Understanding topological phases of matter with statistical methods

  • Topological phases set themselves apart from other phases since they cannot be understood in terms of the usual Landau theory of phase transitions. This fact, which is a consequence of the property that topological phase transitions can occur without breaking symmetries, is reflected in the complicated form of topological order parameters. While the mathematical classification of phases through homotopy theory is known, an intuition for the relation between phase transitions and changes to the physical system is largely inhibited by the general complexity. In this thesis we aim to get back some of this intuition by studying the properties of the Chern number (a topological order parameter) in two scenarios. First, we investigate the effect of electronic correlations on topological phases in the Green's function formalism. By developing a statistical method that averages over all possible solutions of the manybody problem, we extract general statements about the shape of the phase diagram and investigate the stability of topological phases with respect to interactions. In addition, we find that in many topological models the local approximation, which is part of many standard methods for solving the manybody lattice model, is able to produce qualitatively correct phase transitions at low to intermediate correlations. We then extend the statistical method to study the effect of the lattice, where we evaluate possible applications of standard machine learning techniques against our information theoretical approach. We define a measure for the information about particular topological phases encoded in individual lattice parameters, which allows us to construct a qualitative phase diagram that gives a more intuitive understanding of the topological phase. Finally, we discuss possible applications of our method that could facilitate the discovery of new materials with topological properties.

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Author:Thomas MertzORCiDGND
Place of publication:Frankfurt am Main
Referee:Roser ValentíORCiDGND, Falko PientkaORCiD
Document Type:Doctoral Thesis
Date of Publication (online):2022/01/03
Year of first Publication:2021
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Granting Institution:Johann Wolfgang Goethe-Universität
Date of final exam:2022/07/02
Release Date:2022/04/08
Page Number:278
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik
Licence (German):License LogoDeutsches Urheberrecht