Towards quantum glasses

master's thesis

• Title: Towards quantum glasses : Disorder in topological frustrated quantum systems
• Title (croatian): Prema kvantnim staklima : Nered u topološki frustriranim kvantnim sustavima
• Author: Karlo Delić
• Mentor: Salvatore Marco Giampaolo (mentor)
• Mentor: Ivan Kupčić (komentor)
• Committee member: Ivan Kupčić (predsjednik povjerenstva)
• Committee member: Salvatore Marco Giampaolo (član povjerenstva)
• Committee member: Matko Milin (član povjerenstva)
• Committee member: Krešimir Kumerički (član povjerenstva)
• Committee member: Mihael Makek (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Physics); Zagreb
• Defense date and country: 2021-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Physics

Abstract

The one-dimensional XY model in a transverse field is a prototypical exactly solvable quantum system with nontrivial phase diagram, characterised by two different quantum phase transitions, i.e., ones at temperature zero. Using the Jordan-Wigner transformation, the model can always be mapped into a system of free fermions. In the thesis, we focus on the closed XY chain without external magnetic field and with odd number of spins, conducive to the phenomenon of frustration in case of the dominant antiferromagnetic interaction, i.e., the inability of minimising all of the local interactions simultaneously. This system was shown to exhibit interesting properties depending on the value of the anisotropy parameter ϕ determining the relative magnitude and sign of the x and y direction interactions. Also depending on this value, we make distinction between three phases, denoted ordered (1), mesoscopic (2) and incommensurate (3) based on the previously obtained results for each. After solving the XY model for its ground states and spectra via the Jordan- Wigner transformation and Bogoliubov rotation, we focus on calculating the one and two-point static and dynamic correlation functions after the system has undergone a global quench at t = 0, only changing its anisotropy parameter ϕ. In the t = 0 limit, we discuss the agreement of our results with known analytic limiting behaviour. Furthermore, using the dynamic expressions calculated through solving the time-evolution problem for the XY model, we explore the validity of the cluster decomposition hypothesis for all evolutions starting in one of the first two phases. In this way, several interesting results are obtained, namely the emergence of mesoscopic behaviour in the long-time regime for several evolutions and also a usual exponential decay of the cluster decomposition for the remaining ones. We discuss the rates of decay and emphasise that the evolutions starting in the third phase require additional subtlety to be analysed properly due to the breaking of translational invariance and are thus left for an upcoming paper.

Abstract (croatian)

Jednodimenzionalni XY model u transverzalnom magnetskom polju predstavlja prototip egzaktno rješivog kvantnog sustava s netrivijalnim faznim dijagramom kojega karakteriziraju dva kvantna fazna prijelaza na temperaturi nula. Koristeći Jordan- Wignerovu transformaciju, ovaj model se uvijek može preslikati u sustav slobodnih fermiona. U ovom se radu fokusiramo na zatvoreni XY lanac bez vanjskog magnetskog polja i sa neparnim brojem spinova, odnosno sustav koji je podložan frustraciji u slučaju dominantne antiferomagnetske interakcije—nemogućnosti istovremenog minimiziranja globalnog Hamiltonijana i svih lokalnih. Za ovaj sustav je demonstrirano da pokazuje zanimljiva svojstva ovisno o vrijednosti parametra anizotropije ϕ koji određuje relativnu magnitudu i predznak interakcija u x i y smjeru. Uz to, ovisno o vrijednosti tog parametra, razlikujemo tri faze u kojima se sustav može naći: uređenu (1.), mezoskopsku (2.) i nesrazmjernu (3.) temeljem dosad pokazanih rezultata. Nakon pronalaska svojstvenih stanja i spektra modela koristeći JW transformaciju i Bogoljubovljevu rotaciju, okrećemo se računanju korelacijskih funkcija jednog (magnetizacija) i dvaju operatora i to za statički i dinamički vremenski okvir. Dinamički okvir je posljedica globalne promjene parametra ϕ u trenutku t = 0. U statičkom režimu t = 0, referiramo se na poznate asimptotske rezultate i diskutiramo slaganje s izrazima do kojih smo došli. Nastavno na to, koristeći dinamičke izraze koje smo dobili rješavanjem problema vremenske evolucije za XY model, istražujemo valjanost principa dekompozicije nakupina za 6 evolucija koje počinju u nekoj od prve dvije faze. Dolazimo do nekoliko zanimljivih rezultata, posebno postojanja mezoskopskog ponašanja CD za razmjerno velike brojeve spinova. Konačno, diskutiramo brzine trnjenja DN na temelju fizikalnih argumenata i naglašavamo da je analiza evolucija koje započinju u trećoj fazi zahtjevnija zbog narušenja translacijske invarijantnosti sustava te stoga ostavljena za predstojeći članak.

• Keywords: 1D XY model; degenerate ground states; spin chain; frustration; cluster decomposition hypothesis
• Keywords (croatian): 1D XY model; degenerirana osnovna stanja; spinski lanac; frustracija; princip dekompozicije nakupina
• Language: english
• URN:NBN: urn:nbn:hr:217:983766
• Study programme: Title: Physics; specializations in: Research; Course: Research; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra fizike (magistar/magistra fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-11-09 10:29:27