Higher-Order Symmetry Enriched Topological Phases

Frank Pollmann, Ching-Yu Huang, Xie Chen

Shang-Qiang Ning, Zheng-Xin Liu, Peng Ye

Bowen Shi, Yuan-Ming Lu

Bei Zeng, Xie Chen, D. L. Zhou +1 more

Yizhi You, Daniel Litinski, Felix von Oppen

We show that interactions can drive a class of higher order topological superconductors (HOTSCs) into symmetry-enriched topologically ordered phases exemplified by topological quantum error correcting codes. In two dimensions, interacting HOTSCs realize various topologically ordered surface and color codes. In three dimensions, interactions can drive HOTSCs protected by subsystem symmetries into recently discovered fracton phases. We explicitly relate fermion parity operators underlying the gapless excitations of the HOTSC to the Wilson algebra of symmetry-enriched quantum codes. Arrays of crossed Majorana wires provide an experimental platform for realizing fracton matter and for probing the quantum phase transition between HOTSCs and the topologically ordered phase.

Wen, Rui

Despite great success we have achieved in the past two decades on systematically understanding gapped topological phases, a firm foundation for understanding gapless topological phases is still lacking. In this dissertation I develop a general framework to describe, classify and construct gapped and gapless phases with or without symmetry. I then apply this framework to construct new gapless phases, and classify some classes of them completely. The first chapter is a non-technical summary of the subjects, ideas and results of the rest of the dissertation. In the first chapter I introduce the notion of gapless topological phases, and briefly discuss their general structures. We also discuss how topological holography can be useful for classifying and constructing gapless phases with generalized symmetries. In chapter two I develop a framework for understanding general quantum phases and phase transitions with generalized symmetries. After reviewing the established theory of gapped phases, we introduce two key notions: quantum order and symmetry-enriched quantum order. These concepts are the correct generalized of topological order and symmetry-enriched topological orders. I show that these new notions subsume conventional phase transitions, gapless spin liquids, and gapless symmetry-protected topological phases. For a class of symmetry-enriched quantum orders with finite many topological defects, we present a general classification using higher tensor category theory. The rest of the dissertation provides more detailed studies of specific gapless topological phases, including classification and lattice model construction of gapless symmetry-protected topological phases and the connection to the theory of topological holography.