Surface charge theorem and topological constraints for edge states: Analytical study of one-dimensional nearest-neighbor tight-binding models

For a wide class of noninteracting tight-binding models in one dimension, we present an analytical solution for all scattering and edge states on a half-infinite system. Without assuming any symmetry constraints, we consider models with nearest-neighbor hoppings and one orbital per site but the arbitrary size of the unit cell and generic modulations of on-site potentials and hoppings. The solutions are parametrized by determinants that can be straightforwardly calculated from recursion relations. We show that this representation allows for an elegant analytic continuation to complex quasimomentum consistent with previous treatments for continuum models. Two important analytical results are obtained based on the explicit knowledge of all eigenstates. (1) An explicit proof of the surface charge theorem is presented including a unique relationship between the boundary charge $Q_B^{\left(\alpha \right)}$ of a single band $\alpha$ and the bulk polarization in terms of the Zak-Berry phase. In particular, the Zak-Berry phase is determined within a special gauge of the Bloch states such that no unknown integer is left. This establishes a precise form of a bulk-boundary correspondence relating the boundary charge of a single band to bulk properties. (2) We derive a topological constraint for the phase dependence of the edge state energies, where the phase variable describes a continuous shift of the lattice towards the boundary. The topological constraint is shown to be equivalent to the quantization of a topological index $I=\mathrm{\Delta }{Q}_B-\overline{\rho }\in \{-1,0\}$ introduced in an accompanying paper [M. Pletyukhov et al., Phys. Rev. B 101, 161106 (2020)]. Here, $\mathrm{\Delta }{Q}_B$ is the change of the boundary charge $Q_B$ for a given chemical potential in the insulating regime when the lattice is shifted by one site towards the boundary, and $\overline{\rho }$ is the average charge per site (both in units of the elementary charge $e=1$). This establishes an interesting link between universal properties of the boundary charge and edge state physics discussed within the field of topological insulators. In accordance with previous results for continuum systems, we also establish the localization of the boundary charge and determine the explicit form of the density given by an exponential decay and a pre-exponential function following a power law with generic exponent $-1/2$ at large distances.

Figure 1

Sketch of the two half-infinite tight-binding models under consideration. The system is either extending to the right or to the left side, described by the Hamiltonians $H_R$ and $H_L$, respectively. The unit cells are labeled by $n=0,\pm 1,\pm 2,\cdots$, the sites within a unit cell by $j=1,2,\cdots ,Z$. The absolute position of a site is labeled by $m=Z(n-1)+j$. The right/left bar indicates the boundary of $H_{R/L}$. The site indicated between the two bars at $n=0,j=Z$ or $m=0$ is an artificial site where the eigenstate has to fulfill the boundary condition $\psi (n=0,j=Z)=\psi (m=0)=0$.

Figure 2

(a) The function $D\left(\epsilon \right)$ for $v_j=-cos(2\pi j/Z)$ and $t_j=1$. The bands are formed in the regions where $D\left({\epsilon }_k^{\left(\alpha \right)}\right)=cos\left(k\right)\le 1$, with $Z-1$ gaps $\nu =1,\cdots ,Z-1$ in between. The band edges are given by ${\epsilon }_0^{\left(\alpha \right)}$ and ${\epsilon }_{\pi }^{\left(\alpha \right)}$, where ${\epsilon }_0^{\left(\alpha \right)}$ is the band top/bottom for even/odd $\alpha$. ${\epsilon }_{\text{bp}}^{\left(\nu \right)}$ denote the energies where $\frac{d}{d\epsilon }D\left({\epsilon }_{\text{bp}}^{\left(\nu \right)}\right)=0$ and $|D\left({\epsilon }_{\text{bp}}^{\left(\nu \right)}\right)|=|cos\left(k_{\text{bp}}^{\left(\nu \right)}\right)|>1$. They correspond to complex $k$-values $k_{\text{bp}}^{\left(\nu \right)}=\pm i{\kappa }_{\text{bp}}^{\left(\nu \right)}$ and $k_{\text{bp}}^{\left(\nu \right)}=\pm (\pi +i{\kappa }_{\text{bp}}^{\left(\nu \right)})$ for even and odd $\nu$, respectively, with ${\kappa }_{\text{bp}}^{\left(\nu \right)}\ge 0$, see Eq. (68). (b) The band structure for the same parameters as function of the quasimomentum $k$.

