Non-equilibrium dynamics of topological insulators

Ulčakar, Lara (author), Rejec, Tomaž (mentor), Mravlje, Jernej (comentor)

Topological insulators are band insulators with a non-trivial band topology that leads to the presence of in-gap edge states at the boundaries and the associated dissipationless transport. The thesis is focused on non-equilibrium behaviour that arises when a topological insulator is slowly quenched, i.e. is smoothly driven across a topological phase transition. We study a Chern insulator, represented by the Qi-Wu-Zhang model, and a time-reversal symmetric topological insulator, described by the Bernevig-Hughes-Zhang model, and find similar non-equilibrium transport. For slow quenches, the (spin) Hall conductivity approaches that of the final ground state. The deviations from this value diminish as a power-law as the quench becomes slow, which is consistent with the Kibble-Zurek prediction. Conversely, the behaviour of topological invariants differs. The Chern number is conserved under a unitary evolution, while the classification of time-reversal symmetric phases breaks down since the time evolution dynamically breaks the time-reversal symmetry. We also investigate a Chern insulator in ribbon geometry and show that after the system is driven from a trivial to a topological phase, the in-gap states emerge and are populated with electrons. As the Chern invariant remains unchanged, the bulk-boundary correspondence is broken. In order to explore the critical properties and the non-equilibrium dynamics in real space, we introduce a weak disorder to a Chern insulator. In the ground state, the local Chern marker exhibits a critical length scale in its inhomogeneous profile that behaves consistently with the one extracted from the width of the peak in the Berry curvature. During the quench, the length scale grows and saturates to a value that increases with the quench time, as predicted by the Kibble-Zurek mechanism.

• Language: English
• Keywords: topological insulator, quantum phase transition, time-reversal symmetry, quench, Kibble-Zurek mechanism, Landau-Zener model, Hall effect, edge state, topological invariant, local Chern marker, critical exponent
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-122256
• COBISS.SI-ID: 41984003
• Publication date in RUL: 02.12.2020

Secondary language

• Language: Slovenian
• Title: Neravnovesna dinamika topoloških izolatorjev

Abstract

Topološki izolatorji so pasovni izolatorji z netrivialno topologijo pasovne strukture, ki porodi robna stanja z energijo znotraj energijske reže in z njimi povezan transport brez disipacije. Doktorsko delo je posvečeno neravnovesnemu obnašanju topoloških izolatorjev, ki so bili v času zvezno preklopljeni čez topološki fazni prehod. Obravnavamo Chernov izolator, ki ga opisuje Qi-Wu-Zhangov model, in topološki izolator s simetrijo na obrat časa, ki ga predstavlja Bernevig-Hughes-Zhangov model. Za neravnovesni stanji sistemov pokažemo, da imata podobne transportne lastnosti. Po počasnih preklopih se (spin) Hallova prevodnost približa vrednosti v končnem osnovnem stanju, deviacije od le te pa padajo s časom preklopa kot potenčna funkcija. To obnašanje je v skladu s Kibble-Zurekovim mehanizmom. Obnašanje topoloških invariant zavisi od simetrijskega razreda sistema. Chernovo število se ohranja skozi čas, medtem ko časovni razvoj podre simetrijo na obrat časa in tako je klasifikacija z invarianto Z2 nesmiselna. Raziskava preklopov iz trivialne v topološko fazo Chernovega izolatorja v obliki traku pokaže, da se robna stanja pojavijo in so po počasnih preklopih zasedena. Ker ostane Chernovo število nespremenjeno, je po preklopu korespondenca rob-notranjost kršena. Kritične in neravnovesne lastnosti Chernovih izolatorjev smo proučevali tudi v realnem prostoru ob prisotnosti šibkega nereda. V osnovnem stanju se v profilu lokalnega Chernovega markerja pojavi kritična dolžinska skala, ki se sklada s skaliranjem dolžinske skale, ocenjene iz širine vrhu Berryjeve ukrivljenosti. Med preklopom čez fazni prehod dolžinska skala raste in se ustali pri vrednosti, ki se povečuje s časom preklopa po napovedih Kibble-Zurekovega mehanizma.

• Keywords: topološki izolator, kvantni fazni prehod, simetrija na obrat časa, preklop, Kibble-Zurekov mehanizem, Landau-Zenerjev model, Hallov pojav, robno stanje, topološka invarianta, lokalni Chernov marker, kritični eksponent