Zak-Phase Dislocations in Trimer Lattices

Abstract

Wave propagation in periodic media is governed by energy–momentum relations and geometric phases characterizing band topology, such as Zak phase in one-dimensional lattices. We demonstrate that, in the off-diagonal trimer lattices, Zak phase carries quantized screw-type dislocations winding around degeneracies in parameter space. If the lattice evolves in time periodically, as in adiabatic Thouless pumps, the corresponding closed trajectory in parameter space is characterized by a Chern number equal to the negative total winding number of Zak phase dislocations enclosed by the trajectory. We discuss the correspondence between bulk Chern numbers and the edge states in a finite system evolving along various pumping cycles.

1. Introduction

Geometric Berry phase [1] defines topological properties of wave transport in periodic media in condensed matter [2,3,4] and photonics [5,6,7]. In one-dimensional lattices, the Berry phase, introduced by Zak [8], is associated with adiabatic evolution in the Brillouine zone. Zak phase Z was experimentally measured with cold atoms [9] in optical dimer lattices, realizing Su–Schrieffer–Heeger (SSH) [10] and Rice–Mele [11] models. The SSH system exhibits topological transition from trivial ($Z=0$) to nontrivial ($Z=\pi$) [12]; the latter bulk property is manifested in a finite open system by the appearance of a pair of edge states inside the spectral gap. Other lattices without inversion symmetry may posses nonzero phase Z not quantized by $\pi$ [8]; the role of this geometric phase remains obscure.

Here we demonstrate the appearance of screw-type Zak-phase dislocations in the parameter domain of the off-diagonal trimer lattices, i.e., one-dimensional periodic lattices with three equivalent atoms in a unit cell. The family of trimer lattices is spanned by two hopping parameters: intra-cell $J_{1,2}$ and inter-cell $J_3$. The lattice is inversion-symmetric for $J_1=J_2$ with vanishing Zak phase $Z=0$, which may suggest that the system is topologically trivial. However, several studies [13,14,15,16,17] have shown the presence of paired and unpaired chiral edge states and piecewise-continuous Zak phase [16] for $J_1\ne J_2$. Our analysis reveals the winding of Zak phase around the degeneracy in parameter space $J_1=J_2=J_3$, where the gaps close in monatomic lattice.

Phase dislocations were introduced by Nye and Berry [18] as generic features of waves of any nature, e.g., in optics they are associated with quantized vortices [19,20,21] and orbital angular momentum of light [22]. As pointed by Berry [1], the degeneracies act as organizing centers for phase changes, and we demonstrate below how Zak-phase dislocations define the properties of trimer lattices.

Geometric phase manifests itself in the evolution of a system in parameter space [1]. If the evolution is periodic, adiabatic driving leads to quantization of the current, predicted by Thouless [23]. Adiabatic Thouless pumps were demonstrated with cold atoms in optical lattices slowly changing over time [24,25] and with optical waveguides slowly changing in the direction of propagation [26]. The latter system offers novel possibilities for designing periodic media [27,28] from Floquet topological insulators [29] to nonlinear [30] and active [31] topological photonic structures [7]. Adiabatic cyclic parameter can be seen as an additional “synthetic” degree of freedom, thus effectively increasing the dimensions of the system [32,33]. The ideas of the synthetic dimensions in photonics [34] now extend to modal subspaces [35,36] and the emulation of up to seven-dimensional hyper-lattices [37].

One-dimensional photonic lattices modulated in synthetic dimension relate to Aubry–André–Harper (AAH) model [26,38,39], mapped to a two-dimensional integer quantum Hall system [3,4]. The latter is characterized by an integer topological index, the Chern number, determined by the Berry curvature of Bloch bands [2]. In our trimerized commensurate off-diagonal AAH model, the Chern numbers for the three bands are $\{-1,2,-1\}$ [14,17]. Most interesting is the topological transition to double-negative Chern numbers, $\{2,-4,2\}$, predicted numerically for modulation depths exceeding a threshold value [14].

We demonstrate below that the band Chern number in modulated trimer lattices can be explicitly calculated as the negative total winding number of Zak-phase dislocations enclosed by a loop in parameter space. We explain the topological transitions as the crossing of dislocations by adiabatic loop, changing the winding numbers. We suggest a general scheme to construct adiabatic protocols, also with large Chern numbers up to $\{-4,8,-4\}$. We conclude with a version of the bulk–boundary correspondence, linking Chern numbers with the sequence of edge states accompanying evolution.

