Rational boundary charge in one-dimensional systems with interaction and disorder

We study the boundary charge $Q_B$ of generic semi-infinite one-dimensional insulators with translational invariance and show that nonlocal symmetries (i.e., including translations) lead to rational quantizations $p/q$ of $Q_B$. In particular, we find that (up to an unknown integer) the quantization of $Q_B$ is given in integer units of $\frac{1}{2}\overline{\rho }$ and $\frac{1}{2}(\overline{\rho }-1)$, where $\overline{\rho }$ is the average charge per site (which is a rational number for an insulator). This is a direct generalization of the known half-integer quantization of $Q_B$ for systems with local inversion or local chiral symmetries to any rational value. Quite remarkably, this rational quantization remains valid even in the presence of short-ranged electron-electron interactions as well as static random disorder (breaking translational invariance). This striking stability can be traced back to the fact that local perturbations in insulators induce only local charge redistributions. We establish this result with complementary methods including density matrix renormalization group calculations, bosonization methods, and exact solutions for particular lattice models. Furthermore, for the special case of half-filling $\overline{\rho }=\frac{1}{2}$, we present explicit results in single-channel and nearest-neighbor hopping models and identify Weyl semimetal physics at gap closing points. Our general framework also allows us to shed new light on the well-known rational quantization of soliton charges at domain walls.

Figure 1

Illustration of the semi-infinite Hamiltonians $H_{R,n}$ (right) and $H_{L,n}$ (left). The Hamiltonian $H_{R,n}$ is obtained from the bulk Hamiltonian $H_{\text{bulk}}$ by starting it at site $m=n+1$, whereas $H_{L,n}$ is obtained by ending $H_{\mathrm{bulk}}$ at site $m=n$.

Figure 2

Sketch of the envelope function $f\left(m\right)$ with $N\gg M\gg Z,\xi$.

Figure 3

Illustration of the translation by $n$ lattice sites for the left and right semi-infinite lattice. In both cases the lattice is moved to the left. This means that for $H_R\to H_{R,n}$ the lattice is moved by $n$ sites towards the boundary whereas, for $H_L\to H_{L,n}$, the lattice is moved by $n$ sites away from the boundary. As a consequence, the boundary charge increases or decreases by $n\overline{\rho }$, respectively.

Figure 4

Illustration of local inversion by indicating the lattice sites of the unit cells. Inverting the lattice sites within a unit cell via the interchange $m\leftrightarrow Z-m+1$ and turning the system by ${180}^{\circ }$ one finds that $H_R$ is transformed to $H_L$.

Figure 5

Visualization of the nonlocal inversion symmetry ${\mathrm{\Pi }}_n$ for $n=1$ and $Z=4$ for a nearest-neighbor hopping model with one channel per site. $v_j$ denote the on-site potentials and $-t_j$ are the hoppings. The red dashed vertical lines indicate the boundaries between the unit cells and the blue solid vertical line is the symmetry axis of inversion symmetry. In the lower figure, the unit cell is redefined such that the symmetry becomes a local one. However, this is not possible for a tight-binding model since the unit cell contains only one half of a site at the boundaries.

Figure 6

Phase dependence of (a) band structure, (b) boundary charge $Q_B$, and (c) invariant $I\left(\phi \right)=Q_B(\phi +\frac{2\pi }{Z})-Q_B\left(\phi \right)-\overline{\rho }$ for the model (50) with $V=0.5$ and $t=1$ at half-filling $\overline{\rho }=\frac{1}{2}$, and $Z=4,6$ (left/right panel), calculated for a semi-infinite system. In (a), the bands $\alpha =\frac{Z}{2},\frac{Z}{2}+1$ are shown together with edge modes connecting the gap closing points, where $Q_B$ jumps by $\pm \frac{1}{2}$. The invariant is quantized to $I\in \{-1,0\}$. Due to the nonlocal chiral symmetry $v_m\left(\phi \right)=-v_{m\pm \frac{Z}{2}}\left(\phi \right)$, quantization of $Q_B$ occurs in units of $\frac{1}{2}$ for $Z=4$, whereas for $Z=6$ we get $Q_B=\frac{1}{4}\text{mod}(1/2)$. The black symbols (mainly overlaying the lines) in (b) show the case with random disorder for a finite system for additional staggered onsite-disorder drawn from a uniform distribution $(-0.025,0.025]$ for a finite system of $Z\times {10}^5$ lattice sites.

Figure 7

(a) and (b) show the phase-dependence of the band structure and the edge states for $Z=6$ and the same parameters as in Fig. 6 but with additional periodic disorder for the on-site potentials on the scale 0.2 (which is chosen rather strong to visualize the gap openings), calculated for a semi-infinite system. As a result, three edge states run either from the valence to the conduction band or vice versa. In (c), the phase-dependence of the boundary charge $Q_B\left(\phi \right)$ is shown for the two disordered configurations and compared to the clean case, demonstrating stability of the quantization of $Q_B$ between the jumps.

Figure 8

Stability of the rational quantization of the boundary charge upon inclusion of nearest neighbor interaction. The figure shows the boundary charge $Q_B$ for the same parameters (but finite size $N=1000$) as used in Fig. 6 at two values of $\phi =\pi /4$ and 0, where the gap is maximal, for $Z=4$ and 6, respectively.

Figure 9

Stability of Eq. (22) with respect to random disorder. The figure shows $Q_{I,s0}-Q_{B,0}^L-Q_{B,s}^R$ for the same parameters as used in the left column ($Z=4$) of Fig. 6 at $\phi =\pi /4$, where the gap is maximal (but at finite size $N=1000$). The relative shift of the chains left and right to the interface are (a) $s=0$ and (b) 1. We take half-filling instead of $\mu =0$ here. Random disorder drawn from a uniform distribution $[-d/2,d/2)$ is added to the onsite potentials and $p$ describes changes to the interface properties (see main text for details). As the properties of the interface are swept through $Q_{I,s0}-Q_{B,0}^L-Q_{B,s}^R$ only changes $\text{mod}\left(1\right)$.

Figure 10

Stability of Eq. (22) with respect to interactions. The figure shows $Q_{I,s0}-Q_{B,0}^L-Q_{B,s}^R$ for the same parameters as used in the left column ($Z=4$) of Fig. 6 at $\phi =\pi /4$, where the gap is maximal (but at finite size $N=200$ and with larger $V/t=1.2$). The relative shift of the chains left and right to the interface are (a) $s=0$, and (b) $s=1$. We take half-filling instead of $\mu =0$ here. $p$ describes changes to the interface properties (see main text for details). As the properties of the interface are swept through $Q_{I,s0}-Q_{B,0}^L-Q_{B,s}^R$ only changes $\text{mod}\left(1\right)$.

Figure 11

Stability of Eqs. (26) and (31) with respect to (a) random disorder and (b) interaction. We work at half filling such that $\overline{\rho }=1/2$. $Q_{B,1}^R-Q_{B,0}^R=\overline{\rho }\text{mod}\left(1\right)$ is shown to demonstrate Eq. (26), while $Q_{B,0}^L+Q_{B,0}^R=0\text{mod}\left(1\right)$ illustrates Eq. (31). The parameters are the same as in Fig. 9.

Figure 12

The same as the center row of Fig. 6, but for small $N=24$ and two values of the $V$.