Reducing the dimensionality of a physical system can have a profound effect on its properties, as in the ordering of low-dimensional magnetic materials, phonon dispersion in mercury chain salts, sliding phases, and the electronic states of graphene. Here we explore the emergence of quasi-one-dimensional behaviour in two-dimensional artificial spin ice, a class of lithographically fabricated nanomagnet arrays used to study geometrical frustration. We extend the implementation of artificial spin ice by fabricating a new array geometry, the so-called tetris lattice. We demonstrate that the ground state of the tetris lattice consists of alternating ordered and disordered bands of nanomagnetic moments. The disordered bands can be mapped onto an emergent thermal one-dimensional Ising model. Furthermore, we show that the level of degeneracy associated with these bands dictates the susceptibility of island moments to thermally induced reversals, thus establishing that vertex frustration can reduce the relevant dimensionality of physical behaviour in a magnetic system.

Systems of interacting nanomagnets known as artificial spin ice 1-4 have allowed the design, realization and study of geometrically frustrated exotic collective states 5-10 that are absent in natural magnets. We have experimentally measured 11,12 the thermally induced moment fluctuations in the Shakti geometry of artificial spin ice. We show that its disordered moment configuration is a topological phase described by an emergent dimer-cover model 13 with excitations that can be characterized as topologically charged defects. Examination of the low-energy dynamics of the system confirms that these effective topological charges have long lifetimes associated with their topological protection, that is, they can be created and annihilated only as charge pairs with opposite sign and are kinetically constrained. This manifestation of classical topological order 14-19 demonstrates that geometrical design in nanomagnetic systems can lead to emergent, topologically protected kinetics that can limit pathways to equilibration and ergodicity.