京都大学 / 新制・課程博士 / 博士(理学) / 甲第22984号 / 理博第4661号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 藤 定義, 教授 佐々 真一, 教授 早川 尚男 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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.

No description available.

We study anomalous diffusion on fractals with a static external field applied. We utilise the master equation to calculate particle distributions and from that important quantities as for example the mean square displacement <r^2(t)>. Applying different bias amplitudes on several regular Sierpinski carpets we obtain maximal drift velocities for weak field strengths. According to <r^2(t)>~t^(2/d_w), we determine random walk dimensions of d_w<2 for applied external fields. These d_w corresponds to superdiffusion, although diffusion is hindered by the structure of the carpet, containing dangling ends. This seems to result from two competing effects arising within an external field. Though the particles prefer to move along the biased direction, some particles get trapped by dangling ends. To escape from there they have to move against the field direction. Due to the by the bias accelerated particles and the trapped ones the probability distribution gets wider and thus d_w<2. / In dieser Arbeit untersuchen wir anomale Diffusion auf Fraktalen unter Einwirkung eines statisches äußeres Feldes. Wir benutzen die Mastergleichung, um die Wahrscheinlichkeitsverteilung der Teilchen zu berechnen, um daraus wichtige Größen wie das mittlere Abstandsquadrat <r^2(t)> zu bestimmen. Wir wenden unterschiedliche Feldstärken bei verschiedenen regelmäßigen Sierpinski-Teppichen an und erhalten maximale Driftgeschwindigkeiten für schwache Feldstärken. Über <r^2(t)>~t^{2/d_w} bestimmen wir die Random-Walk-Dimension d_w als d_w<2. Dieser Wert für d_w entspricht der Superdiffusion, obwohl der Diffusionsprozess durch Strukturen des Teppichs, wie Sackgassen, behindert wird. Es schient, dass dies das Ergebnis zweier konkurrierender Effekte ist, die durch das Anlegen eines äußeren Feldes entstehen. Einerseits bewegen sich die Teilchen bevorzugt entlang der Feldrichtung. Andererseits gelangen einige Teilchen in Sackgassen. Um die Sackgassen, die in Feldrichtung liegen, zu verlassen, müssen sich die Teilchen entgegen der Feldrichtung bewegen. Somit sind die Teilchen eine gewisse Zeit in der Sackgasse gefangen. Infolge der durch das äußere Feld beschleunigten und der gefangenen Teilchen, verbreitert sich die Wahrscheinlichkeitsverteilung der Teilchen und somit ist d_w<2.

