Termalizacija in kvantni kaos v modelu Holsteinovega polarona

Ulaga, Martin (author), Vidmar, Lev (mentor)

V magistrski nalogi preučujem sklopljen elektronsko-fononski sistem, Holsteinov model za polaron, kot izoliran večdelčni kvantni sistem daleč od ravnovesja. Prvo vprašanje neravnovesne dinamike je problem ekvilibracije in termalizacije. Torej: zakaj in do katere mere lahko po dovolj dolgem času zaprt sistem (izoliran od okolice) opišemo z ansambli statistične fizike? Odgovor je mogoče formulirati s tako imenovano hipotezo o termalizaciji lastnih stanj (ETH) in nastavkom za matrične elemente fizikalno relevantnih opazljivk. Za numerično obravnavo Holsteinovega polarona so razvite metode točne diagonalizacije. Prva motivacija za to delo je primerjati različne približne metode za izgradnjo omejenega Hilbertovega prostora. V numeričnem pristopu smo vedno omejeni s številom delcev, ki lahko zasedejo neko stanje, zato to pravzaprav niso bozoni. Kako to omeji pravilnost opisa visoko vzbujenih stanj, raziščem tako, da v vseh stanjih testiram kazalce kvantnega kaosa in ETH na končnih mrežah. Posebno me je zanimala limita velikega števila fononskih prostostnih stopenj na mestu v periodični verigi fiksne velikosti. To je drugače, kot v fermionskih sistemih, kjer lahko povečujemo le dolžino verige, vsako mesto pa prispeva le nekaj prostostnih stopenj. Nazadnje v celotnem spektru analiziram prepletenostno entropijo med elektroni in fononi.

• Language: Slovenian
• Keywords: Holsteinov polaron, statistika energijskega spektra, kvantni kaos, neravnovesna dinamika, termalizacija lastnih stanj, prepletenostna entropija
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-110368
• COBISS.SI-ID: 3375716
• Publication date in RUL: 14.09.2019

ULAGA, Martin, 2019, Termalizacija in kvantni kaos v modelu Holsteinovega polarona [online]. Master’s thesis. [Accessed 19 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=110368

Secondary language

• Language: English
• Title: Thermalization and quantum chaos in the Holstein polaron model

Abstract

This master thesis examines a coupled electron-phonon system, the Holstein polaron model, as an isolated many-body quantum system far from equilibrium. The elementary question of non-equilibrium dynamics is the problem of equilibration and thermalization. That is to say: why and to what extent are standard statistical descriptions applicable to closed quantum systems after a long enough time? One approach to this question is the so-called eigenstate thermalization hypothesis (ETH), which is a statement about the matrix elements of all physically relevant observables. In this work, we study the Holstein polaron model with a numerical approach with the goal of observing ETH. We apply several numerical methods: exact diagonalization, a general method for many-body systems, and a variational approach to constructing the many-body Hilbert space, developed specifically for the low-energy physics in the Holstein model. We compare the two methods by how well they describe the highly excited states, relevant for nonequilibrium dynamics. We test for indicators of quantum chaos in the spectrum and indicators of thermalization in the eigenstate-to-eigenstate fluctuations in the thermodynamic limit. In finite systems (e.g. spin chains), the thermodynamic limit is usually taken by increasing the number of sites in a periodic chain, but with bosonic degrees of freedom present, it is also possible to increase the maximum occupancy of phononic states, which is the route we choose. In addition to this, we inverstigate the entanglement entropy for the partition along different types of degrees of freedom in the entire spectrum.

• Keywords: Holstein polaron, spectral statistics, quantum chaos, nonequilibrium dynamics, eigenstate thermalization hypothesis, entanglement entropy