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Non-abelian braiding in abelian lattice models from lattice dislocations
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
2014 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Icke-abelsk flätning i abelska gittermodeller genom dislokationer (Swedish)
Abstract [en]

Topological order is a new field of research involving exotic physics. Among other things it has been suggested as a means for realising fault-tolerant quantum computation. Topological degeneracy, i.e. the ground state degeneracy of a topologically ordered state, is one of the quantities that have been used to characterize such states. Topological order has also been suggested as a possible quantum information storage.

We study two-dimensional lattice models defined on a closed manifold, specifically on a torus, and find that these systems exhibit topological degeneracy proportional to the genus of the manifold on which they are defined. We also find that the addition of lattice dislocations increases the ground state degeneracy, a behaviour that can be interpreted as artificially increasing the genus of the manifold. We derive the fusion and braiding rules of the model, which are then used to calculate the braiding properties of the dislocations themselves. These turn out to resemble non-abelian anyons, a property that is important for the possibility to achieve universal quantum computation. One can also emulate lattice dislocations synthetically, by adding an external field. This makes them more realistic for potential experimental realisations.

Abstract [sv]

Topologisk ordning är ett nytt område inom fysik som bland annat verkar lovande som verktyg för förverkligandet av kvantdatorer. En av storheterna som karakteriserar topologiska tillstånd är det totala antalet degenererade grundtillstånd, den topologiska degenerationen. Topologisk ordning har också föreslagits som ett möjligt sätt att lagra kvantdata.

Vi undersöker tvådimensionella gittermodeller definierade på en sluten mångfald, specifikt en torus, och finner att dessa system påvisar topologisk degeneration som är proportionerlig mot mångfaldens topologiska genus. När dislokationer introduceras i gittret finner vi att grundtillståndets degeneration ökar, något som kan ses som en artificiell ökning av mångfaldens genus. Vi härleder sammanslagningsregler och flätningsregler för modellen och använder sedan dessa för att räkna ut flätegenskaperna hos själva dislokationerna. Dessa visar sig likna icke-abelska anyoner, en egenskap som är viktiga för möjligheten att kunna utföra universella kvantberäkningar. Det går också att emulera dislokationer i gittret genom att lägga på ett yttre fält. Detta gör dem mer realistiska för eventuella experimentella realisationer.

Place, publisher, year, edition, pages
2014. , 86 p.
Keyword [en]
Topological order, Topological degeneracy, Ground state degeneracy, Quantum computer, Fault-tolerant quantum computation, Lattice model, Lattice dislocations, Toric code, Non-abelian anyons, Braiding
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:kau:diva-31690OAI: oai:DiVA.org:kau-31690DiVA: diva2:704397
Subject / course
Physics
Supervisors
Examiners
Available from: 2014-03-13 Created: 2014-03-12 Last updated: 2014-03-13Bibliographically approved

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