Figure 3

Two choices for the analytic continuation of the dispersion for the same parameters as in Fig. 2. (a) The analytic continuation defining ${\epsilon }_k$, with ${\epsilon }_k={\epsilon }_k^{\left(\alpha \right)}$ for $(\alpha -1)\pi <\left|k\right|<\alpha \pi$ and ${\epsilon }_k={\epsilon }_k^{\left(Z\right)}$ for $\left|k\right|>Z\pi$ on the real axis. Branch cuts occur at $\pm \nu \pi +i\kappa$, with $\left|\kappa \right|<{\kappa }_{\text{bp}}^{\left(\nu \right)}$ and $\nu =1,\cdots ,Z-1$. (b) and (c) show the analytic continuation of ${\epsilon }_k^{\left(\alpha \right)}$ for $\alpha =2,3$, with ${\epsilon }_k^{\left(\alpha \right)}={\epsilon }_{k+2\pi }^{\left(\alpha \right)}$ taken on the whole real axis. Branch cuts are located at $k_{\text{bc}}^{\left(\nu \right)}+i\kappa$ and ${\left(k_{\text{bc}}^{\left(\nu \right)}\right)}^{*}-i\kappa$ (not shown), with $\left|\kappa \right|>0$ and $\nu =1,2$ (for $\alpha =2$) and $\nu =2,3$ (for $\alpha =3$), together with corresponding ones shifted by multiples of $2\pi$. Between the branch cuts we have indicated in the boxes the relation to the analytic continuation chosen in (a). As one can see the values of the dispersion left and right to the branch cut starting at $k_{\text{bp}}^{\left(2\right)}$ are interchanged for $\alpha =2,3$.

Figure 4

(a) Determination of the edge state energies ${\epsilon }_{\text{e}}^{\left(\nu \right)}$ from the condition $s\left({\epsilon }_{\text{e}}^{\left(\nu \right)}\right)=0$ for $Z=4$, $V=0.5$, $t=1.1$, $\delta t=0.1$, $\phi =1.6\pi$, and three random Fourier coefficients for the real functions $F_v$ and $F_t$ in Eqs. (10) and (11) according to the form (B1), for the precise parameters see Ref. [64]. As explained in the main text the position of the edge state energies is always located in the gaps, each gap hosting exactly one edge state. The sign of $s\left(\epsilon \right)$ is given by ${(-1)}^{\alpha +1}$ in the energy regions of band $\alpha$, see Eq. (78). Depending on whether ${\epsilon }_{\text{e}}^{\left(\nu \right)}\lessgtr {\epsilon }_{\text{bp}}^{\left(\nu \right)}$, the edge states result from the analytic continuation of band $\nu$ or $\nu +1$. (b) The band structure as function of the phase variable $\phi$ for the same parameters as in (a). The band structure is periodic under a change of the phase variable by $2\pi /Z$. The edge state energies ${\epsilon }_{\text{e}}^{\left(\nu \right)}\left(\phi \right)$ for $\nu =1,2,\text{and}3$ are shown by blue solid/dashed lines corresponding to edge states of $H_R/H_L$ with $\text{Im}\left(k_{\text{e}}^{\left(\nu \right)}\right)\gtrless 0$. As can be seen each gap hosts exactly one edge state and the edge states change the boundary when they touch the bands. Approximately in the middle of each gap we show by black dashed lines the phase dependence of the energies ${\epsilon }_{\text{bp}}^{\left(\nu \right)}\left(\phi \right)$ at the branching points for $\nu =1,2,\text{and}3$. For ${\epsilon }_{\text{e}}^{\left(\nu \right)}\lessgtr {\epsilon }_{\text{bp}}^{\left(\nu \right)}$ the edge states belong to the analytical continuation of band $\nu$ or $\nu +1$, respectively. In (b) we have indicated by the vertical dashed line the phase value $\phi =1.6\pi$ used in (a) and where the analytic structure and the position of the edge poles is shown in Fig. 5.

