Emergence and construction of novel gapless topological phases

Emmanuel V. C. Lopes

Jihyeon Park, Gun Sang Jeon

We investigate emergent topological gapless phases in the square-lattice Kitaev model with additional hopping terms. In the presence of nearest-neighbor hopping only, the model is known to exhibit gapless phases with two topological gapless points. When the strength of the newly added next-nearest-neighbor hopping is smaller than a certain value, qualitatively the same phase diagram persists. We find that further increase of the extra hopping results in a new topological phase with four gapless points. We construct a phase diagram to clarify the regions of emergent topological gapless phases as well as topologically trivial ones in the space of the chemical potential and the next-nearest-neighbor hopping strength. We examine the evolution of the gapless phases in the energy dispersions of the bulk as the chemical potential varies. The topological properties of the gapless phases are characterized by the winding numbers of the present gapless points. We also consider the ribbon geometry to examine the corresponding topological edge states. It is revealed that Majorana-fermion edge modes exist as flat bands in topological gapless phases. We also perform the analytical calculation as to Majorana-fermion zero-energy modes and discuss its implications on the numerical results.

Roderich Moessner, Joel E. Moore

Rui Wen, Andrew C. Potter

Intrinsically gapless symmetry-protected topological phases (igSPT) are gapless systems with SPT edge states with properties that could not arise in a gapped system with the same symmetry and dimensionality. igSPT states arise from gapless systems in which an anomaly in the low-energy (infrared) symmetry group emerges from an extended anomaly-free microscopic (ultraviolet) symmetry. We construct a general framework for constructing lattice models for igSPT phases with emergent anomalies classified by group cohomology, and establish a direct connection between the emergent anomaly, group-extension, and topological edge states by gauging the extending symmetry. In many examples, the edge-state protection has a physically transparent mechanism: the extending UV symmetry operations pump lower-dimensional SPTs onto the igSPT edge, tuning the edge to a (multi)critical point between different SPTs protected by the IR symmetry. In two- and three-dimensional systems, an additional possibility is that the emergent anomaly can be satisfied by an anomalous symmetry-enriched topological order, which we call a quotient-symmetry-enriched topological order (QSET) that is sharply distinguished from the nonanomalous UV SETs by an edge phase transition. We construct exactly solvable lattice models with QSET order.

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.

Antônio Sérgio Teixeira Pires