We explore the dynamics of integrable and chaotic quantum many-body systems with
a focus on universal structures known as caustics, which are a type of singularity
categorized by catastrophe theory.
Papers I and II study light cones in quantum spin chains, which we show are
caustics and therefore inherits specific functional forms. For integrable systems, the
edge of the cone is a fold catastrophe, making the wavefunction locally of Airy form.
We also identify the cusp catastrophe in the XY model, thus the secondary light cone
is a Pearcey function. Vortex pairs appear in the dynamics, are sensitive to phase
transitions, and permit the extraction of critical scaling exponents. In paper II we use
a Gaussian wavefront form to distinguish integrable and chaotic models. Writing the
wavefront as exp[−m(x)(x − vt)2 + b(x)t], the scaling of coefficients m(x) and b(x) is
the diagnostic. The local Airy function description in free models leads to a power-law
∼ x^{−n/3} scaling, while for the chaotic case the scaling is exponential ∼ e^{−cx}.
In Paper III, we study the function Fn(t) = <(A(t)B)^n>, a generalization of the
four-point out-of-time ordered correlator (OTOC) F2(t), for an integrable system and
show that the function Fn(t) can be recast as the return amplitude of an effective time dependent chaotic system, exhibiting signals of chaos such as a positive Lyapunov
exponent, spectral statistics consistent with random matrix theory, and relaxation.
In Paper IV we perform a comprehensive investigation of caustics in many-body
systems in (1+1)- and (2+1)-dimensional Fock space and time. We show how a
hierarchy of caustics appear in the dynamics of many-body models, using two- and
three-mode Bose-Hubbard models as guiding systems. We show that, in the case of
the trimer, high dimensional caustics appear and are organized by the catastrophe
X9. / Thesis / Doctor of Philosophy (PhD)
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/27418 |
| Date | January 2022 |
| Creators | Kirkby, Wyatt |
| Contributors | O'Dell, Duncan, Physics and Astronomy |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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