Figure 5

The analytic continuation of (a) ${\left[{\chi }_k\left(j\right)\right]}^2$ and (b) ${\left[{\chi }_k^{\left(2\right)}\left(j\right)\right]}^2$ analog to Figs. 3 and 3 but for the parameters of Fig. 4 at the particular phase $\phi =1.6\pi$ indicated in Fig. 4 by a vertical dashed line. In addition we have indicated the edge pole positions $k_{\text{e}}^{\left(\nu \right)}$, with $\nu =1,2,\text{and}3$. For $\text{Im}\left(k_{\text{e}}^{\left(\nu \right)}\right)\gtrless 0$, they correspond to edge states of either $H_R$ or $H_L$, respectively. In (a), an edge pole lying left (right) to the branch cut located at $\nu \pi$ belongs to the analytic continuation of band $\nu$ ($\nu +1$). The edge poles move around the branch cuts as function of $\phi$, see snapshots and videos provided in Ref. [64].

Figure 6

Edge states of $H_R$ crossing the chemical potential $\mu \left(\phi \right)=\mu (\phi +\frac{2\pi }{Z})$ at $\phi ={\phi }_0$ either from (a) above or (b) below. At ${\phi }_0^{\prime }={\phi }_0-\frac{2\pi }{Z}$ it is not allowed that an edge state of $H_R$ crosses the chemical potential. To the right we show the way how we visualize the two different possibilities by contractions (in blue color).

Figure 7

Visualization of edge states of $H_R$ crossing $\mu \left(\phi \right)$ at $\phi ={\phi }_i$, with $i=1,2,3,\text{and}4$, in terms of sequences of contractions. We defined ${\phi }_i^{\prime }={\phi }_i-\frac{2\pi }{Z}$ such that each contraction has a fixed length $\frac{2\pi }{Z}$.

Figure 8

Graphical rule how to determine the virtual topological charge $\mathrm{\Delta }F\left(\phi \right)$ at some phase $\phi$ lying between two vertices. Drawing a vertical cut at this position (dashed line) one has to take the number of right-going minus the number of left-going contraction lines through this vertical cut.

Figure 9

The allowed configurations for $s=1$, 2, and 3 when all contractions have the same direction. The topological charge $\mathrm{\Delta }F\in \{s-1,s\}$ is indicated between all adjacent vertices. Changing the direction of all contractions leads to the configurations with $s^{\prime }=-s+1=0,-1,-2$, see Eq. (115).

Figure 10

The construction how a pair of two contractions with different directions is inserted into an old configuration, formed by the three boxes. The condition is that between the two right and the two left vertices of the new contractions no other vertex of the old configuration is allowed to appear. Depending on the old configuration it might be necessary to invert the two directions of the new contractions in order to get an allowed configuration after inserting the two new contractions.

Figure 11

Illustration of configurations which arise when one adds subsequently pairs of edge states crossing $\mu$ from above and below. For simplicity we have taken a constant $\mu$ independent of $\phi$. Here, the points $i$ indicate the phase points ${\varphi }_i$ and $i^{\prime }$ the shifted ones ${\varphi }_i^{\prime }={\varphi }_i-\frac{2\pi }{Z}$. (a) Adding the pair (2,3) to an edge state crossing $\mu$ at 1 from above leads to an additional oscillation around $\mu$ with length smaller than $\frac{2\pi }{Z}$ such that the edge state remains to run from the upper to the lower band. (b) Adding another pair (4,5) leads to a second oscillation of length smaller than $\frac{2\pi }{Z}$. Note that $(4^{\prime },5^{\prime })$ lies here between 2 and 3 such that the topological constraint is fulfilled. (c) Two consecutive edge states with one additional oscillation each. Here the shifted points $4^{\prime }$ and $5^{\prime }$ of the right edge state are located between the crossing points 1 and 2 of the left edge state. Note that the topological constraint does not allow $4^{\prime }$ and $5^{\prime }$ to lie between 2 and 3. This gives the rule that if the crossing points $i$ and $j$ are connected by an edge state with a maximum (minimum) in between then the shifted points $i^{\prime }$ and $j^{\prime }$ must be located in an interval where the edge state has also a maximum (minimum). The connection of the edge states between adjacent crossing points is just a guide for the eye, these connections can also be cut (e.g., when $\mu$ is identical to the band edge this is even necessary).

