Classification of topological crystalline insulators based on representation theory

Topological crystalline insulators define a new class of topological insulator phases with gapless surface states protected by crystalline symmetries. In this work, we present a general theory to classify topological crystalline insulator phases based on the representation theory of space groups. Our approach is to directly identify possible nontrivial surface states in a semi-infinite system with a specific surface, of which the symmetry property can be described by 17 two-dimensional space groups. We reproduce the existing results of topological crystalline insulators, such as mirror Chern insulators in the $pm$ or $pmm$ groups, $C_{nv}$ topological insulators in the $p4m,p31m$, and $p6m$ groups, and topological nonsymmorphic crystalline insulators in the $pg$ and $pmg$ groups. Aside from these existing results, we also obtain the following results: (1) there are two integer mirror Chern numbers (${\mathbb{Z}}^2$) in the $pm$ group but only one ($\mathbb{Z}$) in the $cm$ or $p3m1$ group for both the spinless and spinful cases; (2) for the $pmm$ ($cmm$) groups, there is no topological classification in the spinless case but ${\mathbb{Z}}^4$ (${\mathbb{Z}}^2$) classifications in the spinful case; (3) we show how topological crystalline insulator phase in the $pg$ group is related to that in the $pm$ group; (4) we identify topological classification of the $p4m,p31m$, and $p6m$ for the spinful case; (5) we find topological nonsymmorphic crystalline insulators also existing in $pgg$ and $p4g$ groups, which exhibit new features compared to those in $pg$ and $pmg$ groups. We emphasize the importance of the irreducible representations for the states at some specific high-symmetry momenta in the classification of topological crystalline phases. Our theory can serve as a guide for the search of topological crystalline insulator phases in realistic materials.

Figure 1

Schematic plot of the surface BZ of rectangular, square, hexagonal, and rhombic groups in (a), (b), (c), and (d), respectively.

Figure 2

(a) Schematic plot of a semi-infinite system. (b) Schematic plot of the surface states of TR invariant TIs. $\overline{\mathrm{\Gamma }}=(0,0)$ and $\overline{\mathrm{X}}=(\pi ,0)$ are two TI invariant momenta in the surface BZ. (c) Schematic plot of the surface states of trivial insulators.

Figure 3

(a) Schematic plot of the $z$-projection relation between the 3D bulk BZ and the surface BZ. The points in the surface BZ are marked with bar over them. (b) Schematic plot of the surface BZ of $pm$ group with two $m_y$-invariant lines highlighted and denoted by ${\kappa }_{y1}$ and ${\kappa }_{y2}$. (c) Schematic plot of surface states of TCI in $pm$ group on ${\overline{\mathrm{X}}}^{\prime }-\overline{\mathrm{\Gamma }}-\overline{\mathrm{X}}$ with $C_{o1}=-C_{e1}=1$. The red and blue lines denote the bands in even and odd mirror subspaces, respectively. (d) Schematic plot of the BZ of the $p3m1$ group. The MILs are denoted by the red, blue, and green lines. The threefold-invariant points are marked by the yellow triangles.

Figure 4

Schematic plot of surface states of TCIs in the $pmm$ group. The red and blue lines denote the bands with even and odd mirror parities, respectively. (a) The trivial phase in spinless case. The crossing marked by the green circle is not protected and can be gapped. (b) The nontrivial phase in the spinful case.

Figure 5

(a) Schematic plot of nontrivial surface states of TR invariant TIs. (b) Schematic plot of the trivial surface states which are two copies of that in (a). The crossings marked by the green circles are not protected and the gap can be opened. (c) Schematic plot of the nontrivial surface states of the $pmm$ group in the spinful case with $C_M=1$. (d) Schematic plot of nontrivial surface states of the $pmm$ group in the spinful case with $C_M=2$. The red and blue lines in (c) and (d) denote the bands with even and odd mirror parities, respectively.

Figure 6

(a) Schematic plot of a lattice of the $pm$ group with mirror symmetry $m_y$. The lattice constants in $x,y$, and $z$ directions are $a/2,b$, and $c$, respectively. (b) Schematic plot of a lattice of the $pg$ group, which is obtained by a small distortion of the lattice in (a), with glide plane symmetry $g_y$. The atoms in layer A move in the $y$ direction by $r_0$, while the atoms in layer B move by $-r_0$. The yellow regions in (a) and (b) denote a primitive cell of the corresponding lattice. (c) Schematic plot of the surface BZ of the lattice in (a), where ${\overline{\mathrm{X}}}_m=(2\pi /a,0)$ and ${\overline{\mathrm{X}}}_m^{\prime }=(-2\pi /a,0)$. (d) Schematic plot of the surface BZ of the lattice in (b), where ${\overline{\mathrm{X}}}_g=(\pi /a,0)$ and ${\overline{\mathrm{X}}}_g^{\prime }=(-\pi /a,0)$.

