Wilson-loop characterization of inversion-symmetric topological insulators

The ground state of translationally invariant insulators comprises bands which can assume topologically distinct structures. There are few known examples where this distinction is enforced by a point-group symmetry alone. In this paper we show that 1D and 2D insulators with the simplest point-group symmetry, inversion, have a ${\mathbb{Z}}^{\ge }$ classification. In 2D, we identify a relative winding number that is solely protected by inversion symmetry. By analysis of Berry phases, we show that this invariant has similarities with the first Chern class (of time-reversal breaking insulators), but is more closely analogous to the ${\mathbb{Z}}_2$ invariant (of time-reversal invariant insulators). Implications of our work are discussed in holonomy, the geometric-phase theory of polarization, the theory of maximally localized Wannier functions, and in the entanglement spectrum.

Figure 1

Non-Abelian Berry phases for four distinct insulators. Each figure contains three unit cells in an effective 1D lattice along $\widehat{x}$; we track the real-space trajectories of the phases (blue circles) in $\widehat{x}$ as momentum $k_y$ is varied through two adiabatic cycles. (a) An insulator with Chern number $C_1=2$ and zero relative winding ($W$); this is modeled by the Hamiltonian (20) with parameters $m=3$ and $\delta =2$. (b) An insulator with relative winding $W=1$ and zero $C_1$; this is modeled by Eq. (20) with $m=3$ and $\delta =1$. (c) $W=2$ and $C_1=0$; this is modeled by Eq. (24). (d) $W=0$ and $C_1=0$; the insulator has a nontrivial polarization, and is modeled by Eq. (17) with parameters $\alpha =-1.5,\beta =1.5,\delta =-1$.

Figure 2

The Wannier center flows for a 2D insulator with $\mathcal{I}$ symmetry. Each figure contains three unit cells in an effective 1D lattice along $\widehat{x}$; the positions of the primary sites are indicated by black dots. The ground state contains two occupied bands, hence, there are two Wannier centers in each unit cell. (a) This phase is realized in the model Hamiltonian (17), with parameters $\alpha =-1.5$, $\beta =1.5$, and $\delta =1$. Upon $k_y\to k_y+2\pi$, the pair of Wannier centers in each unit cell exchange positions. There is no net transfer of charge between unit cells, hence, $C_1=0$. (b) This phase is realized in the same model with $\alpha =-1.5$ and $\beta =0$. Upon $k_y\to k_y+2\pi$, one Wannier center adiabatically flows to its neighboring unit cell on the left. This represents a quantized Hall current, hence, $C_1=-1$.

Figure 3

The Wannier flow of a 2D insulator with both $\mathcal{I}$ and TR symmetries. (a) This phase has a trivial TR invariant; it is realized in the model Hamiltonian (20), with parameters $m=4.3$, $\delta =0$. (b) This phase has a nontrivial TR invariant; it is realized in the same model, with $m=3$, $\delta =0$.

Figure 4

(a) An $\mathcal{I}$-symmetric phase with relative winding $W=2$; spectral flow is protected by $\mathcal{I}$ symmetry alone. This is the ground state of Hamiltonian (24). (b) Spectral flow is interrupted with an $\mathcal{I}$-breaking perturbation.

Figure 5

Figures in red provide two examples of accidental degeneracies. Upon a gap- and symmetry-preserving perturbation, these accidental degeneracies split, as shown in blue. These Wannier trajectories are calculated from 2D models; the Hamiltonians are not written explicitly. The model that describes the top two figures is a two-band model with $\mathcal{I}$ eigenvalues equal to $\{+1,-1\}$ at all four inversion-invariant momenta. These are mapped to complex-conjugate $\mathcal{W}$ eigenvalues $[\lambda ,{\lambda }^{*}]$, along both $k_y=0$ and $k_y=\pi$ (cf. Sec. 2b). $\lambda$ may vary in the interval $[1,-1]$ by continuous reparametrization of the Hamiltonian that maintains both energy gap and symmetry.

Figure 6

Schematic illustrations of Wannier trajectories for eight occupied bands, where $N_d=4$. (a) All four red trajectories wind with center-of-mass motion, thus $C_1=-4$, $W=0$. (b) One pair of red trajectories relatively wind, so $C_1=-2$, $W=1$. (c) Both pairs of red trajectories relatively wind, hence, $C_1=0$, $W=2$.

Figure 7

Schematic illustrations of Wannier flows with zero relative winding. In figures (a) and (b), $\frac{1}{2}[N_{(-1)}\left(0\right)-N_{(-1)}\left(\pi \right)]=1$ (odd); the Wannier trajectories follow a zigzag pattern. (c) $\frac{1}{2}[N_{(-1)}\left(0\right)-N_{(-1)}\left(\pi \right)]=0$ (even), and the Wannier flows are gapped. (d) $\frac{1}{2}[N_{(-1)}\left(0\right)-N_{(-1)}\left(\pi \right)]=2$ (even).

Figure 8

(a) An $\mathcal{I}$-symmetric phase with zero relative winding; spectral flow persists due to TRS alone. This is the ground state of Hamiltonian (27). (b) Spectral flow is interrupted with a TRS-breaking perturbation.