Figure 12

(a) Two consecutive edge states with two (one) oscillation of the left (right) edge state around $\mu$, analog to the cases discussed in Fig. 11. (b) A more complicated configuration where we insert the pair (9,10) between $6^{\prime }$ and $7^{\prime }$ in (a) and cut the edge mode connecting 3 and 4. As a result we get four consecutive edge states, where the two outer ones connect the upper with the lower band whereas the two middle ones return to the same band, the left (right) one to the upper (lower) band. Note that the crossing points 1 and 2 are essential, otherwise there would be a mismatch between the consecutive points $3^{\prime }$ and $9^{\prime }$ which correspond both to outgoing vertices. The crossing points 4 and 5 are necessary to get a new configuration compared to Fig. 11. Without 4 and 5 one can connect the edge state between 10 and 6 and gets two consecutive edge states oscillating around $\mu$. Furthermore the crossing points 6 and 7 are needed since without $6^{\prime }$ and $7^{\prime }$ there is a mismatch between 3 and 9 and between 10 and 4 corresponding both to incoming or outgoing vertices, respectively. Therefore all crossing points are essential to construct such an involved configuration.

Figure 13

Sketch of the envelope function $f_{N,M}\left(m\right)$ with $N\gg M\gg 1$.

Figure 14

Singularities of the Berry connection in $(k,\phi )$-space at the points where edge states enter or leave band $\alpha$. We have shown one edge state entering the band at ${\phi }_{2+}^{\left(\alpha \right)}$ for $k=\pi$ and two edge states leaving the band at ${\phi }_{1-}^{\left(\alpha \right)}$ and ${\phi }_{3-}^{\left(\alpha \right)}$ for $k=0$. The Chern number $C^{\left(\alpha \right)}={\sum }_i{C}_i^{\left(\alpha \right)}$ is given by the negative sum over all closed integrals over the paths $L_i$ of the Berry connection around the singularities at $(k_0,{\phi }_{i\pm }^{\left(\alpha \right)}$), with $k_0=0$ or $k_0=\pi$. According to Eq. (218), this is identical to the sum over the windings $C_i^{\left(\alpha \right)}$ of the phase of $a_k^{\left(\alpha \right)}\left(1\right)$ around the singularities. The dashed line indicates the path along which the winding number ${\overline{w}}_{\alpha }\left(\phi \right)$ is defined which determines the invariant $I_{\alpha }\left(\phi \right)$ via Eq. (233). When $\phi$ crosses ${\phi }_{i\pm }^{\left(\alpha \right)}$ from below the winding number ${\overline{w}}_{\alpha }\left(\phi \right)$ jumps by $\pm 1$ according to Eq. (220) leading to $C_i^{\left(\alpha \right)}=\mp 1$, see Eq. (219). Note that ${\overline{w}}_{\alpha }\left(\phi \right)$ jumps also by $\mp 1$ at the phase values ${\phi }_{i\pm }^{\left(\alpha \right)}-\frac{2\pi }{Z}$ according to Eq. (220) such that ${\overline{w}}_{\alpha }\left(\phi \right)={\overline{w}}_{\alpha }(\phi +2\pi )$ is fulfilled.

Figure 15

(a) The band structure and edge states analog to Fig. 4 for $Z=6$, $V=0.5$, $t=1$, $\delta t=0.1$ and three random Fourier coefficients for the real functions $F_v$ and $F_t$ in Eqs. (10) and (11) according to the form (B1), see Supplemental Material for the precise parameters [64]. For the gaps $\nu =1,2,\text{and}3$ there are $\nu$ edge states of $H_R$ (blue solid lines) moving upwards whereas, for the gaps $\nu =4\text{and}5$, there are $Z-\nu$ edge states of $H_R$ moving downwards. For band $\alpha =4$ all edge states of $H_R$ are entering the band, $Z-\alpha =2$ from above and $\alpha -1=3$ from below, i.e., in total $Z-1=5$ edge states. (b) The band structure and edge states analog to Fig. 4 for $Z=3$, $V=0.5$, $t=1$, $\delta t=0$ and five random Fourier coefficients for the real function $F_v$ in Eq. (10) according to the form (B1), for the precise parameters see Ref. [64]. Several edge states of $H_R$ return to the same band in both gaps $\nu =1,2$. For band $\alpha =2$, all edge states of $H_R$ are entering the band (disregarding the ones which return to the same band), $Z-\alpha =1$ from above and $\alpha -1=1$ from below, i.e., in total $Z-1=2$ edge states.

