation of half-integer thermal Hall conductance, Nature (559) 205-210, https://doi.org/10.1038/s41586-018-0184-1. 5 C. Wang, A. Vishwanath, and B.I. Halperin, 2018, Topological order from disorder and the quantized Hall thermal metal: Possible applications to the Î½=5/2 state, Physical Review B (98) 045112, https://link.aps.org/doi/10.1103/PhysRevB.98.045112. 6 A.C. Potter, M. Serbyn, A. Vishwanath, 2016, Thermoelectric transport signatures of Dirac composite fermions in the half-filled Landau level, Physical Review X (6) 031026, https://link.aps.org/doi/10.1103/PhysRevX.6.031026. 7 D.T. Son, 2015, Is the composite fermion a Dirac particle? Physical Review X (5) 031027, https://link.aps.org/doi/10.1103/PhysRevX.5.031027. PREPUBLICATION COPYâSUBJECT TO FURTHER EDITORIAL CORRECTION 22

KEY PROBLEMS AND NEXT STEPS Vishwanath asked the workshop speakers to comment on which problems in thermal transport and energy conversion they would most like to see solved. Speakers pointed to a need for better understanding of the thermal Hall effect, phonons, temperature effects, and materials. Vishwanath started the discussion by stating that creating criteria to maximize ZT would be useful for various applications. He added that physics research should be used to improve materials characteristics and find new materials for improved technology. Behnia stated that many aspects of the thermal Hall effect (Îºxy) are still mysterious. Recent studies have demonstrated surprising results with cuprates, Mott insulators, and multiple aspects of Îºxy, all of which were once thought to be well understood. Kapitulnik agreed, adding that the specific classes of materials that can produce Îºxy are unknown. Ong briefly reviewed the case for a phonon-derived Îºxy. An old Grenoble experiment reported the detection in a non-magnetic insulator of a very small thermal Hall conductivity which the authors identified with phonons. The signal is many orders weaker than the results obtained on topological quantum magnets. Ong, Heremans, Behnia, and Vishwanath agreed that finding an insulator with no magnetism that still displays thermal Hall conductivity would be very important. A member of the audience asked if phonons in InSb, which have a thermal magnetoresistance, should also have a thermal Hall effect. Savary answered that for there to be any Hall effect, there has to be a transverse force on the heat carriers, and there is no mechanism that is known to exert such a force on phonons in InSb. A workshop participant noted that semimetals are dominated by electronics, usually at low temperature. In certain experiments, the phonons combined with cyclotron resonance can become circularly polarized, leading to a thermal Hall effect, although it is hard to discern it from the electronic signal, and it is important to determine a sign to know what direction the Hall current is going to go, for example by seeing the result from breaking an electron-hole symmetry. Ong noted that in the superconducting state, the thermal Hall effect comes from Bogoliubov quasiparticles, which are linear combinations of holes and electrons. In a superconductor the thermal Hall effect arises from skew scattering of quasiparticles from vortices. The phonon Hall effect in an insulator may arise from a very weak skew-scattering. However, the process that determines the sign of the phonon Hall effect (if it exists) is not known, he said. Hartnoll asked about searching for strong violation of the Wiedemann Franz (WF) ratio in underdoped cuprates in measurements of Îºxy extended to temperatures much higher than in a previous cuprate experiment.8 There, the WF ratio in an underdoped cuprate was determined by comparing Îºxy and the electrical Hall conductivity (this avoids the phonon contribution). Ong replied that the thermal Hall signals in cuprates are stronger than in the QSL candidates he described. He may consider revisiting his old experiments on Kxy if there is sufficient theoretical motivation and interest. Hartnoll confirmed that recent work posits that if there is scaling, the Lorenz ratio should be independent of temperature because there is a âT.â If it is not constant, then it is nontrivial, but a bigger regime for scaling is still needed. However, there are many unanswered questions, such as the timescale for thermal diffusivity, results at higher temperatures, and T-linear resistivity. Kapitulnik pointed out that we are only just now beginning to realize the fact that materials, such as lead telluride, can have both classical and quantum properties, and what the implications of that are. Heremans agreed and noted that experiments with lead telluride had particularly strong results, perhaps because its anharmonic scattering is very strong. Behnia also agreed, noting that strontium titanate has a much higher thermal conductivity, and added that these are well-known problems that still lack a good explanation. 8 Y. Zhang, N. P. Ong, P. W. Anderson, D. A. Bonn, R. Liang, and W. N. Hardy, "Giant Enhancement of the Thermal Hall Conductivity kxy in the Superconductor YBa2Cu3O7," Phys. Rev. Lett. 86, 890 (2001). PREPUBLICATION COPYâSUBJECT TO FURTHER EDITORIAL CORRECTION 23

