Topological and time-crystalline order in chiral nematic liquid crystals

Liquid crystals have proven to provide a versatile experimental and theoretical platform for studying topological objects such as vortices, skyrmions, and hopfions. In parallel, in hard condensed matter physics, the concept of topological phases and topological order has been introduced in the context of spin liquids to investigate emergent phenomena like quantum Hall effects and hightemperature superconductivity. We bridge these two seemingly disparate perspectives on topology in physics. Combining experiments and simulations, we show how topological defects in liquid crystals can be used as versatile building blocks to create complex, highly degenerate topological phases, which we refer to as ‘Combinatorial Vortex Lattices’ (CVLs). CVLs exhibit extensive residual entropy and support locally stable quasi-particle excitations in the form of charge-conserving topological monopoles, which can act as mobile information carriers and be linked via Dirac strings. CLVs can be rewritten and reconfigured on demand, endowed with various symmetries, and modified through laser-induced topological surgery, an essential capability for information storage and retrieval and other applications.