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Optimised continuous unitary transformations for interacting fermionic systems
Thesis (MSc)--Stellenbosch University, 2025.
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| Format: | Thesis |
| Language: | English |
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Stellenbosch : Stellenbosch University
2025
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| _version_ | 1867613898629185536 |
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| access_status_str | Open Access |
| author | Robson, Damian |
| author2 | Kriel, Johannes N.
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| author_browse | Kriel, Johannes N.
Robson, Damian |
| author_facet | Kriel, Johannes N.
Robson, Damian |
| author_sort | Robson, Damian |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2025. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/132466 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:43:27.297Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2025 |
| publishDateRange | 2025 |
| publishDateSort | 2025 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/132466 Optimised continuous unitary transformations for interacting fermionic systems Robson, Damian Kriel, Johannes N. Kastner, M. Stellenbosch University. Faculty of Science. Dept. of Physics. Unitary transformations Quantum systems -- Mathematical models Many-body problem Fermions -- Mathematical models Quantum theory -- Interactive multimedia Hamiltonian systems Equations UCTD Thesis (MSc)--Stellenbosch University, 2025. Robson, D. 2025. Optimised Continuous Unitary Transformations for Interacting Fermionic Systems. Unpublished masters thesis. Stellenbosch: Stellenbosch University [online]. Available: https://scholar.sun.ac.za/items/5e78f76f-221b-4f32-9851-bfe3d14c58bc ENGLISH ABSTRACT: This thesis is concerned with the so-called flow equation approach, an iterative method to approximately diagonalise Hamiltonians of a quantum system. The general aim of the thesis is to come up with improved variants of the flow equation approach, with a particular focus on applications to quantum systems in which the wave functions are to some degree localised in space. Two main ideas for improvements, both of which affect the approximation error made when truncating terms, are introduced and analysed in this thesis: the choice of basis in which the Hamiltonian is written, and the state with respect to which normal-ordering is done. To quantify the accuracy of the modified flow equation methods, we introduce the second trace invariant within a fixed particle number sector as a probe for the loss of unitarity, and also compare the eigenvalues obtained from the solution of the flow equation to exact numeric results. To benchmark the different improvements to the flow equation approach, we use the XXZ model with magnetic field disorder, which has the desirable property of a tunable disorder strength. The latter allows for control over the degree of localisation of the system’s wave functions. For the flow equation we find that normal-ordering with respect to a density matrix that targets an average particle number density corresponding to half-filling significantly reduces the truncation error in the higher particle-number sectors of the Hilbert space, and yields improved results for the eigenvalues. These improvements were especially noticeable at weaker disorder strengths, where the typical vacuum ordering performs poorly. We show that the choice of a different initial single-particle basis essentially leads to a different choice in generator, and find that single-particle bases which are delocalised in real space reduces the truncation error in the case of weak disorder. However, the benefit of the choice of single-particle bases in reducing the truncation error is lesser when compared to that of normal-ordering. Apart from the truncation error, we observe that a choice of basis may reduce computation time by speeding up the convergence of the flow towards the fixed-point Hamiltonian. By applying the normal-ordering results to Heisenberg’s equation of motion, we demonstrate that an appropriate choice of normal-ordering greatly improves our ability to capture the slow-relaxation dynamics at early times. However, our results eventually deviate from the expected dynamics in the long-time regime. We suggest that normal ordering with respect to the charge density wave state |010101 . . .