The paper is organized as follows: In Section 2 we introduce our model, describe relevant experimental settings, and analyze the parameter domain of the off-diagonal trimer lattices. We demonstrate the winding of geometric Zak phase in the parameter space of the infinite periodic lattices and its relation to the appearance of paired and unpaired edge states in the spectral gaps of finite lattices. Section 3 is devoted to the study of adiabatic Thouless pumps in slowly varying lattices. We analytically establish the relation between Zak-phase winding numbers, Chern numbers of adiabatic loops, and corresponding sequences of the edge states. Section 4 concludes this paper, and Appendix A and Appendix B clarify technical aspects of our derivations.

2. Trimer Lattice and Zak Phase

We introduce trimer lattices using their potential experimental implementation in photonics as a basic example [40]; Figure 1 illustrates the concept. Namely, we consider identical single-mode optical waveguides arranged periodically close enough for the tails of their transverse modes to overlap. The overlap allows for the energy exchange, thus coupling the nearest neighbors. The overlap decays exponentially with the distance between waveguides; varying the separation between waveguides allows control of the three coupling parameters $J_m$, $m=1,2,3$. In particular, the adiabatic modulation of the coupling parameters with the propagation distance z is analogous to the tight-binding model with hopping amplitudes slowly varying over time.

Figure 1. Illustration of a trimer lattice in photonics [40]. Plotted is the refractive index $n_0(x,z)$ for two unit cells, numbered n and $n+1$, of a trimer lattice with intra-cell coupling strengths $J_{1,2}\left(z\right)$ slowly changing with propagation distance z. Dashed line corresponds to the inversion-symmetric lattice with $J_1=J_2$. We count the unit cells from the left (L) to the right (R) edges of the lattice.

Complex amplitudes of the modes in each waveguide are labeled with the three sublattice names and the unit cell number n, as shown in Figure 1,

$$\begin{array}{ccc}& & i\frac{d}{dz}{A}_n+J_1{B}_n+J_3{C}_{n-1}=0,\hfill \\ & & i\frac{d}{dz}{B}_n+J_1{A}_n+J_2{C}_n=0,\hfill \\ & & i\frac{d}{dz}{C}_n+J_2{B}_n+J_3{A}_{n+1}=0.\hfill \end{array}$$

(1)

We distinguish two systems: (i) an infinite periodic lattice with $n\in (-\infty ,\infty )$ and (ii) finite lattice of N unit cells, $n\in (1,N)$, and open boundary conditions, $A_n=B_n=C_n=0$ for $n<1$ and for $n>N$. Note the scaling invariance of Equation (1): introducing an arbitrary propagation scale $z_0$, i.e., $z\to z/z_0$, is equivalent to the scaling $J_m\to z_0{J}_m$. In other words, a longer propagation in waveguides is similar to an increased strength of the couplings between them.

Our basic model is an infinite periodic lattice with constant hopping parameters. Applying the Fourier transform $\{A_n\left(z\right),B_n\left(z\right),C_n\left(z\right)\}={\int }_{-\pi }^{\pi }dq\{a\left(q\right),b\left(q\right),c\left(q\right)\}exp(iqn-i\omega \left(q\right)z)$, we describe a lattice with Bloch state, $\psi ={\{a,b,c\}}^T$ and the crystal quasi-momentum $q=[-\pi ,\pi ]$, defined in a periodic Brillouin zone, $\psi (q+2\pi )=\psi \left(q\right)$. These states satisfy the eigenvalue problem, $\omega \psi =H\psi$, with the Bloch Hamiltonian

$$H=-\left(\begin{array}{ccc}0& J_1& J_3{e}^{-iq}\\ J_1& 0& J_2\\ J_3{e}^{iq}& J_2& 0\end{array}\right).$$

(2)

