Beyond Band Insulators: Topology of Semi-metals and Interacting Phases

TL;DR

This work extends topological insulator concepts to gapless and interacting regimes. It first analyzes Weyl semimetals, detailing stability via Berry monopoles, Fermi-arc surface states, and the Adler–Bell–Jackiw anomaly, then broadens the scope to general semimetal generalizations and candidate materials. It then develops the topology of interacting phases, showing how interactions can merge or create new phases, with 1D SPT phases governed by projective symmetry representations and higher-dimensional bosonic SRE states described by K-matrix/Chern–Simons formalisms and group cohomology. The article concludes with open questions and potential experimental realizations, emphasizing the deep connections between symmetry, topology, and quantum entanglement in both gapless and gapped interacting systems.$

Abstract

The theory of topological insulators and superconductors has mostly focused on non-interacting and gapped systems. This review article discusses topological phases that are either gapless or interacting. We discuss recent progress in identifying gapless systems with stable topological properties (such as novel surface states), using Weyl semimetals as an illustration. We then review recent progress in describing topological phases of interacting gapped systems. We explain how new types of edge states can be stabilized by interactions and symmetry, even though the bulk has only conventional excitations and no topological order of the kind associated with Fractional Quantum Hall states.

Figures (4)

• Figure 1: Surface states of a Weyl semimetal. a)The surface states of a Weyl semimetal form an arc connecting the projections of the two Weyl points to one another. Points on the Fermi arc can be understood as the edge state of a two-dimensional insulator, and the Berry flux of the Weyl points ensures that either the insulators represented by the red or blue planes are integer Hall states, which have edge modes. Here we depict the former. b) A graph of the dispersion of the surface modes (the pink plane) and how this joins to the bulk states (represented by solid red and blue cones)Wan.
• Figure 2: A sketch of the calculated phase diagram of pyrochlore iridates with interaction strength (U) on the horizontal axis. The Weyl semi-metal appears in a regime of intermediate correlations that is believed to be realized in some members of the pyrochlore iridates. It appears as a transition phase between a trivial (Mott) and topological (Axion) insulator. The inset shows the location of the 24 Weyl nodes, labeled by their $\pm$ chirality, as predicted for this cubic system [Wan].
• Figure 3: a) Effects of interactions on classification of topological insulators. A schematic phase diagram (based on two of the phases of Majorana fermion chains) with a parameter ($t$) related to hopping and an interaction parameter ($g$). Along the non-interacting vertical axis, this diagram contains two distinct phases. When a negative interaction is turned on, the two phases connect to each other. b) The group operations on topological phases: addition is represented by layering states, the negative of a state is its mirror image. The last figure shows why this is. The arrows are supposed to indicate that the states may break reflection symmetry.
• Figure 4: Summary of some simple 'integer' bosonic topological phases. 1) A chiral phase of bosons (no symmetry required). An integer multiple of eight chiral bosons at the edge is needed to evade topological order, leading to a quantized thermal Hall conductance $\kappa_{xy}/T=8nL_0$ in units of the universal thermal conductance $L_0=\frac{\pi^2k_B^2}{3h}$. These are bosonic analogs of chiral superconductors. (2) A non-chiral phase of bosons protected by $U(1)$ symmetry (eg. charge conservation). Distinct phases can be labeled by the quantized Hall conductance $\sigma_{xy}=2n\sigma_0$, which are even integer multiples of the universal conductance $\sigma_0=q^2/h$ for particles with charge $q$. These are bosonic analogs of the integer quantum Hall phases.