Topological Quantum Matter: A Field Theoretical Perspective

Jan Carl Budich, Björn Trauzettel

Abstract magnified image Owing to the enormous interest the rapidly growing field of topological states of matter (TSM) has attracted in recent years, the main focus of this review is on the theoretical foundations of TSM. Starting from the adiabatic theorem of quantum mechanics which we present from a geometrical perspective, the concept of TSM is introduced to distinguish gapped many body ground states that have representatives within the class of non‐interacting systems and mean field superconductors, respectively, regarding their global geometrical features. These classifying features are topological invariants defined in terms of the adiabatic curvature of these bulk insulating systems. We review the general classification of TSM in all symmetry classes in the framework of K‐Theory. Furthermore, we outline how interactions and disorder can be included into the theoretical framework of TSM by reformulating the relevant topological invariants in terms of the single particle Green's function and by introducing twisted boundary conditions, respectively. We finally integrate the field of TSM into a broader context by distinguishing TSM from the concept of topological order which has been introduced to study fractional quantum Hall systems. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Thomas Klein Kvorning

Ravi Kumar, Sujit Sarkar

An attempt is made to find different emergent quantum phases for interacting topological state of quantum matter. Our study is based on the quantum field theoretical renormalization group (RG) calculations. The behaviour of the RG flow lines give the emergence of different quantum phases for non-interacting and interacting topological state of quantum matter. We show explicitly electron-electron interaction can turn a topologically trivial phase into a topologically nontrivial one and also topologically nontrivial phase to topologically trivial phase. We show that physics of emergence goes beyond the quantum Berezinskii-Kosterlitz-Thouless transition. We also present the analysis of fixed point and show the behaviour of fixed point changes in presence and absence of interaction. This work provides a new perspective not only from the topological state of interacting quantum matter and but also for the correlated quantum many -body physics.

Christian Spånslätt

States of matter which are described by concepts in topology are of great interest in the community of condensed matter physics. Both from a pure theoretical perspective, where many non-intuitive and mathematically rich results can be found from seemingly simple models, and also since recent experimental developments allows for explicit construction of devices where the underlying topology has observable consequences. This thesis provides a conceptual introduction to the framework describing such topological states of matter (TSM). It starts by explaining ordinary band theory which is followed by a discussion of quantum mechanical symmetries. These two concepts provides a method of classifying all non-interacting fermionic TSM. Following are explicit applications to a few models in one spatial dimension. In the accompanying paper, we study one-dimensional superconductors with Josephson-junctions allowing phase shifts of π. We make a detailed analysis of the junction bound states and their properties in some different settings and compare the behaviour of trivial and topological superconducting junctions. In addition, we provide a phenomenological topological field theory in the low energy limit.

F. D. M. Haldane