The dispersion relation $\omega \left(q\right)$ is derived as the solution to the cubic equation

$$\omega ({\omega }^2-J^2)=-2pcosq,$$

(3)

here $J^2=J_1^2+J_2^2+J_3^2$, and $p=J_1{J}_2{J}_3$; we will use the band index $\mu =1,2,3$ for the three roots ${\omega }_1\le {\omega }_2\le {\omega }_3$. Corresponding normalized Bloch wavefunctions are

$${\psi }_{\mu }\left(q\right)={\Psi }_{\mu }\left(\begin{array}{c}-{\omega }_{\mu }{J}_1+J_2{J}_3{e}^{-iq}\\ {\omega }_{\mu }^2-J_3^2\\ -{\omega }_{\mu }{J}_2+J_1{J}_3{e}^{iq}\end{array}\right),$$

(4)

here ${\Psi }_{\mu }^{-2}=(3{\omega }_{\mu }^2-J^2)({\omega }_{\mu }^2-J_3^2)$.

Parameter J can be scaled to 1 by choosing $z_0=J$ in Equation (1) and transforming $J_m\to J{J}_m$ and $\omega \to J\omega$, thus reducing the dimensions of parameter space from 3 to 2. On the sphere $J=const$, the only parameter that defines the energy $\omega \left(q\right)$ in Equation (3) is $p=J_1{J}_2{J}_3$. Therefore, the dispersion relation $\omega \left(q\right)$ does not change if we follow a line $p=const$ on the sphere J. These “dispersionless” trajectories are closed loops around the degeneracy axis $J_0=J\left\{1,1,1\right\}/\sqrt{3}$ (monatomic lattice); several such level contour lines of the surface $p=const$ are drawn on a sphere in Figure 2a. The loops originate from a degeneracy point $J_0$ with a maximal value $p_{max}=3^{-3/2}$, and, as $p\to 0$, they converge to the boundary of the spherical triangle in Figure 2a, where one or two hopping parameters vanish at the edges or vertices, respectively.

Figure 2. (a) Parameter domain $J_{1,2,3}\ge 0$ on sphere J. Different sectors shaded in accordance with the number of edge states in a finite lattice: the gray-shaded sector has no edge states, the blue and white sectors contain left-edge states, and the right-edge states appear in red and white sectors; thus, the white sector contains both edge states. The dispersion Equation (3) of an infinite lattice is plotted for (b) $q=0$ and (c) $q=\pi$; the gaps close at the degeneracy axis $J_0\equiv j_3$ with corresponding bands forming Dirac cones. The contour lines correspond to parameter $p=0.01,0.04,0.07,0.1$ (dashed), $0.13,0.16$, and $0.19$ (closest to the maximal value $p_{max}=3^{-3/2}$ at the degeneracy $J_0$).

We would like to separate two degrees of freedom: the spectral parameter p and the orthogonal dimension along the contours in Figure 2. Evidently, while the second cyclic parameter does not influence $\omega \left(q\right)$, it modifies the eigenvectors Equation (4) and generates a family of states. In order to emphasize the role of the cyclic parameter, which does not influence energy–momentum relations, we introduce the new parameters $\{j_1,j_2,j_3\}$ by identifying $j_3$ with the degeneracy axis $J_0$; we identify $j_1$ ($j_2=0$) with the inversion-symmetric lattice $J_1=J_2$; we choose azimuthal angle $\varphi ={tan}^{-1}(j_2/j_1)$ as the cyclic parameter, see Appendix A.

The new “coordinates” $j_m$ are also useful for visualization of the dispersion relation Equation (3) in Figure 2b,c. The degeneracy axis $J_0\equiv j_3$ is placed at the center of the spherical triangle, $j_1=j_2=0$, where the three bands join into a single band of a monatomic lattice. This is not an energy–momentum relation of course because the eigenvalue $\omega$ is parameterized here by the coupling parameters. Therefore, the Dirac cones in Figure 2b,c do not relate directly to the Dirac cones in, e.g., graphene or other two-dimensional lattices [41]. However, their appearance is not incidental and demonstrates the relation of the families of one-dimensional lattices to their two-dimensional counterparts. We will demonstrate that the additional parameter of a one-dimensional lattice family can be treated as a synthetic dimension along which quasi-momentum can be introduced, thus effectively expanding the family into a two-dimensional Floquet lattice.

In the following we study the family of trimer lattices in the parameter domain $J_m\ge 0$ with a focus on the parameterization by the cyclic parameter $\varphi$, tracing a particular contour along $p=0.1$ dashed curves in Figure 2 as a representative example. Similar to Figure 2 we will compare the appearance of the edge states in a finite lattice with the bulk property of the eigenstates in the bands of an infinite lattice, namely, the Zak phase.

