magistrsko delo
Domen Zevnik (Author), Tomaž Prosen (Mentor)

Abstract

V tej magistrski nalogi raziskujemo kvantno integrabilnost in kaos skozi študij 2D Heisen- bergovega modela, ki je pomembna razširitev dobro poznanega 1D Heisenbergovega modela. Kvantno integrabilnost definiramo s pomočjo Algebraičnega Bethejevega nastavka in dokažemo, da je 2D Heisenbergov model integrabilen, če upoštevamo samo vodoravne interakcije. Poleg tega kvantno integrabilnost definiramo tudi preko Poissonove statistike, kjer sistem velja za integrabilen, če statistika njegovega spektra sledi Poissonovi porazdelitvi. Kvantni kaos pa definiramo z uporabo teorije naključnih matrik. Pravimo, da je sistem kaotičen, če statistika spektra sledi eni izmed Wigner-Dysonovih porazdelitev. Obe definiciji sta veljavni le v primeru, ko so v sistemu odpravljene vse prostorske simetrije. Na primeru 2D Heisenbergovega modela opišemo simetrije in pojasnimo, kako jih odpraviti. Na koncu se osredotočimo na numerične izračune in s pomočjo porazdelitev razmerij raz- mikov sosednjih nivojev spektra ter spektralnega oblikovnega faktorja pokažemo, da v sistemu pride do zloma integrabilnosti ob povečanju moči vertikalnih interakcij, kar ponazarja prehod v kaotično dinamiko.

Keywords

algebraični Bethejev nastavek;2D Heisenbergov model;kvantna integrabilnost;kvantni kaos;spektralni oblikovni faktor;

Data

Language: Slovenian
Year of publishing:
Typology: 2.09 - Master's Thesis
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: [D. Zevnik]
UDC: 530.145
COBISS: 206425091 Link will open in a new window
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Downloads: 23
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Other data

Secondary language: English
Secondary title: Quantum integrability and chaos in the 2D Heisenberg model
Secondary abstract: In this master’s thesis, we explore quantum integrability and chaos through the study of the 2D Heisenberg model, which is an important extension of the well-known 1D Heisenberg model. Quantum integrability is defined using Algebraic Bethe ansatz, and we demonstrate that the 2D Heisenberg model is integrable when considering only horizontal interactions. We give an alternative definition of quantum integrability using Poisson statistics, where a system is considered integrable if the statistics of its spectrum follow the Poisson distribution. Quantum chaos is defined using random matrix theory, where a system is said to be chaotic if the spectral statistics follow one of the Wigner-Dyson distributions. These definitions are valid only when all spatial symmetries in the system have been removed. We describe the symmetries of the 2D Heisenberg model and explain how to remove them. Finally, we focus on numerical analysis and demonstrate, through the use of level spacing ration and the spectral form factor, that integrability breaks down in the system as the strength of the vertical interactions increases, indicating a transition to chaotic dynamics.
Secondary keywords: algebraic Bethe ansatz;2D Heisenberg model;quantum integrability;quantum chaos;spectral form factor;
Type (COBISS): Master's thesis/paper
Study programme: 0
Thesis comment: Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za fiziko
Pages: 70 str.
ID: 24910795