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New paradigms for correlated quantum matter: Hierarchical topology, Kondo topological metals, and deep learning

Periodic Reporting for period 2 - PARATOP (New paradigms for correlated quantum matter:Hierarchical topology, Kondo topological metals, and deep learning)

Reporting period: 2019-07-01 to 2020-12-31

We study quantum materials with properties that were previously unknown. Understanding quantum materials is of foundational importance for the future developments of in electronics, be it for data storage, electronic chip architectures, sensors, quantum computers and more. In recent years, materials with so-called topological properties came into the focus of research. They are characterized by stable conducting states on their boundary (edge or surface), while their interior is insulating. Due to these properties, they are envisioned for low-power electronic devices and for various applications in metrology.

In PARATOP we study theoretically materials with qualitatively new behaviors. One of the main new directions are so-called higher order topological insulators, which are three-dimensional crystals that are insulating in the bulk, on the surface, but not on the hinges. Rather, on the crystal hinges we find one-dimensional conducting channels. These channels have the special property that current running through them is dissipationless, that means no power ist lost when the current flows. Such channels may be useful in future ultra-low-power electronics. Aside from predicting new materials with these intriguing properties, develop numerical methods to study the hardest problems of crystals with many strongly interacting electrons. To be able to compute their properties with unprecedented precision, and to predict quantum phase diagrams, we employ neural networks. The neural network represents the quantum mechanical state of the system.
For higher-order topological insulators, we developed a mathematical framework that allows to predict which materials have this property. Based on our prediction we found that elementary bismuth is an iconic representative of this new phase of matter. Together with experimental groups, we collected evidence that our prediction is actually true. From this starting point, we developed the concept of higher-order topology in many further directions. Among others, we found criteria under which two-dimensional crystals (like graphene) have corner charges (the lower dimensional version of the conducting hinge modes). Furthermore, we studied higher-order topological superconductors, which have corner or hinge modes that are called Majorana fermions and may be relevant for building quantum computers. These developments are flanked by many other subprojects and methodological progress. Most prominently, we work on a code package called Wannier Berri, which allows to calculate response functions of real materials, based on band structure calculations, with extremely high precision. We tested the code on several occasions in collaboration with experimental groups or when we predict new topological quantum materials.

For the numerical investigation of strongly interacting phases of matter, we have mostly focused on code development and benchmarking of the method of neural network quantum states. In that way, one of the PARATOP team members, Kenny Choo, contributed to NetKet, the most prevalent software package in this filed. The results are very promising. We tested the new method on 1D systems, and on the 2D J1-J2 Heisenberg model on the square lattice, which is a iconic model of frustrated magnetism. We found that ground state energy and low-lying excitation energies with new records in numerical precision. More broadly, we studied various applications of machine learning technology to condensed matter physics.
All the topological material properties that we studied so far assumed that the energy is conserved in the system, in which case it is mathematically described by a Hermitian operator. Going forward, we plan to extend the new topological concepts to the domain of non-Hermitian topology. This concerns the description of lossy systems, either trough coupling to some environment or materials in which quasi-particles have a finite lifetime due to interactions. In this case, completely new mathematical structures and physical concepts emerge. We want to study the fate of higher-order and related topology under this new paradigm.

On the forefront of neural network quantum states, we are ready to take the next step and explore systems which have not been studied with comparably powerful methods before, because of computational limitations. In particular, we want to look at 3D frustrated spin systems, for which very few numerically unbiased results exist.

Aside from these two major directions, we will continue to improve our understanding of topological matter in various respects, and improve our code package Wannier Berri.
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