$N$-band Hopf insulator

We study the generalization of the three-dimensional two-band Hopf insulator to the case of many bands, where all the bands are separated from each other by band gaps. The obtained $\mathbb{Z}$ classification of such an $N$-band Hopf insulator is related to the quantized isotropic magnetoelectric coefficient of its bulk. The boundary of an $N$-band Hopf insulator can be fully gapped, and we find that there is no unique way of dividing a finite system into bulk and boundary. Despite this nonuniqueness, we find that the magnetoelectric coefficient of the bulk and the anomalous Hall conductivity of the boundary are quantized to the same integer value. We propose an experiment where the quantized boundary effect can be measured in a nonequilibrium state.

Figure 1

The band structure of a band insulator, where the existence of a single band gap is guaranteed (a). The same as in panel (a) for the case of two bands (b). The band insulator can have $N-1$ band gaps that divide the bands into $N$ disjoint sets (c). The same as in panel (c) for the case when each of $N$ sets contains exactly one band (d).

Figure 2

A finite $N$-band Hopf insulator has the bulk with the total (all the bulk bands combined) isotropic magnetoelectric polarizability coefficient equal to $N_{\text{Hopf}}$ (where $e=h=1$) with the gapped boundary that has Hall conductivity equal to $-N_{\text{Hopf}}$. The division between the bulk and the boundary is achieved by representing the bulk by a set of exponentially localized Wannier functions.

Figure 3

The Hilbert space, spanned by the orbitals of the slab's supercell, is divided at each $(k_x,k_z)$ point into the three mutually orthogonal subspaces corresponding to the bulk and the two surfaces. The bulk hybrid WFs $|w_{k_x{R}_y{k}_{z}n}\rangle$ are continuous in the $(k_x,k_z)$ space. On the other hand, for a nontrivial $N$-band Hopf insulator, there is an obstruction in finding continuous surface WFs $|w_{k_x{R}_y{k}_{z}n}^{\text{surf}}\rangle$.

Figure 4

The adiabatic process of a ribbon corresponding to $N$-band Hopf pump. The ribbon's supercell is divided into the three regions, where the middle (gray) region is spanned by the bulk WFs. Whereas the bulk WFs perform periodic motion, some WFs of the edge subsystems get shifted to the left or right.

Figure 5

(a) Energy bands of a ribbon, finite in the $y$ direction, as a function of the $y$ coordinate, where all the states are fully filled with electrons. (b) After applying a gate voltage, an equilibrium state is reached where the electrons from the middle of the ribbon are empty (i.e., they move to the drain gate). (c) After switching off the gate voltage, a nonequilibrium state is obtained with the states close to the edges filled.

Figure 6

(a) The charge density $|w_{(0,0,0)1}\left(\vec{x}\right){|}^2$ of the Wannier function $|w_{(0,0,0)1}\rangle$ localized at the center of the lattice, for $N_y=11, N_x=N_z=5$. (b) The absolute value of the matrix elements of the projector in Eq. (18), $|{\mathcal{P}}_{k_x{k}_z}^L(\vec{x},{\vec{x}}^{\prime })|$, for $N_y=31$ with cutoff $L=8$. (c) The surface polarization along the $x$ direction $P_{k_z}^{\text{surf}}$ for $k_z$ between 0 and $2\pi$ for $N_y=31$ with cutoff $L=8$.

Figure 7

Two different adiabatic processes. The full and empty dots denote $|\vec{R}1\rangle$ and $|\vec{R}2\rangle$ orbitals. If the orbital $|\vec{R}1\rangle$ is occupied, the electron is adiabatically transferred (red arrow) to the orbital $|\vec{R}2\rangle$; see Eq. (29). At the end of such process the orbital $|\vec{R}1\rangle$ is empty while the orbital $|\vec{R}2\rangle$ is occupied. The very same adiabatic process (a) transfers the electron from the orbital $|\vec{R}2\rangle$ to the orbital $|\vec{R}1\rangle$ as indicated by blue arrow. The second adiabatic process corresponds to an incomplete transfer between the two orbitals (b). In that case the final occupied state is a superposition of orbitals $|\vec{R}1\rangle$ and $|\vec{R}2\rangle$.

Figure 8

Two-level periodic drive made of 4 steps of equal duration. Panel (a) shows adiabatic evolution for the initial states $|1\rangle$. The same as panel (a) for the initial state $|2\rangle$ (b).

Figure 9

The spectrum ${\xi }_{\vec{k}}$ of the 1-cycle unitary operator $U_{k_x{k}_{y}T}^{\text{F}}$ defined in Eq. (36) for different values of $BT$. (a) $BT=3$. (b) $BT=5$, around which the quasienergy gap closes. (c) $BT=9$, after reopening of the quasienergy gap.

Figure 10

(a) Adiabatic time evolution of the upper edge subsystem ${\mathcal{P}}_{k_{x}t}^{\text{edge}}$. (b) The same as panel (a) for the lower edge subsystem ${\mathcal{P}}_{k_{x}t}^{{\text{edge}}^{\prime }}$. The orbitals of the ribbon supercell are labeled by $|n\rangle , n=1,\cdots ,8$, as shown in panel (a).

Figure 11

Three-level periodic drive made of 6 steps of equal duration. At the first and fourth stages, we consider an incomplete rotation between the orbitals $|\vec{R}1\rangle$ and $|(\vec{R}+{\vec{a}}_x)3\rangle$, such that the area enclosed by the trajectory of the third orbital is zero. This is controlled by the parameter $\delta \in [0,1)$.

Figure 12

Redefinition of the unit cell. We consider a unit cell consisting of 4 orbitals.