School of Science Department of Physics 59 Angle Formula for Evaluating the Quantum Geometric Tensor Supervisor: PO Hoi Chun / PHYS Student: CHEUNG Man Yin / PHYS-IRE POON Pak Hei Andersen / MATH-PMA Course: UROP 1100, Summer UROP 1100, Summer Quantum Geometric Tensor (QGT) is a quantity that describes the geometry of the parameter space of a system. It is found to be related to the quantum metric and Berry curvature, which both determine physical properties of a system. In this report, we review the QGT formula in terms of Bloch angles for a two-level system, and then derive a general angle formula for QGT. Moreover, we use computational method to verify the formula and validate its accuracy. Coupled Wire Construction for Two-dimensional Metals Supervisor: PO Hoi Chun / PHYS Student: QIAN Yuchen / PHYS WONG Kin Ka / PHYS-IRE Course: UROP 1000, Summer UROP 1000, Summer In coupled-wire consturctions, two-dimensional metals could be obtained by coupling free-fermion wires in a network pattern. Building upon commonly-used network models consisting of two sets of chiral wire, we devise a similar four-wire setup. The allowed independent parameters in the model are analyzed by imposing symmetry restrictions. We show that various Fermi surfaces can be obtained by tuning the allowed parameters, highlighting how the features of the Femri surfaces depend on the paramaters. Diffusion of Nanoparticles in a Potential Energy Landscape Supervisor: WONG Michael Kwok Yee / PHYS Student: CHEUNG Ho Tin / PHYS Course: UROP 4100, Fall The lifetime distribution of nanoparticles diffusing in a potential energy landscape is highly relevant to the efficacy of signal transmission between neurons in the neural system, as it determines the density of neurotransmitter receptors staying in the synaptic cleft. In our previous studies, we formulated the theoretical model of the lifetime distribution in one-dimensional square potential well, and simulated the diffusion motion in two-dimensional potential energy landscapes. In the present project, we extend our study to nanoparticle diffusions in three-dimensional geometrical wells, and explore the extent to which the geometrical constraint is similar to the potential energy constraint. The study of the geometrical case also enabled us to compare with experimental data provided by the experimental branch of our team. We found that the deviations caused by changing the nanoparticle size are insignificant in experimental observations. We observed discrepancies between theory and preliminary experimental data, and will conduct further comparisons when more experimental data become available.
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