Weyl points on non-orientable manifolds

TL;DR

This paper explores how Weyl points behave on non-orientable manifolds, revealing new topological invariants and circumventing traditional no-go theorems, with experimental realization in photonic systems.

Contribution

It introduces the concept of Weyl points on non-orientable manifolds, showing they can have non-zero total chirality and carry a new $ ext{Z}_2$ invariant, challenging existing topological constraints.

Findings

Weyl points on non-orientable manifolds can have non-zero total chirality.

A new $ ext{Z}_2$ topological invariant characterizes these Weyl points.

Experimental realization demonstrated in a photonic platform.

Abstract

Weyl fermions are hypothetical chiral particles that can also manifest as excitations near three-dimensional band crossing points in lattice systems. These quasiparticles are subject to the Nielsen-Ninomiya "no-go" theorem when placed on a lattice, requiring the total chirality across the Brillouin zone to vanish. This constraint results from the topology of the (orientable) manifold on which they exist. Here, we ask to what extent the concepts of topology and chirality of Weyl points remain well-defined when the underlying manifold is non-orientable. We show that the usual notion of chirality becomes ambiguous in this setting, allowing for systems with a non-zero total chirality. This circumvention of the Nielsen-Ninomiya theorem stems from a generic discontinuity of the vector field whose zeros are Weyl points. Furthermore, we discover that Weyl points on non-orientable manifolds carry an additional $\mathbb{Z}_2$ topological invariant which satisfies a different no-go theorem. We implement such Weyl points by imposing a non-symmorphic symmetry in the momentum space of lattice models. Finally, we experimentally realize all aspects of their phenomenology in a photonic platform with synthetic momenta. Our work highlights the subtle but crucial interplay between the topology of quasiparticles and of their underlying manifold.