School of Science Department of Physics 69 Theoretical Modeling of the Motion of Particles in Energy Walls Supervisor: PARK Hyo Keun / PHYS Student: XU Haohan / PHYS-IRE Course: UROP 1100, Summer This study models the diffusion of inhibitory neurotransmitter receptors in shallow energy traps formed by postsynaptic scaffolding proteins using single nanoparticles in PDMS geometric wells. Experimental and simulation results show bimodal dwell time distributions in log-binned time scale, showing that the particles have short-term and long-term behaviors. They either escaping rapidly or lingering in the well region. Geometric wells prolong survival, shifting the secondary dwell time peak to longer scales compared to “no well” conditions. Analytical solutions for survival probability, discretized to account for microscope frame rate, align with observations. These findings support geometric wells as a model for potential energy traps, advancing understanding of synaptic receptor dynamics and brains’ short-term plasticity. Tensor-Network Representation of Gapless Frustration-Free Free Fermions Supervisor: PO Hoi Chun / PHYS Student: CHEN Xinpeng / PHYS-IRE Course: UROP 1100, Summer This article aims to utilize the Quantics Tensor Train (QTT) method to solve the one-dimensional Schrödinger equation with high efficiency and controlled accuracy. We begin by reviewing tensor-network representations of high-dimensional data, with emphasis on the tensor train decomposition and its rank structure, storage complexity, and rounding operations. We then describe how continuous functions and discretized operators can be mapped to the QTT format via dyadic reshaping, enabling exponential compression on uniform grids. Building on these ingredients, we present a QTT-based discretization of the Hamiltonian and briefly discuss how to solve the one-dimensional Schrödinger equation using QTT. Tensor-Network Representation of Gapless Frustration-Free Free Fermions Supervisor: PO Hoi Chun / PHYS Student: CHEN Yifan / PHYS Course: UROP 1100, Summer Frustration freeness refers to the existence of a ground state which simultaneously minimizes the energy of all the local terms in a Hamiltonian. Such ground states are naturally amenable to a tensor-network representation. In this project, we study how the tensor network representation captures the physical properties of a free-fermion system when the model is frustration free and gapless. But in practice, we find that whether the system is frustration free actually depends on the local hamiltonian we choose. So although finding a tensor network is our initial intention, I ended up learning a method to build the frustration free gapless hamiltonian out of any MPS for the system. And I further extend this method from a spin chain to a spinless fermion chain.
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