# Topological phases by momentum space braiding

Lead Research Organisation:
University of Cambridge

Department Name: Physics

### Abstract

The development of the theory of quantum mechanics entails one of the most influential achievements in Physics and, arguably, science as a whole. Standing at the basis of the field of condensed matter, one of its earlier triumphs was to shed light on the question why some materials behave as insulators, while others exhibit metallic properties. Upon utilising the wave interpretation of particles, it was readily found that electrons in a periodic potential give rise to energy bands; the spectrum shows bands of continuous energy levels separated by gaps. Hence, filling up the 'Fermi Sea' such that an integer amount of bands are filled ensures that there is an excitation gap, and thus insulating behavior up to that energy scale, whereas filling a band fractionally gives rise to a metallic phase.

Although band theory has been extraordinarily successful, it has been reinvigorated the past years due unexpected connections with the mathematical domain of topology. Topology in essence characterises properties of objects that are preserved under smooth deformations. This is usually exemplified by the topological equivalence of a coffee cup and doughnut. Without tearing or poking holes one can be deformed into the other and their general class may be quantified in terms of a so-called invariant, being the integer that counts the number of holes. Rather remarkably, this principle has been found to be of pivotal importance in phases of electronic matter, where the wavefunctions can tie distinctive collective knots, topologically distinguishing different classes of insulators and metals. These topological insulators and metals are not only appealing from a purely theoretical point of view, but also exhibit remarkable physical phenomena such as protected metallic edge states that could shape next-generation electronics, or excitations that can store quantum information, making topological materials a potentially key component of quantum computing platforms.

While the impact of topological materials has been underpinned by a vast research interest and a rapid advancement of the field, it was discovered the past year that a whole new class of topological metals exists. These systems feature bands that are gapped everywhere except for special points at which bands pairwise touch. The resulting band nodes furthermore carry exotic kinds of topological charges that can be altered in a highly non-trivial manner. Namely, when such nodes between different sets of bands are braided along each other in momentum space, their charges are converted, inducing specific phase factors in the collective wave function that cannot be untangled. As a result, a new topological structure emerges that can be quantified by a novel type of invariant, known as Euler class. There are however clear indications that these results comprise the tip of the iceberg and that a whole new class of such Euler metals exists, especially when other crystalline symmetries are present that enforce new conditions on the topological classification. This programme aims to exploit these timely indications and investigate these new exciting forms of matter. This articulates around three main pillars that aim to (i) advance the theoretical understanding of these Euler phases, (ii) uncover their physical properties and (iii) design concrete pathways to bring them to the experimental domain. For the latter objective this includes an explicit integration of experimental and ab-initio project partners, with whom we intend to foster long-term alliances, thereby creating a strong programme in the prominent field of topological materials.

Given the strong indications that these new Euler phases host exotic physical properties that, apart from their immense scientific potential, could culminate impact future technologies, we anticipate that this programme will generate profound impact, thereby further underpinning the strong research position of the UK.

Although band theory has been extraordinarily successful, it has been reinvigorated the past years due unexpected connections with the mathematical domain of topology. Topology in essence characterises properties of objects that are preserved under smooth deformations. This is usually exemplified by the topological equivalence of a coffee cup and doughnut. Without tearing or poking holes one can be deformed into the other and their general class may be quantified in terms of a so-called invariant, being the integer that counts the number of holes. Rather remarkably, this principle has been found to be of pivotal importance in phases of electronic matter, where the wavefunctions can tie distinctive collective knots, topologically distinguishing different classes of insulators and metals. These topological insulators and metals are not only appealing from a purely theoretical point of view, but also exhibit remarkable physical phenomena such as protected metallic edge states that could shape next-generation electronics, or excitations that can store quantum information, making topological materials a potentially key component of quantum computing platforms.

While the impact of topological materials has been underpinned by a vast research interest and a rapid advancement of the field, it was discovered the past year that a whole new class of topological metals exists. These systems feature bands that are gapped everywhere except for special points at which bands pairwise touch. The resulting band nodes furthermore carry exotic kinds of topological charges that can be altered in a highly non-trivial manner. Namely, when such nodes between different sets of bands are braided along each other in momentum space, their charges are converted, inducing specific phase factors in the collective wave function that cannot be untangled. As a result, a new topological structure emerges that can be quantified by a novel type of invariant, known as Euler class. There are however clear indications that these results comprise the tip of the iceberg and that a whole new class of such Euler metals exists, especially when other crystalline symmetries are present that enforce new conditions on the topological classification. This programme aims to exploit these timely indications and investigate these new exciting forms of matter. This articulates around three main pillars that aim to (i) advance the theoretical understanding of these Euler phases, (ii) uncover their physical properties and (iii) design concrete pathways to bring them to the experimental domain. For the latter objective this includes an explicit integration of experimental and ab-initio project partners, with whom we intend to foster long-term alliances, thereby creating a strong programme in the prominent field of topological materials.

Given the strong indications that these new Euler phases host exotic physical properties that, apart from their immense scientific potential, could culminate impact future technologies, we anticipate that this programme will generate profound impact, thereby further underpinning the strong research position of the UK.

### Publications

Bennett D
(2023)

*Polar meron-antimeron networks in strained and twisted bilayers.*in Nature communications
Bennett D
(2023)

*Theory of polarization textures in crystal supercells*in Physical Review Research
Hamara D
(2023)

*Ultrafast helicity-dependent photocurrents in Weyl Magnet Mn3Sn*in Communications Physics
Lange G
(2023)

*Spin texture as a bulk indicator of fragile topology*in Physical Review Research
Park H
(2022)

*Topological phase transitions of non-Abelian charged nodal lines in spring-mass systems*in Physical Review B
Rüegg L
(2023)

*Pairing of composite electrons and composite holes in ? T = 1 quantum Hall bilayers*in Physical Review ResearchDescription | New direction on polar sturctures. |

Organisation | Harvard University |

Country | United States |

Sector | Academic/University |

PI Contribution | With Dr Chaudhdary and myself we have started to work on topological polar structures. This has resulted already in a preprint https://arxiv.org/abs/2210.10786 that has been accepted by Nature Communications. We are now pursing new topological directions as predicted in the proposal. This collaborative efforts involves a PhD in my group, Mr Jankowski, A postdoc at Harvard, Dr Benner, and Prof Ghosez. |

Collaborator Contribution | These parters are experts in DFT, while we ad theorists describe the theory. As such this is a good example of a nice symbiosis. |

Impact | https://arxiv.org/abs/2210.10786 |

Start Year | 2022 |