A Noncommutative Sigma Model

We replaced the classical string theory notions of parameter space and world-time with noncommutative tori and consider maps between these spaces. The dynamics of mappings between different noncommutative tori were studied and a noncommutative generalization of the Polyakov action was derived. The quantum torus was studied in detail as well as *-homomorphisms between different quantum tori. A finite dimensional representation of the quantum torus was studied and the partition function and other path integrals were calculated. At the end we proved existence theorems for mappings between different noncommutative tori.Comment: The thesis was based on an article by Vargese Mathai and Jonathan Rosenberg with the same titl

Interplay of Soundcone and Supersonic Propagation in Lattice Models with Power Law Interactions

We study the spreading of correlations and other physical quantities in quantum lattice models with interactions or hopping decaying like $r^{-\alpha}$ with the distance $r$. Our focus is on exponents $\alpha$ between 0 and 6, where the interplay of long- and short-range features gives rise to a complex phenomenology and interesting physical effects, and which is also the relevant range for experimental realizations with cold atoms, ions, or molecules. We present analytical and numerical results, providing a comprehensive picture of spatio-temporal propagation. Lieb-Robinson-type bounds are extended to strongly long-range interactions where $\alpha$ is smaller than the lattice dimension, and we report particularly sharp bounds that are capable of reproducing regimes with soundcone as well as supersonic dynamics. Complementary lower bounds prove that faster-than-soundcone propagation occurs for $\alpha<2$ in any spatial dimension, although cone-like features are shown to also occur in that regime. Our results provide guidance for optimizing experimental efforts to harness long-range interactions in a variety of quantum information and signaling tasks.Comment: 20 pages, 8 figure

A noncommutative sigma model.

We replace the classical string theory notions of mapping between parameter space and world-time with noncommutative tori mapping between these spaces. The dynamics of mappings between different noncommutative tori are studied and noncommutative versions of the Polyakov action and the Euler-Lagrange equations are derived. The quantum torus is studied in detail, as well as C*-homomorphisms between different quantum tori. A finite dimensional representation of the quantum torus is studied, and the partition function and other path integrals are calculated. At the end we prove existence theorems for mappings between different noncommutative tori.This abstract was presented at the ‘Studentesimposium in die Natuurwetenskappe 2011’, presented under the protection of the Suid- Afrikaanse Akademie vir Wetenskap en Kuns. The symposium was held at the University of South Africa on 27–28 October 2011.http://www.satnt.ac.zaam201

Dynamics of long-range interacting quantum spin systems

Thesis (PhD)--Stellenbosch University, 2015.ENGLISH ABSTRACT: In this thesis we study the time evolution of correlation functions in quantum lattice models in the presence of long-range interactions or hopping decaying asymptotically as a power law. For a large class of initial conditions, exact analytic results are obtained in arbitrary lattice dimension for the longrange Ising model. In contrast to the nearest-neighbour case, we find that correlations decay like stretched or compressed exponentials in time. Provided the long-range character of the interactions is sufficiently strong, pronounced prethermalization plateaus are observed and relaxation timescales are widely separated. Starting from uncorrelated states that are easily prepared in experiments, we show the dynamical emergence of correlations and entanglement in these far-from-equilibrium interacting quantum systems. We characterize these correlations by the entanglement entropy, concurrence, and squeezing, which are inequivalent measures of entanglement corresponding to different quantum resources. For interaction exponents larger than the lattice dimensionality, a Lieb- Robinson-type bound effectively restricts the spreading of correlations to the interior of a causal region, but allows supersonic (faster than linear) propagation. Using tools of quantum metrology, for any exponents smaller than the lattice dimension, we construct Hamiltonians giving rise to quantum channels with capacities not restricted to any causal region. An analytic analysis of long-range Ising models illustrates the disappearance of the causal region and the creation of correlations becoming distance-independent. In all models we analyzed the spreading of correlations follows a power law, and not the exponential increase of the long-range Lieb-Robinson bound. Lieb-Robinson-type bounds are extended to strongly long-range interactions where the interaction exponent is smaller than the lattice dimension, and we report particularly sharp bounds that are capable of reproducing regimes with soundcone as well as supersonic dynamics. Our results provide guidance for optimizing experimental efforts to harness long-range interactions in a variety of quantum information and signaling tasks.AFRIKAANSE OPSOMMING: In hierdie tesis studeer ons die tydevolusie van korrelasiefunksies in kwantumroostermodelle in die teenwoordigheid van lang-afstand interaksies of spronge wat asimptoties in `n magswet verval. Vir `n groot versameling begintoestande word presiese analitiese resultate vir die lang-afstand-Ising-model verkry. In teenstelling met die naaste-naasliggende-interaksie-Ising-model vind ons dat korrelasies soos uitgerekte of saamgepersde eksponentiele funksies in tyd verval. Indien die lang-afstand gedrag van die interaksies lank genoeg is word lanklewende kwasi-stationere toestande gevorm as gevolg van wyd verspreide tydskale wat in die ontspanningsgedrag van die korrelasiefunksies voorkom. Wanneer ongekorreleerde begintoestande, wat eenvoudig in die laboratorium voorberei kan word, gebruik word kan die dinamiese opkoms van verskillende maatstawe van verstrengeling analities bereken word. Gevolglik kan hierdie verstrengelde toestande gebruik word om kwantumberekeninge mee uit te voer. Vir interaksie-eksponente groter as die roosterdimensie bestaan daar sogenaamde Lieb-Robinson-grense wat die verspreiding van korrelasies beperk tot die binneruim van `n oorsaaklike-kegel, maar steeds supersoniese verspreiding (vinniger as line^er) toelaat. Deur gebruik te maak van gereedskap uit kwantummeetkunde kan ons, vir enige interaksie-eksponent kleiner as die roosterdimensie, Hamiltoniane konstrueer wat oorsaak gee aan kwantumkanale met kapasiteite wat nie beperk word deur enige oorsaaklike-kegel nie. Die analitiese resultate van die lang-afstand-Ising-model illustreer die verdwyning van die oorsaaklike-kegel sowel as die vorming van afstand onafhanklike korrelasies. In al die modelle waar ons die verspeiding van korrelasies bestudeer het, het die speiding magswette gevolg, en nie die eksponentiele toename wat deur die lang-afstand Lieb-Robinson-grense voorspel word nie. Lieb-Robinson-grense word uitgebrei na kwantumsisteme met sterk-lang-afstand interaksies waar die interaksie-eksponent kleiner is as die roosterdimensie. Besonderse skerp grense word voorgedra wat die oorgang van line^ere na supersoniese verspreiding vasvang. Die resultate wat in hierdie tesis vasgevang is verskaf riglyne vir die optimering van eksperimentele pogings on lang-afstnad interaksies in te span in `n verskeidenheid kwantuminformasie en -seintake