Typical thermodynamic systems have a small number of minimum-energy ground states into which the system strives to arrange itself when the temperature is sufficiently low. In some cases, however, competition between interactions leads to the suppression of conventional long-ranged magnetic ordering, through a mechanism known as frustration. One of the typical features is the appearance of a large number of degenerate (or nearly degenerate) minimal energy states, that can exhibit topological order. The dynamical properties of topological systems, in particular those that host fractionalised excitations, is currently a topic of intense research activity.

This thesis focuses on one specific instance of a topological frustrated magnetic system: spin ice. Spin ice is one of the most thoroughly researched frustrated models, exhibiting exciting phenomena typical of topological matter – including an emergent gauge field description with fractionalised excitations that take the form of emergent magnetic monopoles. Highly accurate numerical methods and analytical techniques have been developed to model spin ice materials, and have facilitated a detailed understanding of their static properties.

The primary aim of this thesis is to address some of the outstanding questions about the dynamics of classical spin ice materials. The focus will be on two dynamical properties: the magnetic relaxation time and the magnetic noise spectrum. Several independent experiments, some dating back over two decades, have found that the magnetic relaxation time in these materials diverges rapidly upon cooling – more rapidly than what any previous theory can explain without invoking extrinsic contributions such as surface effects, disorder, and temperature dependent microscopic time scales. Previous theories also predict that the magnetic noise spectrum of spin ice should be Lorentzian, with a power spectral density that decays as ν^(-2) with the frequency, ν. Here, noise spectra measured on a single crystal of Dy₂Ti₂O₇ are presented and shown to instead decay as ν^(−α), with an anomalous exponent α ≈ 1.5. Through a combination of extensive numerical modelling and fundamental arguments about the motion of the magnetic monopoles, I establish that the conventional model of spin ice dynamics cannot explain the anomalous magnetic noise.

I resolve these issues by introducing a new model for the dynamics, which includes the effects of local symmetry on the flip rate of spins. The main insight the new model leads to is the emergence of dynamical fractal structures on which the magnetic monopoles are constrained to move. It is by hosting the monopole motion that the fractals bequeath the magnetic noise with an anomalous exponent and cause a slow-down of the relaxation. The spatial properties of the fractals thus manifest themselves in the time dependence of the magnetisation – and the anomalous dynamics observed in Dy₂Ti₂O₇ are evidence of fractal objects in a disorder-free, bulk crystal. Indeed, the new model dynamics quantitatively captures the experimentally observed noise spectrum and relaxation times, without the introduction of any additional fitting parameters.

The dynamical fractals cannot be imaged directly, but do influence other dynamical properties, besides the already discussed magnetic noise and relaxation time. Using simulations and simplified models of analogous systems, I investigate how the fractals affect both equilibrium and out-of- equilibrium transport properties. The response of spin ice to an oscillating magnetic field is identified as one particularly promising area to explore in future experiments, with suggestions made for the field and temperature regimes in which fractals are likely to manifest themselves.

An alternative spin ice model capable of reproducing the experimentally observed relaxation time and anomalous magnetic noise, but incompatible with the thermodynamics of spin ice materials, is also introduced. The model includes a specific choice of third-nearest-neighbour exchange interactions, which are frustrated independently of the usual spin ice interactions. These additional interactions are found to induce a phase transition between the normal spin ice phase and a spin nematic phase – a phase displaying rotational symmetry breaking without long-ranged order. In the “nematic spin ice” phase, low energy paths are formed along which magnetic monopoles move freely. These paths form anisotropic fractal clusters with properties akin to the dynamical fractals mentioned previously. Thus, the anomalous noise spectrum is again found to originate from the motion of magnetic monopoles on fractals.