Zak phase is defined for each band as [8]

$$Z_{\mu }={\int }_{-\pi }^{\pi }{A}_{\mu q}dq,$$

(5)

with Berry connection [1,2]

$$A_{\mu q}=i⟨{\psi }_{\mu }|{\partial }_q{\psi }_{\mu }⟩=(J_2^2-J_1^2)J_3^2{\Psi }_{\mu }^2,$$

(6)

here we used the explicit solution for the Bloch spinor in Equation (4). $A_{\mu q}$ manifestly vanishes in the inversion-symmetric lattice $J_1=J_2$, marking it as topologically trivial [8], see Appendix B.1. However, our further analysis challenges this conclusion.

In Figure 3 we study the finite open system solved numerically by diagonalization of the discretized Hamiltonian in Equation (2): as expected, the bands are completely flat in synthetic dimension $\varphi$, i.e., for every band state ${\partial }_{\varphi }{\omega }_{\mu }\left(q\right)=0$. Flat spectral bands appear naturally in systems with particle–hole symmetry with odd numbers of bands [41,42], and they exhibit interesting sensitivity to disorder and interactions [43,44]. Suppression of dispersion in synthetic dimension may enhance Thouless pumping [45]; here, it is achieved simply by choosing particular modulation of hopping parameters in Figure 3a.

Figure 3. Family of states along the dashed curve $p=0.1$ in Figure 2. (a) Hopping parameters $J_m\left(\varphi \right)$ help the eye connect the dashed grid lines at $\varphi =\pi s/3$ ($s=1,2,4,5$) with noncentered inversion symmetries $J_3=J_1$ and $J_3=J_2$ [46]. (b) Numerically obtained spectra for a chain of $N=50$ unit cells with open boundaries. The bands are shaded gray, the gap states are marked (R, red) for the right-edge and (L, blue) for the left-edge. (c) Zak phases Equation (5) are plotted mod$\left[-\pi ,\pi \right]$; note that $Z_{\mu }\equiv 0$ for $J_1=J_2$ at $\varphi =0,\pi$.

The most revealing topological property is the appearance of edge states in the spectral gaps of insulators [3,4]. Edge modes in Figure 3b bifurcate from bands at specific values of $\varphi$, connected with noncentered inversion symmetries of the lattice [46,47,48,49,50]. The domains of their existence [17] can be summarized as follows: (R) right-edge modes for $J_3>J_1$ and (L) left-edge modes for $J_3>J_2$. This definition is sufficient to describe the whole parameter domain $J_m>0$ in Figure 2a. Indeed, an overlap of both domains, $J_3>J_{1,2}$, contains both edge modes, and it is marked white in Figure 2a. If none of the conditions are satisfied, $J_3<J_{1,2}$, there are no edge states, and corresponding area in Figure 2a is shaded gray. The red-colored sector $\pi /3<\varphi <2\pi /3$ ($J_1<J_3<J_2$) corresponds to a single right-edge state in each spectral gap, while the sector with a single left-edge state, $4\pi /3<\varphi <5\pi /3$ ($J_2<J_3<J_1$), is shaded blue.

At the same time, the inversion-symmetric lattice exhibits sharp transition along $J_1=J_2$ from a domain with no edge states at $J_3<J_1=J_2$ ($\varphi =0$) to a symmetric pair of gap states on both edges at $J_3>J_1=J_2$ ($\varphi =\pi$). This transformation closely resembles the topological transition in SSH model [10,12], accompanied by a change in Zak phase from 0 to $\pi$. However, as we demonstrated above, in our model the Zak phase is trivial in the whole domain with inversion symmetry, $J_1=J_2$, which naturally includes topologically trivial monoatomic lattices with $J_1=J_2=J_3$. Nevertheless, we observe in Figure 3c that any other crossing of degeneracy from $\varphi$ to $\varphi +\pi$ is accompanied by Zak-phase jumps close to $\pi$ for $Z_{1,3}$ and $2\pi$ for $Z_2$, thus confirming observations of Ref. [16].

