**Electro-Magnetic Duality, Magnetic Monopoles and Topological Insulators**

**ABSTRACT**

The Maxwell equations of electrodynamics acquire an additional symmetry if one assumes the existence of hypothetical particles-magnetic monopoles, carrying a magnetic charge. The additional internal symmetry is the electro-magnetic duality generated by the rotations in the space of electric and magnetic charges.

In this project we revise the electromagnetic duality in his global aspect starting with the celebrated Dirac monopole, a singular solution in a slightly modified Maxwell theory. We then take account of the new insight on the duality in the broken SO(3) gauge theory where the magnetic monopoles arose as finite-energy smooth solution (found by ’t Hooft and Polyakov). The stability of these monopoles is guaranteed by the conservation of topological invariants, i.e., these are topologically protected states.

The spectrum of the gauge theory states enjoys a symmetry between the electrically charged gauge boson and the magnetic monopole, manifesting a quantum electro-magnetic duality which turns out to be a part of larger SL(2; Z)-group symmetry acting on the 2-dimensional charge lattice.

Recently the idea of magnetic monopoles and dyons was revived by the discovery of new kind of materials known as topological insulators. The theoretical considerations in the modified axion electrodynamics show that the electric charges on the boundary of a topological insulator induce mirror images carrying magnetic charges. We consider carefully the mirror images in the case of topological insulator with planar and spherical boundary. We then provide a description of the induced mirror images in a manifestly SL(2; Z)-covariant form.

**TABLE OF CONTENTS**

1 Introduction

2 Electro-Magnetic Duality in Maxwell Theory 5

2.1 Maxwell Theory with Magnetic Charges . . . . . . . . . . 5

2.2 Dirac Monopole . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Angular Momentum of EM field . . . . . . . . . . . . . . 9

2.4 CP-symmetry and Dyon Quantization . . . . . . . . . . . 11

3 ’t Hooft-Polyakov monopole 15

3.1 Georgi-Glashow model . . . . . . . . . . . . . . . . . . . 15

3.2 Finite Energy solutions . . . . . . . . . . . . . . . . . . . 18

3.3 t’ Hooft-Polyakov ansatz . . . . . . . . . . . . . . . . . . 18

3.4 Topological Charges . . . . . . . . . . . . . . . . . . . . 22

3.5 Bogomol’nyi-Prasad-Sommerfeld(BPS) state . . . . . . . 26

4 Duality Conjectures 31

4.1 Montonen-Olive Conjecture . . . . . . . . . . . . . . . . 31

4.2 The Witten Effect . . . . . . . . . . . . . . . . . . . . . . 32

4.3 SL(2; Z)- Duality . . . . . . . . . . . . . . . . . . . . . . 34

5 Topological Insulators 37

5.1 Axion Electrodynamics . . . . . . . . . . . . . . . . . . . 37

5.2 Topological Insulators with Planar Boundary . . . . . . . 38

5.3 Spherical Topological Insulators . . . . . . . . . . . . . . 41

5.4 Topological insulator and SL(2;Z) . . . . . . . . . . . . 48

Bibliography 53

**CHAPTER ONE**

**Introduction**

This thesis will consider abelian U(1) and non-abelian SO(3) gauge theories allowing for states carrying units of magnetic charge, the so called magnetic monopoles. The existence of such magnetic monopoles was first suggested by Dirac as a speculation and an outcome of a thought experiment [3]. It attracts so much attention because if a monopole with magnetic charge g exists in nature, would automatically imply the quantization of the electric charge. In fact, the requirement that the wave-function solving the Schrodinger in the presence of monopole is single-valued function implies the Dirac quantization condition eg = nh n 2 Z where h stays for the Plank constant h = 2~ and then all electric charges are multiples of a minimal electric charge e = h g.

The presence of magnetic charges would restore the broken symmetry between electric and magnetic charges, and the extended Maxwell equations enjoys electromagnetic duality, an exchange symmetry of the electric and magnetic components of the electromagnetic field. The electrodynamics with magnetic monopoles becomes highly symmetric, reducing the difference between “electric” and “magnetic” to a matter of convention.

The Dirac monopole suffers from one defect, it is described by a singular potential. It is only in the ’70 after the advent of the non-abelian gauge theories when ’t Hooft [9] and Polyakov [7] independently discovered that in a model with non-abelian gauge group G spontaneously broken to a U(1) through the Higgs mechanism there exists a non-singular non-perturbative solution with a finite energy which is looking from outside like a Dirac monopole. This finite energy solution is called ’t Hooft-Polyakov monopole, it is a static field configuration with potential non-vanishing at spatial infinity and Higgs field asymptotically approaching one of the Higgs vacuums. The stability of the finite energy solution is guaranteed by the conservation of topological charges, these are topological invariants of the field configuration. The magnetic charge of the ’t Hooft-Polyakov monopole is a topological charge. The mass of the magnetic monopole has a lower bound, found by Bogomol’nyi [1]; the states that saturate the bound are called BPS-states (after Bogomol’nyi-Prasad-Sommerfeld). Montonen and Olive put forward a conjecture [5] that there should exist a dual “magnetic” gauge theory in which the roles of the massive gauge bosons and the magnetic BPS-monopoles are exchanged. This is an attractive possibility, since due to the Dirac quantization condition if the coupling constant e of the original theory is small, the coupling constant g = h e of the dual “magnetic” theory must be large, and vice versa. Therefore the strong coupling regime of a gauge theory will be controlled by the weak coupling regime of its dual gauge theory. An additional term in the Lagrangian of the gauge theory describes the Witten effect, forcing the magnetic monopoles to pick up non-trivial electric charges. Thus the excitation of the theory become dyons, these are particles with both electric and magnetic charge. The Montonen-Olive conjecture exchanging electric charged states with magnetic ones provides a quantum electromagnetic duality which can be enhanced to a larger group SL(2;Z) operating on the lattice of dyons charges, that is, the two-dimensional lattice spanned by the quantized electric and

magnetic charges.

Magnetic monopoles came back to the scene in a new guise after the discovery of the topological insulators. Topological insulators are new electronic materials insulating in bulk but having gapless edge or surface states

which are protected against the opening up of a gap as long as the time-reversal symmetry is respected. Recently, Xiao-Liang Qi, Taylor Hughes and Shou-Cheng Zhang [10] proposed that the topological band insulator can provide a condensed-matter example of axionic electromagnetism. The axion field is now disguised as a parameter of the medium, together with the permittivity and the permeability . Only two disconnected value of are compatible with the time-reversal symmetry of the problem; these are = 0 corresponding to an ordinary (trivial) insulator and = corresponding to a topological insulator. In the work [11] it was argued that an electric charge near the interface of topological insulators (with an ordinary insulator) induces as a mirror image a magnetic monopole in the bulk.

We do a careful analysis of the induces charges of a topological insulator with a planar and spherical boundary. Finally we apply the ideas of the Montonen-Olive duality in the context of topological insulators providing a description of the induced mirror images into a SL(2;Z)-covariant form clarifying the meaning of the image charges.

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