⟩ may further improve the description of the slow-relaxation dynamics. This thesis presents new results for improving the numerical implementation of continuous unitary transformations for interacting fermionic systems, as well as a novel probe for the loss of unitarity, the second trace invariant within a fixed particle number sector, which has not appeared in the literature before. AFRIKAANSE OPSOMMING: Hierdie tesis handel oor die sogenaamde vloeivergelyking benadering, 'n iteratiewe metode om die Hamiltonian van 'n kwantumstelsel te diagonaliseer. Die algemene doel van die tesis is die ontwikke- ling in verbeterde variante van die vloeivergelyking benadering, met 'n spesifieke fokus op toepassings op kwantumstelsels waarin die golffunksies tot 'n mate in die ruimte gelokaliseer is. Twee hoofidees vir verbeteringe word in hierdie tesis bekendgestel en ontleed: die keuse van basis waarin die Hamiltoniaan geskryf word, en die toestand wat betrekking tot die normaal-ordering uitgevoer word. Beide hierdie verbeteringe be¨ınvloed die benaderingsfout wat onstaan as gevolg van die afkapping van terme. Om die akkuraatheid van die gemodifeerde vloeivergelyking metodes te kwantifiseer, s tel o ns voor d ie tweede spoor-invariante binne 'n vaste deeltjiegetalsektor as 'n getuie vir die verlies aan eenheid, en vergelyk ook die eiewaardes vanaf die oplossing van die vloeivergelyking met presiese numeriese resultate. Om die verskillende verbeteringe te meet op die vloeivergelyking benadering, gebruik ons die XXZ-model met magnetiese veldversteuring, wat die voordelige eienskap van 'n stelbare versteuringssterkte het. Hierdie laat toe om die mate van lokalisering van die stelsel se golffunksies te beheer. Vir die vloeivergelyking met normaal-ordening met betrekking tot 'n digtheidsmatriks wat 'n gemiddelde deeltjiegetaldigtheid stel ooreenstemmend met halfvulling, vind ons 'n aansienlikke vermindering in die afkappingsfout vir die hoe¨r deeltjiegetal sektore van die Hilbert-ruimte, asook verbeterde resultate vir die eiewaardes. Hierdie verbe- teringe was veral opmerklik by swakker wanorde-sterktes, waar die tipiese vakuum-ordering swak vaar. Ons wys dat die keuse van 'n ander aanvanklike enkel-deeltjie basis lei tot 'n ander keuse in generator. Ons vind dat die afkappingsfout word verminder deur enkel-deeltjie basisse waar die werklike ruimte gede- lokaliseer is, in die geval van swak wanorde. Die voordeel van die keuse van enkel-deeltjie basisse om die afkappingsfout te verminder, is egter minder in vergelyking met die´ van normaal-ordening. Afgesien van die afkappingsfout, die keuse van basis mag die berekeningstyd verminder deur middel van die konvergen- sie, van die vloei na die vastepunt Hamiltoniaan, te versnel. Deur die resultate van normaal-ordening toe te pas op Heisenberg se bewegingsvergelyking, demonstreer ons dat 'n gepaste keuse van normaal-ordening lei tot 'n verbeterde vermoe¨ om die stadige-ontspanningsdinamika vas te vang in vroee¨ tye. Alhoewel, ons resultate wyk egter vanaf die verwagte dinamika in die langer tye. Ons stel voor dat normaal-ordening met betrekking tot die lading-digtheid golf toestand 010101 . . . mag verder die beskrywing van die stadige- ontspannings dinamika verbeter. Hierdie tesis bied aan nuwe resultate en verbeteringe oor die numeriese implementasie van deurlopende uniteˆre transformasies vir fermioniese sisteme met interaksies, sowel as 'n nuwe getuie vir die verlies van uniteˆriteit. Masters 2025-06-09T10:16:55Z 2025-06-09T10:16:55Z 2025-03 Thesis https://scholar.sun.ac.za/handle/10019.1/132466 en Stellenbosch University xii, xiv, 49 pages : illustrations application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Unitary transformations Quantum systems -- Mathematical models Many-body problem Fermions -- Mathematical models Quantum theory -- Interactive multimedia Hamiltonian systems Equations UCTD Robson, Damian Optimised continuous unitary transformations for interacting fermionic systems |
| title | Optimised continuous unitary transformations for interacting fermionic systems |
| title_full | Optimised continuous unitary transformations for interacting fermionic systems |
| title_fullStr | Optimised continuous unitary transformations for interacting fermionic systems |
| title_full_unstemmed | Optimised continuous unitary transformations for interacting fermionic systems |
| title_short | Optimised continuous unitary transformations for interacting fermionic systems |
| title_sort | optimised continuous unitary transformations for interacting fermionic systems |
| topic | Unitary transformations Quantum systems -- Mathematical models Many-body problem Fermions -- Mathematical models Quantum theory -- Interactive multimedia Hamiltonian systems Equations UCTD |
| url | https://scholar.sun.ac.za/handle/10019.1/132466 |
| work_keys_str_mv | AT robsondamian optimisedcontinuousunitarytransformationsforinteractingfermionicsystems |