Figure 7

(a) Illustration of the BZ folding process from the lattice of $pm$ group to $pg$ group. The range of the first BZ of the lattice of $pm$ group in the $k_x$ direction is $[-2\pi /a,2\pi /a]$, and that of $pg$ group is $[-\pi /a,\pi /a]$. The solid lines are the bands of $pm$ group in the bulk gap, and the dashed lines are the bands that appear after the BZ folding. The band with glide even (odd) parity is drawn in red (blue) color. (b) The band dispersion of the model of the $pm$ group in a slab configuration with one Dirac cone on ${\overline{\mathrm{X}}}_m^{\prime }-\overline{\mathrm{\Gamma }}-{\overline{\mathrm{X}}}_m$. We only show the surface bands in the bulk gap on one surface of the slab. The parameters are $a=b=c=1,m_0=-2,t_0^{\prime }=1.5,t_0=t_0^{\prime \prime }=t=t^{\prime }=t^{\prime \prime }=1,\varphi =0.4$. The number of layers in the $z$ direction is $N=60$. (c) The band dispersion of the model of the $pg$ group in a slab configuration. The parameters are the same with those in (b). The perturbation term $t_3=0.2$. (d) The band dispersion of the model of the $pm$ group with one Dirac cone on ${\overline{\mathrm{X}}}_m^{\prime }-\overline{\mathrm{\Gamma }}-{\overline{\mathrm{X}}}_m$, and one Dirac cone on ${\overline{\mathrm{M}}}_m-\overline{\mathrm{Y}}-{\overline{\mathrm{M}}}_m^{\prime }$. The parameters, which are different with those in (b), are $m_0=-0.9,t_0^{\prime }=0.5$. (e) The band dispersion of the model of the $pg$ group with the same parameters in (d) and $t_3=0.2$. (f) The band dispersion of the model of the $pg$ group got by adding a surface potential $V=0.5\mathrm{eV}$ on the first layer of the slab. The other parameters are the same with those in (e).

Figure 8

(a) Schematic plot of a lattice of the $pgg$ group. The yellow region denotes a primitive cell. (b) The energy bands of a 2D lattice of the $pgg$ group. $A_i$ and $B_i$ ($i=1,2$) denote the Irreps of the bands at $\overline{\mathrm{\Gamma }}$. (c) One gapless surface state with all the bands belonging to Irreps $A_i$ at $\overline{\mathrm{\Gamma }}$. (d) Two copies of gapless surface states in (c). (e) One gapless surface state with all the bands belonging to Irreps $B_i$ at $\overline{\mathrm{\Gamma }}$. (f) Two copies of gapless surface states in (e). (g) One gapless surface state with some bands belonging to Irreps $A_i$ and some bands belonging to $B_i$ at $\overline{\mathrm{\Gamma }}$. (h) Two copies of gapless surface states in (g). The crossings marked by the green circles can be gapped with perturbations that do not break the symmetry.

Figure 9

Band dispersions around $\overline{\mathrm{X}}$ of the model of TCI in the $pgg$ group in a slab configuration. $A=1.7$ is fixed and $B$ is tuned from $0.2$ to $0.8$ in (a)–(e). We only show the surface states on one surface of the slab. The red (blue) lines denote the bands with even (odd) glide parity. (f) Schematic plot of the positions of Weyl nodes (marked by the red dots) in the surface BZ when $B=0.4$. (g) The density of states calculated by iterative Green functions on the line connecting two Weyl nodes with $B=0.4$ in a semi-infinite configuration. (h) The density of states of the Fermi surface passing through the nodal points with $B=0.4$. The other parameters in all of the figures are $f_{1a}=f_{2a}=-f_{1b}=f_{2b}=0.1,f_{3a}=f_{3a}^{\prime }=f_{3b}=f_{3b}^{\prime }=0, {\theta }_{1a}={\theta }_{2a}={\theta }_{1b}={\theta }_{2b}=0, f_1=f_2=1, f_4=0.1, {\theta }_1=\pi /2,{\theta }_2=0$, and $n=2$.

Figure 10

The band structure of the slab with a 2D layer on the top surface on $\overline{\mathrm{Y}}-\overline{\mathrm{\Gamma }}-\overline{\mathrm{X}}$. The original slab is in ${\chi }_1=-2$ phase. (a) All the bands of the 2D layer belong to Irreps $A_i$ at $\overline{\mathrm{\Gamma }}$. (b) All the bands of the 2D layer belong to Irreps $B_i$ at $\overline{\mathrm{\Gamma }}$. (c) The bands of the 2D layer belonging to Irreps $A_i$ and $B_i$ coexist at $\overline{\mathrm{\Gamma }}$. (d) The surface states are gapped by adiabatically tuning the parameters in (c).

Figure 11

Band dispersions of the model of $p4m,p31m$, and $p6m$ group in the spinful case in a slab configuration. We only show the surface states on one surface of the slab. The red (blue) lines denote the bands with even (odd) mirror parity. (a) A trivial phase of the model of $p4m$ group, with $f_{AB}=0.8i$; (b) a TCI phase of the model of $p4m$ group with $f_{AB}=1.5i$; (c) a trivial phase of the model of $p31m$ group with $l_2=0.7$; (d) a TCI phase of the model of $p31m$ group with $l_2=2.5$; (e) a trivial phase of the model of $p6m$ group with $f_{AB1}=0.5e^{-i\pi /6}$; (f) a TCI phase of the model of $p6m$ group with $f_{AB1}=e^{-i\pi /6}$.

Figure 12

Band dispersions around $\overline{\mathrm{X}}$ of the model of the $p4g$ group in the spinless case in a slab configuration. The parameter $l_1=0.3,1.2,0.8$ in (a), (b), and (c), respectively. We only show the surface states on one surface of the slab. The red (blue) lines denote the bands with even (odd) glide parity. (d) The density of states calculated by iterative Green functions on the line connecting two Weyl nodes (N1-N2) with $l_1=0.8$ in a semi-infinite configuration. The line N1-N2 parallel to $\overline{\mathrm{\Gamma }}-\overline{\mathrm{X}}$ with the middle point on $\overline{\mathrm{X}}-\overline{\mathrm{M}}$.