# Research Overview

## Symmetry-preserving gapping of Weyl and Dirac semimetals

Dirac semimetals are 3D materials that have electrons that behave as if they are massless. A well-known 2D example is graphene. It is possible to make these electrons massive by subjecting the material to a magnetic field, or by proximity to a superconductor. This is because certain symmetries of the semimetal are broken. It is natural to ask if electronic states in semimetals can be made massive (gapped) without breaking any symmetries. In this work, we describe an exactly solvable model that makes these states massive without breaking any symmetries, by using ideas from topology and introducing many-body interactions.

Such a symmetry-preserving gapping is of interest as it leads to 3D topological order, exotic point-like and loop-like quasiparticle excitations which can be used for topological quantum computation. The methods and construction developed in this work would be useful tools for understanding many-body interactions in a whole range of 3D topological phases. There have been numerous field-theoretical discussions on possible properties of topologically ordered phases in 3D. However, unlike the 2D case there are no materials at all that exhibit topological order in 3D. Our work argues that an interacting Weyl or Dirac semimetal is a good candidate for realizing topologically ordered phases in 3D.

https://arxiv.org/abs/1711.05746

*[ppt presentation]** *[Poster pdf]

## Interacting Dirac nodal superconductors

The topological nodal superconductors are the superconducting analog of topological semimetals. The topological nodal superconductors possess nodal points or lines in the Brillouin zone, which have a vanishing superconducting gap. We consider the exactly solvable model of the Dirac nodal superconductor in the presence of many-body interactions. We construct various many-body interactions that result in a gapped phase. For odd copies of the Dirac nodes, we find a gapping potential that supports non-trivial topological order which can be used for topological quantum computation. More interestingly, when 16 copies of the nodal superconductor exist, we find a special form of interaction that trivially gaps out the system without supporting any non-local excitations.

## Classification of nodal band theories with combined symmetries

Band theories that describe electronic systems can be classified based on what symmetries they preserve. A well-known classification of topological phases of matter is the ten-fold way where the classification is based on the three non-spatial symmetries (time-reversal, particle-hole and chiral symmetries) in various dimensions [1]. In this work, we study nodal band theories which can be invariant under combined symmetries. We consider spatial lattice symmetries such as mirror, twofold rotation, and parity combined with one of the three non-spatial symmetries (time-reversal, particle-hole and chiral symmetries). For example, a system may not preserve a mirror or time-reversal symmetry individually but it can be invariant under a composition of these two symmetries and lead to a non-trivial topological phase. In this work, we systemically classify topological semimetals and nodal superconductors with different composite symmetries in any dimensions. We also discuss topological invariant characterizations as well as potential material realizations.

## Stacking disorder in Topological Insulators and Dirac/Weyl semimetals

Topological states were first understood using band theories, where lattice momenta are conserved. Subsequently, there have been theoretical studies on disordered topological states [1] but applications to materials have been relatively limited. In this work, we construct a model of stacked layers which have a semimetallic phase transition. We study the effects of stacking disorder on the phase transition. The goal is to understand stacking disorder in layered materials like Ge_{2}Sb_{2}Te_{5 }[2] which are used to make high-speed computer memories.

## Machine Learning topological phases

This is an ongoing project which involves applying machine learning (ML) techniques to better understand topological phase transitions in interacting topological systems. In recent years, ML has advanced significantly and machines can be trained to distinguish between pictures and identify patterns in large data sets. The question then arises if we can train machines to learn many-body wave-functions and distinguish between different quantum phases. Pioneering works in the last couple of years [1,2]have opened up the use of ML techniques in various fields of condensed matter theory, ranging from many-body interacting physics to topological phases. In this project, we use supervised learning to distinguish between disordered topological phases. The models we are currently working with are the recently identified topological phases in random matrix networks [3]. We use variational Monte-Carlo and train the machine using techniques such as ‘Quantum Loop Topography’ [4] to distinguish between disordered topological phases.

## Coupled-wire models of 3D non-Abelian Topological order

Topological order corresponds to patterns of long-range entanglement in the ground-states. The point-like and line-like excitations in 3D topologically ordered states can have fractional charge and spin degrees of freedom giving rise to fractional quantum statistics. In this work, we study 3D non-Abelian topological orders via coupled-wire constructions. By putting the wire configurations on closed manifolds, we study the properties of ground states and the Wilson algebra of various excitations. We study topological order of previously proposed coupled-wire models of symmetry-preserving gapped Dirac semimetal and Dirac superconductor. We also study new non-Abelian topologically ordered states that inherit their topological properties from conformal field theories like the SO(3)_3 case.