Figure 16

Sketch of the branch cut separating band $\nu$ from $\nu +1$, corresponding to the analytic continuation of Fig. 5. The position of $k_{{\mu }_{\nu }}$ corresponds to ${\mu }_{\nu }$ and is defined via ${\epsilon }_{k_{{\mu }_{\nu }}}={\mu }_{\nu }$ and $\text{Im}\left(k_{{\mu }_{\nu }}\right)>0$. The positions of $k_e^{\left(\nu \right)}\left({\phi }_1\right)$ and $k_e^{\left(\nu \right)}\left(\widetilde{{\phi }_1}\right)$ correspond to the edge state poles at the phases ${\phi }_1$ and ${\tilde{\phi }}_1={\phi }_1+\frac{2\pi }{Z}$, respectively. Two possibilities are shown in (a) and (b) for their position. Obviously, for both cases, one can choose the orientation of the path $k_e^{\left(\nu \right)}\left(\phi \right)$ of the edge state pole for all phases ${\phi }_1<\phi <{\tilde{\phi }}_1$ in such a way that no crossing through $k_{{\mu }_{\nu }}$ occurs (red part of the contour). We have not indicated in the figure the weak dependence of $k_{{\mu }_{\nu }}$ and $k_{\text{bp}}^{\left(\nu \right)}$ on $\phi$.

Figure 17

In (a) and (b), we show the phase dependence of the band structure of the first two bands and the edge states of $H_R$ in the first gap (blue lines) for $Z=3$, $V=0.1$, $t=1$, $\delta t=0.1$ for two functions $F_v$ and $F_t$ in Eqs. (10) and (11) taken from (B4) with fixed and random parameters for $v_j\left(0\right)$ and $t_j\left(0\right)-t$ at phase $\phi =0$ but different choices for the phase dependence in between [via the random parameters $v_j^{\left(1\right)}$ and $t_j^{\left(1\right)}$ used in (B10)], see Ref. [64] for the concrete parameters. In (a), one edge state appears running from the lower to the upper band whereas in (b) two edge states run from the upper to the lower band. In (c), the phase dependence of $Q_B$ is shown for the two choices of (a) and (b) with ${\mu }_1$ located in the first gap [see dashed line in (a) and (b)]. At $\phi =0$ and $\phi =2\pi /3$, we get the change $\mathrm{\Delta }{Q}_B\left(0\right)=\mathrm{\Delta }{Q}_B\left(\frac{2\pi }{3}\right)=\overline{\rho }=\frac{1}{3}$ (see the two left arrow). For the first choice, $Q_B$ increases on average by $\frac{1}{Z}=\frac{1}{3}$ on both intervals whereas for the second choice $Q_B$ decreases on average by $\frac{1}{Z}-1=-\frac{2}{3}$ and obtains a jump by $+1$ at $\phi ={\phi }_{1,2}$ where an edge state moves below ${\mu }_1$. For $\phi =4\pi /3$, we get $\mathrm{\Delta }{Q}_B\left(\frac{4\pi }{3}\right)=\overline{\rho }-1=-\frac{2}{3}$ (see right arrow). Here, for the first choice, an edge state moves above ${\mu }_1$ at $\phi ={\phi }_3$ leading to a jump of $Q_B$ by $-1$, whereas for the second choice no edge state is involved.

Figure 18

Four scenarios how edge states can enter/leave a band for phase $\phi$ and $\tilde{\phi }=\phi +\frac{2\pi }{Z}$. We have shown a band top but the same behavior occurs for a band bottom. The quasimomentum at the band edge can be either $k_0=0$ or $k_0=\pi$. For (a) and (d), an edge state enters at $\phi$ and $\tilde{\phi }$, respectively, such that the sign of $\frac{d}{d\phi }{d}_1^{\left(\alpha \right)}$ is given by ${(-1)}^{k_0/\pi }$ at the entering point according to (E10). Analog, for (b) and (c) the edge state leaves such that the sign is given by $-{(-1)}^{k_0/\pi }$ at the leaving point. Together with (241) and (E9) this gives the correct correlation to the jump of $I_{\alpha }$ by $-1$ for (a) and (b) and by $+1$ for (c) and (d), see explanation in the main text.

