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Thermal and Electrical Transport in Correlated Quantum Materials

Periodic Reporting for period 2 - TRANSPORT (Thermal and Electrical Transport in Correlated Quantum Materials)

Reporting period: 2021-07-01 to 2022-12-31

The last decade has witnessed a revolution in the prediction and understanding of new phases of matter characterized by topology and/or entanglement. This includes topological insulators, semimetals, superconductors, quantum spin liquids, topological Kondo systems, etc.. However, most of the theory is directed at the classification of ground state properties and elementary excitations, and most successful experiments are spectroscopic.

The big frontier is transport. The latter is the key for nearly all applications in condensed matter. It is also the most sensitive probe, and gives access to the lowest frequency, lowest energy properties of materials.

Yet its understanding remains much more primitive than that of various spectroscopies, at a time when the body of transport experiments in extreme regimes is rapidly growing. Responding to the need to tackle this problem extensively, this ERC aims to provide a broad and detailed understanding of heat and electrical transport in a wide variety of quantum systems in different regimes: quantum magnets at low and high temperatures, near critical points, and semimetals. It will provide key tools and guidelines to study and understand transport in experiments.

The ERC focuses on two goals that are most direcly needed for the understanding of transport in modern quantum materials: (i) Developping the theory of thermal conductivity in complex magnets, on several fronts, both in quantum spin liquids and more generally at high temperatures and quantum critical points, (ii) Investigating transport in novel conducting and superconducting states with controllable topology and entanglement induced by spontaneous symmetry breaking phenomena in correlated systems --- topics at the frontier of topological metals and semimetals.
Achieving these goals will enable informed interpretation of a growing body of modern experiments on highly entangled states.

The work adds to fundamental theory, develops new tools and methodologies, and forges ties to specific experiments, laying the groundwork for future applications. It trains students and postdoctoral researchers in cutting-edge theory and the art of applying it to quantum materials.
- In a set of substantial manuscripts, we have achieved a substantial fraction of the proposed work in the first part of the ERC (referred to as (i) above):

In most solids, heat is carried by phonons, which are the elastic waves in the lattice. The speed of heat transport is typically controlled by scattering. We have shown that phonons scattering off of electronic degrees of freedom can induce novel phenomena like thermal Hall effect when phonons scatter more "to the left" than "to the right" or vice versa.

We have provided a very general formulation to calculate the scattering of phonons with an arbitrary quantum degree of freedom Q, and from it we derived the consequences on the thermal conductivity tensor of the phonons, i.e. both its longitudinal and Hall components. This should apply to any material—conductor, insulator, magnet etc—for any correlations of the degrees of freedom (the Q may in particular be a strongly interacting field), provided only that the material is not too disordered. A central result is that the thermal Hall conductivity is proportional to a four-point correlation function of Q, which we gave explicitly. This shows how chiral/handed scattering probes highly non-trivial structure of correlations.

As an illustration of the method, we applied our results to the case where the fluctuating field Q arises from magnon excitations of an ordered antiferromagnet. For a reasonable set of parameters, we found that the Hall angle can be of the order of magnitude of that in systems where it is lauded as "large", and obtained various power-law regimes of the longitudinal and Hall thermal conductivities.

- In a related project we have provided methodology, symmetry constraints etc to understand the phonon Hall viscosity, which is another mechanism through which a nonzero thermal Hall effect can appear, and shown that its size will generally be small compared with experimental measurements.

- We have also investigated how unusual magnetic phases could arise in several materials, both two- and three-dimensional, and with spin-1 and spin-1/2. In the first case, we showed that the phase diagram for a small set of parameters was particularly rich and in particular observed that a macroscopically-degenerate phase arose at the mean-field level, which could signal a quantum spin liquid in a more "quantum" calculation. Developing new methodology, we also derived the neutron scattering excitation spectra, to which experiments can be directly compared. In the second case, we showed that an antisymmetric (Dzyaloshinskii-Moriya) interaction arose in the half-Heusler compounds and contributed to the appearance of noncollinear/noncoplanar phases, which we expect will yield an anomalous Hall effect in the compounds in this class where there exist itinerant electrons.

- We have established unbiased methodology to carry out renormalization group calculations in flat band systems, and applied it to twisted bilayer graphene (TBG), where a number of correlated phases have been shown to appear. We found that at half-filling TBG ought to favor nematic order, consistent with experimental observations, and moreover showed that our procedure was "controlled", a rare occurrence of such calculations.
Team members have made headway in a number of directions, and progress beyond the state of the art: in particular, the novel methodology developed in the papers on thermal conductivity represent a major advance in the understanding of thermal conductivity in insulators, and as we discovered, a way to probe non-Gaussian fluctuations which are otherwise not directly accessible in experiment.
We have also discovered a new macroscopically-degenerate phase in a "simple" model of spin-1 on the triangular lattice, paving the way for the investigation of new material candidates for quantum spin liquids.
The determination of the ingredients necessary for the appearance of non-collinear magnetic phases in a large class of materials has allowed us to pursue know where to look for a complex-magnetism-driven anomalous Hall effect in these compounds, and more generally what the conditions might be for the appearance of the latter.
Finally, given the growing number of flat band materials grown in the laboratory and the flurry of theoretical predictions associated with the latter, we expect the novel methodology we established for renormalization group calculations in such a setting to inevitably prove particularly useful. The "emergent" "controlled" nature of the approach is a rare occurrence which appeared as a surprising byproduct of our detailed calculations.

By the end of the project, I expect that almost all items covered in the proposal will have been achieved, despite the fact that results on topics not described in the proposal were also obtained, as new topics emerged in the community and as our research opened new promising avenues for discoveries.