Colloquia IFIC - Severo Ochoa
The 2016 Nobel prize: Topological phases of matter and topological phase transitions
by Prof. Alfredo Segura (Universidad de Valencia)
Tuesday, 11 April 2017 from to (Europe/Madrid)
The 2016 Nobel Prize in Physics of 2016 was awarded to David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz, for their "theoretical discoveries on topological phase transitions and topological phases of matter." The aim of the talk is presenting a simplified version of the main innovations introduced by the winners and showing the vast influence that their ideas have had in different fields of condensed matter physics. After introducing some basic notions of topology, especially the concept of topological invariant, we will discuss in what context they can be applied to the description of the electronic structure of solids and how they affect some of the properties that derive from that structure. Both the analysis of the electronic densities in the "real" space (position space) and the electron state densities in the reciprocal space (wave vector space) require the use of "surfaces" that can be described and classified in basic topological terms. Nevertheless, topological concepts introduced by the 2016 Nobel laureates in the description of electronic states are far more subtle and refer to the topological structure of the system Hilbert space. In addition to quantum numbers derived from basic symmetries, it is possible to introduce topological quantum numbers: while the former are extremely sensitive to defects in materials, the latter are "robust" and hardly affected by defects. They give rise to phenomena such as the quantum Hall effect in which, regardless of the device design and defects, the physical response is controlled by a universal constant (the von Klitzing resistance). The concept of phase transition associated with the breaking of a given symmetry, applies to transitions between "ordinary" states of condensed matter. It cannot encompass some physical effects that occur in low dimensional systems (electronic, magnetic, superfluid, etc.), associated to changes in topological structure of defects or elementary excitations of the system. This seminal idea, introduced by Thouless, Haldane and Kosterlitz can describe and systematize the great variety of exotic states discovered in low dimensional systems (melting of 2D solids, superfluid or superconducting thin films, 2D networks of Josephson junctions, 1D and 2D spin systems, etc).