Figure 19

phase dependence of $Q_B^{\alpha }$, $Q_F^{\left(\alpha \right)}$, $Q_P^{\left(\alpha \right)}$ and $I_{\alpha }$ for $\alpha =1,3,\text{and}4$ in (a), (b), and (c), respectively, using the parameters of Fig. 15.

Figure 20

Phase dependence of boundary charge $Q_B(\phi ,{\mu }_{\nu })$ and invariant $I(\phi ,{\mu }_{\nu })$ for $Z=3$ and (a) $\nu =1$ and (b) 2 for the parameters of Fig. 15.

Figure 21

Phase dependence of boundary charge $Q_B(\phi ,{\mu }_{\nu })$ and invariant $I(\phi ,{\mu }_{\nu })$ for $Z=4$ and (a) $\nu =1$, (b) 2, and (c) 3 for the parameters of Fig. 4.

Figure 22

Phase dependence of boundary charge $Q_B(\phi ,{\mu }_{\nu })$ and invariant $I(\phi ,{\mu }_{\nu })$ for $Z=6$ and (a) $\nu =1$, (b) 2, (c) 3, (d) 4, and (e) $\nu =5$ for the parameters of Fig. 15.

Figure 23

(a) The two vertices 1 and 2 are assumed to be a pair of left-going vertices with shortest distance where 1 is an incoming and 2 an outgoing vertex. As shown it is not possible that two other right-going vertices $3^{\prime }$ and $4^{\prime }$ occur between 1 and 2 since the two vertices 3 and 4 would form a pair with shorter distance compared to the pair (1,2). (b) Here we show the two right-going vertices $1^{\prime }$ and $2^{\prime }$ corresponding to 1 and 2 of (a) which have the same distance. As shown it is not possible that two left-going vertices 3 and 4 can appear in between since they would have a shorter distance compared to the pair (1,2).

Figure 24

Closing of the integration contour in the upper half via the path $\gamma$ to calculate the Friedel density ${\rho }_F^{\left(2\right)}(n,j)$ of band $\alpha =2$, corresponding to the analytic continuation shown in Fig. 5. The two contributions from $3\pi /2\to 3\pi /2+i\infty$ and $-\pi /2+i\infty \to -\pi /2$ cancel each other due to the periodicity of ${\chi }_k^{\left(2\right)}={\chi }_{k+2\pi }^{\left(2\right)}$. The asymptotic part from $3\pi /2+i\infty \to -\pi /2+i\infty$ is zero. Using the analytic properties from Fig. 5 the integration along $\gamma$ is the same as the sum of the two integrals along ${\gamma }_{\text{bc}}^{\left(1\right)}$ and ${\gamma }_{\text{bc}}^{\left(2\right)}$ surrounding the two branch cuts, together with the integral along ${\gamma }_{\text{e}}^{\left(2\right)}$ encircling the edge pole at $k_{\text{e}}^{\left(2\right)}$.

Figure 25

The phase dependence of the edge state energy (solid blue line for $H_R$ and dashed blue line for $H_L$) in the first gap $\nu =1$ for $Z=4$ and $V=0.3$, $t=1$, $\delta t=0.2$, for various functions $F_v$ and $F_t$ in Eqs. (10) and (11) taken from (B4) with fixed and random parameters for $v_j\left(0\right)$ and $t_j\left(0\right)-t$ at phase $\phi =0$ but different choices for the phase dependence in between [via the random parameters ${\gamma }_j^{\left(1\right)}$ for $\gamma =v,t$ used in (B10)], see Ref. [64] for the concrete parameters. The chemical potential ${\mu }_1$ is indicated by a dashed black line. In (a) we have chosen ${\gamma }_j^{\left(1\right)}=0$ which leads to a gap closing at $\phi =\frac{\pi }{4}$. In (c) the ${\gamma }_j^{\left(1\right)}$ are by a factor of 2 larger compared to (b). (d) and (e) correspond to (b) and (c), respectively, but the sign of all ${\gamma }_j^{\left(1\right)}$ has been reversed. As can be seen for the choices in (d) and (e) no edge state of $H_R$ crosses ${\mu }_1$ in the phase interval $[0,\frac{2\pi }{Z}]=[0,\frac{\pi }{2}]$.