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Topological degeneracy and vortex dynamics in the Kitaev honeycomb model
Get PDFWe investigate the loop symmetries of Kitaev’s honeycomb lattice model. These provide a natural framework to study the abelian topological phase. We show that in the thermodynamic limit, the abelian phase on a torus is topologically degenerate to all orders of the Brillioun-Wigner expansion. We then demonstrate that these symmetries correspond to the evolution of fermions in closed loops. Importantly, we demonstrate that these fermions are made from energetically confined pairs of vortices and that their transport happens with no additional energy cost. This has important implications for the gapped and gapless phases
Aspects of the Holographic Study of Flavor Dynamics
Get PDFThis thesis is dedicated to the holographic study of flavor dynamics. The technique employed is a D7–brane probing of various D3–brane backgrounds. The first topic covered studies the influence of an external magnetic field on a flavored large N Yang– Mills theory. The theory exhibits spontaneous chiral symmetry breaking. A discrete self–similar structure of the spontaneous symmetry breaking mechanism is observed, this structure is reflected in the meson spectrum. The meson spectrum exhibits Zeeman splitting and characteristic GMOR relation. The second topic examines thermal properties of the dual gauge theory. The study reveals a first order phase transition associated to the melting of mesons. The critical ratio of the bare quark mass and temperature at which the transition happens is computed. The third topic studies the phase structure of the finite temperature dual gauge theory in the presence of magnetic field. A phase diagram of the theory is obtained. The temperature restores the chiral symmetry while the magnetic field has a freezing effect on the meson melting. The meson spectrum exhibits Zeeman splitting and characteristic GMOR relation. Thermodynamic quantities such as free energy, entropy, and magnetization are computed. The fourth topic studies the addition of an external electric field. The observed effect is dissociation of the bound quarks, favoring the meson melting. For sufficiently strong electric fields a global electric current is induced. Thus, the dissociation of mesons corresponds to an insulator/conductor phase transition. This transition persists at vanishing temperature. The fifth topic studies the addition of an R–charge chemical potential via brane probing of the spinning D3–brane geometry. The corresponding phase diagram is obtained. The chemical potential favors the dissociation of mesons. For high chemical potential, a finite phase difference between the bare quark mass and the quark condensate is induced. The last topic explores universal properties of gauge theories dual to the Dp/Dq system. A universal discrete self–similar behavior associated to the insulator/conductor phase transition is observed and the corresponding scaling exponents are computed. A similarity between the electric field and R–charge chemical potential cases is discussed
Constraints on an asymptotic safety scenario for the Wess–Zumino model
Get PDFUsing the nonrenormalization theorem and Pohlmeyer’s theorem, it is proven that there cannot be an asymptotic safety scenario for the Wess-Zumino model unless there exists a non-trivial fixed point with (i) a negative anomalous dimension (ii) a relevant direction belonging to the Kähler potential
Non-locality of non-Abelian anyons
Get PDFTopological systems, such as fractional quantum Hall liquids, promise to successfully combat environmental decoherence while performing quantum computation. These highly correlated systems can support non-Abelian anyonic quasiparticles that can encode exotic entangled states. To reveal the non-local character of these en-coded states we demonstrate the violation of suitable Bell inequalities. We provide an explicit recipe for the preparation, manipulation and measurement of the desired correlations for a large class of topological models. This proposal gives an operational measure of non-locality for anyonic states and it opens up the possibility to violate the Bell inequalities in quantum Hall liquids or spin lattices
Compatible associative products and trees
Get PDFWe compute dimensions and characters of the components of the operad of two compatible associative products, and give an explicit combinatorial construction of the corresponding free algebras in terms of planar rooted trees
Parking functions and vertex operators
Get PDFWe introduce several associative algebras and families of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we define a family of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct families of vector spaces whose dimensions are Catalan numbers and Fuss–Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Cherednik algebras
The discrete Feynman integral
Get PDFWe construct a genuine Radon measure with values in B(l^2(Z^d)) on the set of paths in Z^d representing Feynman’s integral for the discrete Laplacian on l^2(Z^d), and we prove the Feynman integral formula for the solutions of the Schrödinger equation with Hamiltonian H = (-1/2)∆ + V , where ∆ is the discrete Laplacian and V is an arbitrary bounded potential
Boundary states and edge currents for free fermions
Get PDFWe calculate the ground state current densities for 2+1 dimensional free fermion theories with local, translationally invariant boundary states. Deformations of the bulk wave functions close to the edge and boundary states both may cause edge current divergencies, which have to cancel in realistic systems. This yields restrictions on the parameters of quantum field theories which can arise as low energy limits of solid state systems. Some degree of Lorentz invariance for boosts parallel to the boundary can be recovered, when the cutoff is removed
