Topological Quantum Matter

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)

F. D. M. Haldane

Xiao-Gang Wen

It has long been thought that all different phases of matter arise from symmetry breaking. Without symmetry breaking, there would be no pattern, and matter would be featureless. However, it is now clear that for quantum matter at zero temperature, even symmetric disordered liquids can have features, giving rise to topological phases of quantum matter. Some of the topological phases are highly entangled (that is, have topological order), whereas others are weakly entangled (that is, have symmetry-protected trivial order). This Review provides a brief summary of these zero-temperature states of matter and their emergent properties, as well as their importance in unifying some of the most basic concepts in nature.

Wei-Feng Tsai, Hsin Lin, Arun Bansil

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.