Anomalous spin dynamics in low-dimension: superdiffusion, subdiffusion, and solitons

In Part I of this thesis we examine solitons – local, non-dissipative excitations – in the dynamics of spin systems. We open, in Ch. 1, with a short account of the history of solitons, from their first observation, to the theories of shallow water and the Korteweg-De Vries model; their appearance in field theories like the sine-Gordon model; to the general description of integrable systems, such as the Toda lattice. We pay particular attention, of course, to solitons in spin models – especially those obtained by Ishimori in an integrable classical spin chain which bears his name. In Ch. 2 we present our work which establishes the existence of solitons in non-integrable spin chains. We begin by constructing exact static solitons in the Heisenberg chain, which we connect to the static Ishimori solitons via an adiabatic interpolation. We then use this adiabatic transform to construct moving Heisenberg solitons, which show no sign of having a finite lifetime. We further show that the interactions of these solitons are remarkably similar to the integrable case, and we establish their presence in low temperature thermal states – which will have important consequences in Part II. Ch. 3 considers a different set-up, where we study the dynamics of domain walls in anisotropic spin chains. Our work shows a striking co-existence of linear and non-linear phenomena – to wit, we show that the free propagation and subdiffusive spreading of domain walls can be captured by non-interacting, linear spin wave theory; but that these domain walls are unstable to decay via the emission of topological solitons. In Part II we will show how the solitons we have discovered play a hydrodynamic role, and find that superdiffusion, far from being limited to the special cases where the model is integrable, may be observed in non-integrable spin chains for (arbitrarily) long times, at low – but non-zero – temperatures. We will, however, preface this with a review of the literature on superdiffusion