2. - Symmetry and Topology

Summary

In this chapter, we shall discuss the interplay of symmetry and topology that are essential in understanding the topological protection rendered by the inherent symmetries and how the topological invariants are related to physical quantities.

Introduction

Point set topology is a disease from which the human race will soon recover.

—H. Poincaré (1908)

Poincaré conjecture was the first conjecture made on topology which asserts that a three-dimensional (3D) manifold is equivalent to a sphere in 3D subject to the fulfilment of a certain algebraic condition of the form f (x, y, z) = 0, where x, y and z are complex numbers. G. Perelman has (arguably) solved the conjecture in 2006 [4]. However, on practical aspects, just the reverse of what Poincaré had predicted happened. Topology and its relevance to condensed matter physics have emerged in a big way in recent times. The 2016 Nobel Prize awarded to D. J. Thouless, J. M. Kosterlitz, F. D. M. Haldane and C. L. Kane and E. Mele getting the Breakthrough Prize for contribution to fundamental physics in 2019 bear testimony to that.

Topology and geometry are related, but they have a profound difference. Geometry can differentiate between a square from a circle, or between a triangle and a rhombus; however, topology cannot distinguish between them. All it can say is that individually all these shapes are connected by continuous lines and hence are identical. However, topology indeed refers to the study of geometric shapes where the focus is on how properties of objects change under continuous deformation, such as stretching and bending; however, tearing or puncturing is not allowed. The objective is to determine whether such a continuous deformation can lead to a change from one geometric shape to another. The connection to a problem of deformation of geometrical shapes in condensed matter physics may be established if the Hamiltonian for a particular system can be continuously transformed via tuning of one (or more) of the parameter(s) that the Hamiltonian depends on. Should there be no change in the number of energy modes below the Fermi energy during the process of transformation, then the two systems (that is, before and after the transformation) belong to the same topology class. In the process, something remains invariant. If that something does not remain invariant, then there occurs a topological phase transition.