Date of Award

12-24-2025

Date Published

January 2026

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics

Advisor(s)

Arindam Chakraborty

Second Advisor

Christian Santangelo

Keywords

Computation;Computational efficiency;Light-matter interaction;Quantum dots;Tensor contraction

Subject Categories

Condensed Matter Physics | Physical Sciences and Mathematics | Physics

Abstract

Quantum dots are a class of materials in the nanometer scale that have unique optical and electronic properties due to the quantum confinement effect. Traditional electronic structure computational techniques used for calculating these properties fall to scalability issues as the size of the quantum dot increases, or introduce significant errors into the final result. This makes using computational methods to predict new materials, or verify properties of existing materials, challenging to use for experimental applications of quantum dots. This gap between experiment and computation is due to large basis sets, expensive integrals, and storage and manipulation of tensors that grow nonlinearly with system size. In this work, we present new methods to bridge this gap and allow for theoretical calculations to be performed on systems large enough to be grown in a lab. The first work presented is a diagrammatic technique used for calculation of the polarization propagator, an equation that governs the response of a system to external perturbations. This is used for calculation of the absorption spectra of two types of semiconductor quantum dots, Cadmium Sulfide (CdS) and Lead Sulfide (PbS) of varying sizes from $0.5 - 2$ nanometers in diameter. Calculations were performed using a basis defined by an inverted Krylov expansion around a given incoming photon energy $\omega$. This not only makes the basis $\omega$ dependent, leading to enhanced accuracy of results of excited states near the photon frequency, but also makes the size of the basis customizeable and scalable based on the computational hardware available to the user. We then use these basis vectors in a folded spectrum calculation to solve for eigenvalues of the superoperator resolvent near the photon frequency $\omega$ without explicit inversion of the superoperator matrix. The folded spectrum method allows for variationally stable calculations near a given eigenvalue, making the expensive tensor contractions in the polarization propagator computationally efficient. Diagrammatics are used to discover noncontributing terms in the second order expansion of the polarization propagator, and then the techniques discussed above are applied only on the contributing diagrams. The second work presented is the Stratified Stochastic Tensor Contraction method (SSTC), a method developed for use in quantum chemistry and quantum physics, but that has broad applications in many fields. The key idea of the SSTC method is to utilize the structure of a tensor to make efficient and accurate approximations of a tensor contraction result. This is done using a contracted index scheme, in which the tensor indices are flattened into a vector, and then stratified across a given sampling space into some number of segments. The contracted indices are then stochastically chosen and assigned to each segment based on a given mapping function, where ther kernel of the tensor is computed and the variance inside of each segment is analyzed. The most accurate result is then the mapping function that minimizes the variance on the segment means, which implies a functional minimization over the mapping function space. This functional minimization is performed over a finite set of user defined mapping functions, and the best mapping function is chosen to give the result of the tensor contraction. The SSTC method is demonstrated on four examples important in quantum chemistry and quantum physics. We show that results obtained using the SSTC method are good approximations of the exact, sequential results with low error. These results position the SSTC method as an efficient technique for application in any areas where challenging tensor contractions are performed.

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Open Access

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