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 in integrable spin chains in Ch. 4. Ch. 5 presents our work on the existence of superdiffusion in non-integrable spin chains – with a particular focus, again, on the classical Heisenberg chain. We show that the Heisenberg chain exhibits long-lived superdiffusion of spin – with a striking scaling collapse of the correlation function onto the KPZ function across three decades of time at low temperature – but only ordinary diffusion of energy. We present an argument that explains this phenomenology in terms of the solitons we established in Part I. Further, we examine how the time-scales and temperature-scales of superdiffusion depend on the degree of integrability breaking, by considering the model which interpolates between the Ishimori and Heisenberg chains (and which built the solitons of Ch. 2); and, furthermore, show examples of other non-integrable spin chains evincing the same spin superdiffusion at low temperatures. We turn, in part III, to the opposite kind of anomalous dynamics – subdiffusion. We briefly survey this type of slow dynamics in Ch. 6, describing various mechanisms by which it can arise, including kinetic constraints, disorder, and higher-moment (e.g., the dipole moment) conservation of some charge density. Ch. 7 contains our work on bond-disordered classical Heisenberg chains; the main contribution here is that we provide an interacting model with a continuously tune-able subdiffusive exponent, which we obtain analytically from a related, solvable phenomenological model. This also allows us to obtain the leading corrections to the asymptotic behaviour, clarifying the role of large sub-leading terms in hydrodynamic transport. Now, Parts I – III of this thesis are concerned either with the structure of single excitations above the ground state – an effectively zero temperature regime – or the dynamics of the spins in thermal equilibrium, finding anomalous hydrodynamics both faster and slower than ordinary diffusion. In Part IV, however, we will forswear the canonical ensemble entirely. In Ch. 8, we study the classical version of a boundary-driven quantum spin chain which was the subject of recent experiments by Google Quantum AI. We show that the observed dynamical regimes are not inherently quantum-mechanical, since the classical variant evinces the entire phenomenology observed in the quantum experiments. Moreover, we show that the classical chain is analytically tractable, and that, depending on the degree of anisotropy, either ballistic transport, subdiffusion, or localisation may be found. We then go beyond the direct comparison with the quantum version and introduce quenched random couplings to the classical model. We find, most strikingly, that the ballistic transport regime survives, so long as the disorder is not strong enough to completely sever the chain. We further show how, if we do allow for very strong disorder, different subdiffusive exponents may be obtained. In Ch. 9, we address the consequences of non-reciprocal interactions – in essence, an evasion of Newton’s third law – in periodically driven systems. This question emerges from the spin dynamics studied in the previous parts of this thesis because one of the main numerical methods we have used to calculate the time evolution is, intrinsically, a non-reciprocal periodic drive. Whilst Floquet theory – the study of periodically-driven Hamiltonian systems – is by now a well-developed field, non-reciprocal systems cannot be described by any Hamiltonian, time-dependent or static, and so the techniques of Floquet theory do not, a priori, apply. The high-frequency regime of Floquet systems typically features long-lived meta-stable (prethermal) states, which has allowed the techniques of Floquet-engineering to produce novel prethermal phases of matter which have no equilibrium counterpart – but the theorem which establishes the prethermal plateau explicitly uses the Hamiltonian formalism. Nevertheless, by combining the ingredients of non-reciprocity and periodic driving in the context of many-body spin dynamics, we uncover a new class of long-lived prethermal states – independently of dimensionality, support of interactions, or lattice geometry – indicating that non-reciprocal systems may offer a propitious arena to generate new material properties via Floquet-engineering