Classification of topological quantum matter with symmetries

Topological materials have become the focus of intense research in recent years, since they exhibit fundamentally new physical phenomena with potential applications for novel devices and quantum information technology. One of the hallmarks of topological materials is the existence of protected gapless surface states, which arise due to a nontrivial topology of the bulk wave functions. This review provides a pedagogical introduction into the field of topological quantum matter with an emphasis on classification schemes. Both fully gapped and gapless topological materials and their classification in terms of nonspatial symmetries, such as time reversal, as well as spatial symmetries, such as reflection, are considered. Furthermore, the classification of gapless modes localized on topological defects is surveyed. The classification of these systems is discussed by use of homotopy groups, Clifford algebras, K theory, and nonlinear sigma models describing the Anderson (de)localization at the surface or inside a defect of the material. Theoretical model systems and their topological invariants are reviewed together with recent experimental results in order to provide a unified and comprehensive perspective of the field. While the bulk of this article is concerned with the topological properties of noninteracting or mean-field Hamiltonians, a brief overview of recent results and open questions concerning the topological classifications of interacting systems is also provided.

Figure 1

The eight real symmetry classes that involve the antiunitary symmetries $T$ (time-reversal) and/or $C$ (particle-hole) are specified by the values of $T^2=\pm 1$ and $C^2=\pm 1$. They can be visualized on an eight-hour clock. Adapted from [446].

Figure 2

Topological defects characterized by a $D$ parameter family of $d$-dimensional Bloch-BdG Hamiltonians. Line defects correspond to $d-D=2$, while point defects correspond to $d-D=1$. Temporal cycles for point defects correspond to $d-D=0$. Adapted from [446].

Figure 3

(a) Spatial configuration of the $\mathbf{d}$ vector around a half-quantum vortex of a $p+ip$ SC. (b) Zero-energy Majorana modes of a half-quantum vortex (HQV) and a full quantum vortex (FQV).

Figure 4

(a) Dislocation on a square lattice. (b) Two inequivalent $\mathrm{\Omega }=-\pi /2$ disclinations. (c) A $\mathrm{\Omega }=\pm \pi /2$ disclination dipole.

Figure 5

Zero-energy MBSs (yellow dots) in heterostructures: (a) superconductor (SC)—magnet (M) domain wall along a QSH edge or a Chern insulator interface; (b) a flux vortex across a superconducting interface between a 3D topological (TI) and trivial insulator (I).

Figure 6

Energy spectrum of the spin-orbit coupled nanowire (3.132) in a magnetic field.

Figure 7

Gapless spectra inside the bulk gap: (a) chiral Dirac modes, (b) helical Dirac mode, (c) chiral Majorana modes, and (d) helical Majorana mode. From [363].

Figure 8

Line defect (yellow line) at a heterostructure. A, B, and C are different bulk gapped materials put together so that there is no gapless surface modes along interfaces of any pairs. The mass term $m{\mathrm{\Gamma }}_0(\varphi )$ wraps nontrivially around the line interface which results in a gapless 1D excitation localized at the line defect.

Figure 9

Heterostructure cross section on the $x\text{−}y$ plane. $\mathrm{AF}=\text{antiferromagnetic}$, $\mathrm{I}=\text{trivial insulator}$, $\mathrm{TI}=\text{topological}\text{insulator}$, $\mathrm{SC}=\text{superconductor}$, and $\mathrm{TSC}=\text{class DIII}\text{topological superconductor}$. (a) The chiral Dirac mode $\odot$ protected by winding of the magnetoelectric $\theta$ angle. (b) The helical Dirac mode $\odot \otimes$ separating opposite polarization insulating domains. (c) The chiral Majorana mode. (d) The helical Majorana mode between SC domains with TRS pairing phase $\phi =0$, $\pi$.

Figure 10

The 27 symmetry classes with reflection symmetry can be visualized as “the extended Bott clock.”

Figure 11

The classification of stable Fermi surfaces depends on how the Fermi surfaces transform under nonspatial antiunitary symmetries and hence their location in the Brillouin zone. Here $d$ denotes the spatial dimension (the dimension of the Brillouin zone) and $p$ is the codimension of the Fermi surface. The blue circles and spheres represent the contour on which the topological invariant is defined. (a) Each Fermi surface (red points and lines) is left invariant under nonspatial symmetries. (b) Different Fermi surfaces are pairwise related by the nonspatial symmetries which map $\mathsf{k}\leftrightarrow -\mathsf{k}$. Adapted from [80].

Figure 12

Surface spectrum of the Weyl semimetal (5.4) for the (100) face in the presence of the Zeeman term $\mathrm{\Delta }{\tau }_0{s}_3$. The surface and bulk states are colored in green and gray, respectively. (a) Surface spectrum as a function of surface momenta $(k_y,k_z)$. (b), (c) Surface spectrum as a function of surface momentum $k_y$ and $k_z$ with fixed $k_z=\pi /4$ and $k_y=0$, respectively. (d) Bulk Fermi surface and surface Fermi arc at the energy $E=0.1$. The Fermi arcs located within the interval $\pi /3<|k_z|<2\pi /3$ are protected by the nonzero Chern number $\mathrm{Ch}(k_z)=-1$; see Eq. (5.6). The surface modes with $|k_z|<\pi /3$ are unstable and can be gapped out by surface perturbations [e.g., by the term $\cos (3k_z/2){\tau }_1{s}_3$].

Figure 13

Surface spectrum of the time-reversal symmetric (i.e., without Zeeman term) Dirac semimetal (5.4) for the (100) face. The surface and bulk states are colored in green and gray, respectively. (a)–(c), (d)–(f) The surface spectrum in the presence and absence of the surface perturbation $+g\sin k_z{\tau }_1{s}_3$, respectively, which breaks reflection and chiral symmetry. (b), (e) The surface bands as a function of surface momentum $k_y$ with fixed $k_z=\pi /4$ and $k_z=0$, respectively. (c), (f) The bulk and surface states for a fixed energy $ϵ=0.1$. The surface Dirac cone of (a) is protected by a ${\mathbb{Z}}_2$ invariant, while the surface Fermi arc of (d) is protected by the mirror winding number ${\nu }^{+}$ (see Sec. 5b2a).

Figure 14

(a) The honeycomb lattice is a bipartite lattice composed of two interpenetrating triangular sublattices $A$ (black dots) and $B$ (blue dots). The vectors connecting nearest-neighbor and next-nearest-neighbor sites are denoted by ${\mathbf{s}}_i$ (green) and ${\mathbf{d}}_i$ (red), respectively, where ${\mathbf{s}}_1=(-1,0)$, ${\mathbf{s}}_2=\frac{1}{2}(1,\sqrt{3})$, ${\mathbf{s}}_3=\frac{1}{2}(1,-\sqrt{3})$, and ${\mathbf{d}}_1=-{\mathbf{d}}_4=\frac{1}{2}(3,\sqrt{3})$, ${\mathbf{d}}_2=-{\mathbf{d}}_5=\frac{1}{2}(3,-\sqrt{3})$, ${\mathbf{d}}_3=-{\mathbf{d}}_6=(0,-\sqrt{3})$. The mirror line $x\to -x$ is indicated by the green line. (b) Energy spectrum of a graphene ribbon with (10) edges (i.e., zigzag edges) and $(t_1,t_2)=(1.0,0.1)$. A linearly dispersing edge state (red trace) connects the Dirac points at $k_y=2\pi /3$ and $k_y=4\pi /3$ in the edge BZ. Adapted from [80].