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Author:Ervasti, Mikko
Title:Topological phases of matter and the fractional quantum Hall effect
Aineen topologiset faasit ja murtolukuinen kvantti-Hall-ilmiö
Publication type:Master's thesis
Publication year:2011
Pages:105      Language:   eng
Department/School:Teknillisen fysiikan laitos
Main subject:Fysiikka (laskennallinen fysiikka)   (Tfy-105)
Supervisor:Nieminen, Risto
Instructor:Harju, Ari ; Tölö, Eero
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Location:P1 Ark Aalto  97   | Archive
Keywords:topological phase
topological order
topological quantum field theory
topological quantum computation
topological entanglement entropy
quantum Hall effect
topologinen faasi
topologinen järjestys
topologinen kvanttikenttäteoria
topologinen kvanttilaskenta
topologinen entropia
kvantti-Hall-ilmiö
Abstract (eng): The subject of this work is to study the topological phases of matter in (2+1)-dimensions theoretically and numerically.
As the theory is not yet complete, we use various gapped model systems to probe the signature behaviour that arise in the topological phase.
Most notably, we study the fractional quantum Hall systems, since they are well-established to exist in nature.
The main motivation is that the topological phases can be used for topological quantum computation.

The first part of the Thesis studies the topological phase from theoretical principles.
This consists of defining modular tensor categories to describe consistent quasiparticle systems with arbitrary number of quasiparticles, and topological quantum field theories to describe the low energy, long wavelength effective field theories for the topological phases.
There are also other viewpoints that are covered, namely ground state degeneracy that is due to topological properties and not by any symmetry-breaking, and also topological phases forming universality classes of different longranged entanglements.

We solve a fractional quantum Hall system on a sphere with Coulomb interaction numerically and evaluate the topological entanglement entropy.
This amounts to finding the ground states with various system parameters by an exact diagonalization technique, evaluating the entanglement entropies based on bipartite splits, and extrapolating the entropies to the thermodynamic limit.
The subleading term in the entropy then matches the logarithm of the total quantum dimension, which characterizes the quasiparticles.
The results match the theoretical predictions.
Abstract (fin): Työssä käsitellään aineen topologisia faaseja (2+1)-ulottuvuudessa sekä teoreettisesti että laskennallisesti.
Näissä systeemeissä on aina energiaero perustilan ja alimpien viritystilojen välillä.
Topologisten faasien teoria ei ole vielä valmis, joten esimerkkisysteemien ominaisuuksia tutkimalla saadaan vinkkejä laajemmasta kuvasta.
Erityisesti tutkimme murtolukuista kvantti-Hall-ilmiötä, koska se on todellinen toisin kuin monet muut mallit.
Topologisten faasien pääasiallinen sovelluskohde on topologinen kvanttilaskenta.

Työn ensimmäinen osa tutkii topologisia faaseja teoreettisesti.
Määrittelemme modulaarisia tensorikagetorioita, jotka kuvaavat monen kvasihiukkasen systeemejä johdonmukaisesti, ja topologisia kvanttikenttäteorioita, jotka kuvaavat topologisten faasien matalan energian efektiivisiä teorioita.
Muitakin näkökulmia ja karakterisoivia ominaisuuksia pohditaan, muun muassa perustilan degeneraatiota, joka riippuu topologiasta eikä symmetrian rikkoutumisesta.
Topologisia faaseja voidaan tulkita myös ekvivalenssiluokiksi, joita erottavat erilaiset pitkän kantaman lomittumisominaisuudet.

Ratkaisemme murtolukuisen kvantti-Hall systeemin Coulombin vuorovaikutuksella pallon pinnalla numeerisesti ja määritämme niin sanotun topologisen lomittumisentropian.
Tämä tarkoittaa sitä, että löydämme perustilan aaltofunktiot eksaktilla diagonalisaatiolla monilla systeemin parametrien arvoilla, määritämme entropiat tietyillä kahtiajaoilla, ja ekstrapoloimme entropiat termodynaamiselle rajalle.
Tällöin entropian skaalauksessa vakiotermi vastaa kvasihiukkasten kollektiivisen kvantti-dimension logaritmia.
Tulokset vastaavat teoreettisia ennusteita.
ED:2011-09-22
INSSI record number: 42793
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