Submitted by Sandra Maria Neri Santiago (sandra.neri@ufpe.br) on 2016-04-15T18:46:34Z No. of bitstreams: 2 license_rdf: 1379 bytes, checksum: ea56f4fcc6f0edcf0e7437b1ff2d434c (MD5) Dissertação_Tiago Aécio Grangeiro de Souza Barbosa Lima.pdf: 2215610 bytes, checksum: 8993869b89fc394d9e8171a017cfee6e (MD5) / Made available in DSpace on 2016-04-15T18:46:34Z (GMT). No. of bitstreams: 2 license_rdf: 1379 bytes, checksum: ea56f4fcc6f0edcf0e7437b1ff2d434c (MD5) Dissertação_Tiago Aécio Grangeiro de Souza Barbosa Lima.pdf: 2215610 bytes, checksum: 8993869b89fc394d9e8171a017cfee6e (MD5) Previous issue date: 2010-07-23 / Neste trabalho investigamos o problema do caminhante aleatório unidimensional como modelo para encontrar que distribuição de probabilidades é a melhor estratégia a ser utilizada na busca por sítios-alvos aleatoriamente distribuídos, cuja localização é desconhecida, na situação em que o buscador tem informação limitada sobre sua vizinhança. Embora tal problema tenha surgido na década de 1960, uma nova motivação surgiu nos anos 1990 quando dados empíricos mostraram que várias espécies de animais, sob condições gerais (especialmente escassez de comida), não usam estratégias brownianas de busca, mas sim distribuições de Lévy. A principal diferença entre elas é que as distribuições de Lévy decaem muito mais lentamente com a distância (com cauda do tipo lei de potência no limite de longos passos), não obedecendo, portanto, ao Teorema do Limite Central, e apresentam propriedades interessantes, como fractalidade, superdifusão e autoafinidade. Estes experimentos, juntamente com conceitos evolucionistas, levantaram a suspeita de que tal escolha pode ter sido adotada por ser mais vantajosa para o buscador, uma idéia conhecida como Lévy Flight Foraging Hypothesis. Em nosso estudo, definimos a eficiência da busca e obtemos a sua expressão analítica para o modelo. Utilizamos métodos computacionais para comparar as eficiências associadas às distribuições de Lévy e duas outras dentre as mais citadas na literatura, a gama e a "stretched exponential", concluindo que a de Lévy representa a melhor estratégia. Finalmente, empregamos métodos variacionais de extremização e obtemos a equação de Euler do problema. / In this work we study the one-dimensional random walk problem as a model to find which probability distribution function (pdf) is the best strategy when looking for randomly istributed target sites whose locations are not known, when the searcher has only limited information about its vicinity. Although research on this problem dates back to the 1960’s, a new motivation arose in the 1990’s when empirical data showed that many animal species, under broad conditions (especially scarcity of food), do not use Brownian strategies when looking for food, but Lévy distributions instead. The main difference between them is that the Lévy distribution decay much slower with distance (with a power-law tail in the long-range limit), thereby not obeying the Central Limit Theorem, and present interesting properties, like fractality, superdiffusivityand self-affinity. These experiments, coupled with evolutionary concepts, lead to suspicions that this choice might have been adopted because it is more advantageous for the searcher, an idea now termed as the Lévy Flight Foraging Hypothesis. To study the problem, we define a search efficiency function and obtain its analytical expression for our model. We use computational methods to compare the efficiencies associated with the Lévy and two of the most cited pdfs in the literature, the stretched exponential and Gamma distributions, showing that Lévy is the best search strategy. Finally, we employ variational extremization methods to obtain the problem’s Euler equation.

No description available.

We study anomalous diffusion on fractals with a static external field applied. We utilise the master equation to calculate particle distributions and from that important quantities as for example the mean square displacement <r^2(t)>. Applying different bias amplitudes on several regular Sierpinski carpets we obtain maximal drift velocities for weak field strengths. According to <r^2(t)>~t^(2/d_w), we determine random walk dimensions of d_w<2 for applied external fields. These d_w corresponds to superdiffusion, although diffusion is hindered by the structure of the carpet, containing dangling ends. This seems to result from two competing effects arising within an external field. Though the particles prefer to move along the biased direction, some particles get trapped by dangling ends. To escape from there they have to move against the field direction. Due to the by the bias accelerated particles and the trapped ones the probability distribution gets wider and thus d_w<2. / In dieser Arbeit untersuchen wir anomale Diffusion auf Fraktalen unter Einwirkung eines statisches äußeres Feldes. Wir benutzen die Mastergleichung, um die Wahrscheinlichkeitsverteilung der Teilchen zu berechnen, um daraus wichtige Größen wie das mittlere Abstandsquadrat <r^2(t)> zu bestimmen. Wir wenden unterschiedliche Feldstärken bei verschiedenen regelmäßigen Sierpinski-Teppichen an und erhalten maximale Driftgeschwindigkeiten für schwache Feldstärken. Über <r^2(t)>~t^{2/d_w} bestimmen wir die Random-Walk-Dimension d_w als d_w<2. Dieser Wert für d_w entspricht der Superdiffusion, obwohl der Diffusionsprozess durch Strukturen des Teppichs, wie Sackgassen, behindert wird. Es schient, dass dies das Ergebnis zweier konkurrierender Effekte ist, die durch das Anlegen eines äußeren Feldes entstehen. Einerseits bewegen sich die Teilchen bevorzugt entlang der Feldrichtung. Andererseits gelangen einige Teilchen in Sackgassen. Um die Sackgassen, die in Feldrichtung liegen, zu verlassen, müssen sich die Teilchen entgegen der Feldrichtung bewegen. Somit sind die Teilchen eine gewisse Zeit in der Sackgasse gefangen. Infolge der durch das äußere Feld beschleunigten und der gefangenen Teilchen, verbreitert sich die Wahrscheinlichkeitsverteilung der Teilchen und somit ist d_w<2.