# Strongly Correlated and Topological Electron Systems - from microscopic descriptions to effective theories

Project leader

Funding source

Swedish Research Council - Vetenskapsrådet (VR) |

Project Details

Description

Background: Up until the early 1980-ies, phases of matter were classified according to the pattern of symmetry breaking. In a crystal the full translational symmetry is broken to a discrete subgroup, and in a magnet the rotational symmetry is broken to a single rotation around the axis of magnetization. The discovery of the quantum Hall effect, and later topological insulators and superconductors, showed that this is not sufficient, and the new concepts of topological and quantum order were introduced to describe the new states of matter. Sometimes it is possible to ignore the particle – particle interaction, and in these cases have a good understanding of what are the possible phases, and what are their properties. Often, however, the interactions are crucial, and these so called fractional topological phases have exotic properties typically hosting fractionally charged quasiparticles obeying fractional statistics. The paradigmatic example is the fractional quantum Hall effect. For these fractional phases there is neither an exhaustive classification, nor any general theoretical tools for analyzing individual examples. In this project we will study important examples of topological matter, mostly in two dimensions (surfaces/ interfaces), but also in one dimension (wires) and three dimensions. Aims and importance: The main trust of our research is to construct and study effective theories for these interacting exotic states of matter, and to find out how they are related to each other and how they can be derived from more detailed microscopic models. An effective theory is rather insensitive to many microscopic properties of a particular system and can thus captures only certain low energy phenomena. It is however precisely this insensitivity to details that allows us to use them to sharply distinguish different phases of matter. Effective theories are of different types – some describe only topological properties, while others keep some information about low energy dynamics – and in one of our projects we will study how such effective descriptions for exotic two-dimensional superconductors are related to each other, and then, based on this understanding, generalize to three dimensions where no such theory is yet proposed. If successful, this research will deepen our general understanding of exotic matter, and since our work proceeds by studying particular system, we are also likely to learn new things about them. A second main theme is the fractional quantum Hall effect, where one project aims at understanding how a general high level scheme for constructing wave functions is related to the underlying microscopic theory. In yet another line of study we will consider various network configurations of one-dimensional topological wires, with the eventual goal of constructing a general network theory. Methods: In our work, we use a variety of theoretical techniques, mainly quantum filed theory, but also standard many-body quantum theory. Computer algebra is often an important tool in the calculations. Most of our work is not directly aimed at interpreting or proposing experiments, but in an initial state of the project on topological wires, where we study simple “component” setups, we will keep in close contact with an experimental group. An important part of our research is about the “hierarchy states” which is a class of quantum Hall liquids for which we have developed methods to calculate explicit many body wave functions. The expressions are however complicated, and are usually computationally hard to evaluate. During the last decade new numerical methods based on insights from quantum information theory has been successfully applied to simple quantum Hall states. We aim to adopt these “matrix product state” methods, which is employing state-of-the-art computer calculations, to be useful also for the hierarchy states.