Browsing by Subject "Symmetry"

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    Applications of moving frames to group foliation of differential equations
    (2013-10) Thompson, Robert
    The classical group foliation algorithm uses the continuous symmetries of a differential equation to aid in its integration. This is accomplished by transforming the differential equation into two alternative systems, called the resolving and automorphic systems. Incorporating the theory of equivariant moving frames for Lie pseudogroups, a completely symbolic and systematic version of the group foliation algorithm is introduced. In this version of the algorithm, the resolving system is derived using only knowledge of the structure of the differential invariant algebra, requiring no explicit formulae for differential invariants. Additionally, the automorphic system is replaced by an equivalent reconstruction system, again requiring only symbolic computation. The efficacy of this approach is illustrated through several examples. Further applications of aspects of group foliation are given, including the construction of Backlund transformations using resolving systems and a reconstruction process for an invariant submanifold flow corresponding to a given invariant signature evolution.
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    Phonon and Thermal Dynamics of Kitaev Quantum Spin Liquids
    (2022-01) Feng, Kexin
    Quantum spin liquids (QSL) is a novel state of magnetic material, where magnetic order is absent down to very low temperature. This novel state has massive quantum entanglement, and can host emergent quasiparticle excitations, which carry fractionalized quantum numbers and display anionic statistics. In this thesis, I focus on the phonon and thermal dynamics of the Kitaev QSL, explore detectable signatures of QSL phase. More specifically, I will new propose observables, including sound attenuation coefficient, Hall viscosity and Fano effects in the optical phonon Raman spectroscopy, which are shown to encode information of the fractionalized excitations, namely Z2 gauge fluxes and itinerant fermions. The key technique to deal with spin-phonon couplings largely relies on symmetry considerations, which involves group and representation theory. The main numerical technique to simulate flux thermodynamics are Markov Chain Monte Carlo and stratified Monte Carlo. The latter is a new efficient algorithm which I designed specifically for the Kitaev model, based on my phenomenological study of the Z2 flux thermodynamics.