One of the most powerful and prized concepts in physics is that of universal behaviour. Universality allows us to make predictions for entire classes of systems without requiring knowledge of the microscopics, and can be found in classical and quantum systems in both equilibrium and in their dynamics. Often in many-body systems, this universal behaviour is found in regimes where effects at macroscopic length scales dominate over microscopic fluctuations, which is particularly true at a phase transition. In this thesis, we will address universality in quantum many-body physics, and its connection to the branch of mathematics known as catastrophe theory (CT). In CT, singularities in a theory take on several universal forms, known as catastrophes, which can be shown to manifest themselves in classical mechanical trajectories. We extend the concept of catastrophes to their wave variants, known as diffraction integrals, and identify how these universal features appear in many-body wavefunctions and observables. Specifically, in Chapter 2, we examine how the wavefunction of a $\delta$-kicked Hamiltonian can be mapped exactly onto the Pearcey function, along with the effects of a phase transition on the diffraction. In Chapter 3, we examine the free-Fermion representation of the one-dimensional transverse-field Ising model in a similar vein, and identify the presence of catastrophes away from and through the critical point. / Thesis / Master of Science (MSc) / In this thesis, we investigate how certain universal structures, known as catastrophes, manifest themselves in quantum many-body systems. These catastrophes, which are associated with singularities somewhere in the theory, have been shown to be universal structures, and are expected to emerge in our study of condensed-matter systems. We will identify the presence of catastrophes, how they arise in specific classes of physical theories, and how they are affected in regimes where other universal behaviour is known to arise: near locations of quantum criticality.
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/22333 |
| Date | 17 November 2017 |
| Creators | Kirkby, Wyatt |
| Contributors | O'Dell, Duncan, Physics |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
Page generated in 0.0028 seconds