טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentOvdat Omrie
SubjectScale and Parity Symmetry Breaking in Graphene:
Universality, Discrete Scale Invariance and Vacuum
Charge
DepartmentDepartment of Physics
Supervisor Professor Eric Akkermans


Abstract

One of the most interesting predictions resulting from quantum physics is the violation of classical symmetries, collectively referred to as anomalies. A remarkable class of anomalies occurs when the continuous scale symmetry of a scale-free quantum system is broken into a discrete scale symmetry for a critical value of a control parameter. This is an example of a (zero temperature) quantum phase transition. Such an anomaly takes place for the quantum inverse square potential known to describe ‘Efimov physics’.

The first part of this thesis is dedicated to the study this transition in the framework of a scale invariant systems. We explore its universal features, ingredients and realizations in relevant cases. Specifically, we demonstrate the existence and universality of this quantum phase transition for a massless fermion in an attractive Coulomb potential and present convincing experimental evidence of it as realized around a charged vacancy in graphene. Furthermore, we consider an infinite class of scale invariant hamiltonians allowing for anisotropic scaling between space and time. We show that the transition to discrete scale invariance is realized as a generic feature in the landscape of these hamiltonians. We formulate a renormalization group picture and demonstrate that close to the critical point, the discrete scale invariant phase is characterized by an isolated, closed, attracting trajectory in renormalization group space (a limit cycle). Moving in appropriate directions in the parameter space of couplings this picture is altered to one controlled by a quasi periodic attracting trajectory (a limit torus) or fixed points. We identify a direct relation between the critical point, the renormalization group picture and the power laws characterizing the zero energy wave functions.

An additional type of anomaly occurs when the parity symmetry of the ground state of a massless Dirac system is broken in the presence of a magnetic flux. In this case, the flux induces zero energy bound states and a fractional vacuum charge with abnormal parity. As a result, the Index of the corresponding Dirac operator acquires non-zero values proportional to the flux as well as to the vacuum charge.

The second part of this thesis is devoted to the study of neutral vacancies in graphene and their relation to the aforementioned physics of gauge field induced vacuum charge. A single vacancy induces a localized stable charge of order unity interacting with other charges of the conductor through an unscreened Coulomb potential. It also breaks the symmetry between the two triangular graphene sub-lattices hence inducing zero energy states at the Dirac point. Here we show the fractional and pseudo-scalar nature of this vacancy charge. A continuous Dirac model is presented which relates zero modes to vacuum fractional charge and to a parity anomaly. This relation constitutes an Index theorem and is achieved by using particular chiral boundary conditions, which map the vacancy problem onto edge state physics. This essential difference makes vacancy physics relatively easy to implement and an interesting playground for topological charge switching.