Savary agreed with Behnia that thermal Hall conductivity has not been studied enough at the theoretical level. QSL phases, if achieved, could be used in actual quantum computing applications. Heremans reiterated that thermal Hall is an extraordinarily difficult measurement. Not only is the theory not fully developed, but the technology is very difficult to master. However, he believes there is a solution, likely involving circulation and spin. He suggested that we may see a new principle to define it within five years. ADDITIONAL CONSIDERATIONS As the workshop drew to a close, participants were encouraged to ask questions or contribute new ideas. Topics ranged from chemistry to experimental challenges to real-world applications. One participant asked if including comparative electronegativity would give any insight into the spin-orbit coupling. Vishwanath noted that that is more chemical detail than physicists usually work with, although ab initio calculations of band structures are performed, but it is unclear how accurate the comparisons are. Heremans added that anharmonicity is also important, because the Born effective charge is critical to anharmonicity and shows up in thermal conductivity at high temperatures. Another participant asked Ong to elaborate on the difficulties of thermal Hall effect experiments. Ong replied that the main difficulty is cleaving in the crystals, which can ruin an experiment. RuCL3, for example, cleaves very easily and, as a result, Ongâs team is having difficulty reproducing the Kasahara data. Heremans shared that copper has a very high Îºxy that can leak into the sample from instrumentation. Also, the thermometers used are larger than the samples themselves, and can create heat leaks. Another participant asked how much overlap there was between the quasiparticle picture and the proposed hydrodynamic regime. Hartnoll noted that there was some overlap but stressed that they are not the same. Some experiments are trying to show that the hydrodynamic flow in graphene is with quasiparticles. These two can co-exist, which is not true of a ballistic regime. Behnia agreed that they are related but stressed that there is not one answer to both of them. The participant then asked if anharmonicity should be a consideration when making thermally conductive materials, given its ability to create strong interactions and strong ZT. Behnia and Heremans agreed that it should be considered to the extent possible, but there are other considerations as well. Another participant asked if there were good ways of measuring anharmonic phonon effects, and Raphael Hermann, Oak Ridge National Laboratory, described several methods to measure anharmonicity in materials with low thermal connectivity. A participant pointed out that these were indirect probes. Behnia agreed but added that they still work in most insulators, and Heremans noted that this measurement is under good control theoretically and experimentally. Vishwanath asked participants to share ideas for turning the fundamental aspects discussed, such as wave propagations and the nonlinear thermal transport coefficient, into technologies. Heremans believes that there are potential uses for thermal heat switches and rectifiers using thermal heat conductance. In principle, control over the thermal conductors enables the creation of a heat engine, which could have multiple applications for energy conversion. Kapitulnik shared that his lab is working with materials, at non-cryogenic temperatures, that violate the WF law in the largest way yet seen. Next, they will attempt to find the phenomenon associated with strong electron-phonon interactions. Phase transitions might be promising, Heremans noted, but they fix the temperature at which you work, whereas electric diodes work at any range of voltages. Underscoring the importance of temperature control, Ong pointed out that if something could be designed that allows phonon conductivity in only one direction, that could be very helpful. Another participant suggested using state-of-the-art 3D printing to create custom materials for use in these experiments, and Vishwanath agreed that a certain level of material customization and control is very important. PREPUBLICATION COPYâSUBJECT TO FURTHER EDITORIAL CORRECTION 24