Dipole representation of half-filled Landau level

We introduce a variant of dipole representation for composite fermions in a half-filled Landau level, taking into account the symmetry under exchange of particles and holes. This is implemented by a special constraint on composite fermion and composite hole degree of freedom (of an enlarged space), that makes the resulting composite particle, dipole, a symmetric object. We study an effective Hamiltonian, that commutes with the constraint on the physical space, and fulfills the requirement for boost invariance on the Fermi level. The calculated Fermi liquid parameter F2 is in a good agreement with numerical investigations in [Phys. Rev. Lett. 121, 147601 (2018)].Comment: 9 pages, 4 figure

Harmonic and quasiconformal mappings, quasi - isometries and the curvature

U ovoj tezi se razmatraju različite osobine običnih harmonijskih preslikavanja, kvazikonformnih preslikavanja i harmonijskih preslikavanja u odnosu na zadatu konformnu metriku u slici. Dobijeni su odgovori na mnoga pitanja koja se odnose na određivanje onih osobina tih klasa funkcija, koje su od esencijalnog značaja za validnost rezultata poput onih koji uopštavaju čuvene nejednakosti Švarc-Pikovog tipa. Prednost je data geometrijskom pristupu, analizirnjem osobina Gausove krivine konformnih metrika sa kojima operišemo.This thesis considers various properties of Euclidean harmonic mappings, quasiconformal mappings and generalized harmonic mappings, which are harmonic with respect to the conformal metric on the image surface. In particular, we obtained the answers to many questions concerning these classes of functions and that are related to the determination of different properties that are of essential importance for validity of the results such as those that generalize famous inequalities of the Schwarz-Pick type. The approach used was geometrical in nature, via analyzing the properties of the Gaussian curvature of the conformal metrics we are dealing with