The discontinuities of Zak phases in Figure 3c and Ref. [16] are features known as branch cuts in twisted phases around screw-type wave dislocations [18]; in complex fields phase $\sim e^{il\varphi }$ is quantized by an integer winding number l, often called topological charge. In other words, while the straight path from $\varphi$ to $\varphi +\pi$ in Figure 2b,c involves crossing degeneracy with gap closure, the same phase jump is gained on the path around the degeneracy point. Therefore, we calculate Zak phases in the whole parameter domain; Figure 4 reveals their twists around the degeneracy with well-defined winding numbers, $l_{\mu }=\{1,-2,1\}$. In contrast to complex fields, e.g., optical vortices [19], there is no singularity at the origin $j_{1,2}=0$ because Zak phases are well-defined zeros along the $j_1$ axis in Figure 4.

Figure 4. Zak phases Equation (5): (a,c) $Z_{1,3}$ with winding number $l_{1,3}=+1$ and (b,d) $Z_2$ with winding number $l_2=-2$. The dashed contour $p=0.1$ corresponds to Figure 3c. The branch cuts are evident in gray scale in (a,b), and they almost vanish if we plot $Z_{1,3}$ mod$[0,2\pi ]$ and $Z_2$ mod$[-2\pi ,2\pi ]$, or use cyclic colormap, such as hue in (c,d). The data plotted in (a,b) and (c,d) are exactly the same; only the colormaps are different.

It is interesting to connect the winding numbers $l_{\mu }$ with the Dirac cones in Figure 2b,c. While the first and the third bands contain only one Dirac cone at $q=\pi$ and $q=0$, respectively, the second band is connected to both Dirac cones with opposite circulation [1]; thus, $l_2=-2$. We recall the “wormhole” argument by Haldane [51], namely, that the two Dirac singularities couple the bands by a “Berry flux loop”, where Berry curvature flux passes from one band to the other through the first Dirac point and then returns through the second.

3. Thouless Pump and Chern Numbers

So far, we considered a family of lattices in the parameter space. A natural question is the role of collective features, such as Zak-phase dislocations, in the adiabatic evolution along the family sequence [14,16,17,26]. We assume an arbitrary closed trajectory $\overrightarrow{\tau }$, parameterized by its length $\tau =|\overrightarrow{\tau }|$ and given by $J_m\left(\tau \right)$ functions, governing the energy Equation (3) and eigenstates Equation (4) along the loop. The Chern number is defined as the surface integral over two-dimensional Berry curvature [1,2],

$$C_{\mu }={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }dq\oint d\tau \left({\partial }_q{A}_{\mu \tau }-{\partial }_{\tau }{A}_{\mu q}\right),$$

(7)

with the Berry connection $A_{\mu \tau }=i⟨{\psi }_{\mu }|{\partial }_{\tau }{\psi }_{\mu }⟩$ on $\overrightarrow{\tau }$,

$$A_{\mu \tau }={\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial }{\partial q}}ln({\omega }_{\mu }^2-J_3^2){\displaystyle \frac{d}{d\tau }}ln{\displaystyle \frac{J_1}{J_2}},$$

(8)

where we used the group velocity expression, $\partial {\omega }_{\mu }/\partial q=2psinq/(3{\omega }_{\mu }^2-J^2)$, derived from Equation (3). Equation (8) shows that contribution of $A_{\mu \tau }$ to Equation (7) is zero on any contour $\overrightarrow{\tau }$ because the integral ${\int }_{-\pi }^{\pi }{\partial }_q{A}_{\mu \tau }dq$ vanishes due to periodicity $\omega (q+2\pi )=\omega \left(q\right)$. Reordering $\oint {\partial }_{\tau }\left({\int }_{-\pi }^{\pi }{A}_{\mu q}dq\right)d\tau$ in Equation (7), we obtain

$$C_{\mu }=-{\displaystyle \frac{1}{2\pi }}\oint {\displaystyle \frac{\partial Z_{\mu }}{\partial \tau }}d\tau ,$$

(9)

i.e., the Chern number for each band is the negative winding number of the Zak phase on the trajectory $\overrightarrow{\tau }$.

The first result of Equation (9) is that Zak-phase dislocations in Figure 4 define Chern numbers $C_{\mu }=-l_{\mu }=\{-1,2,-1\}$ on any contour around dislocation, thus confirming the numerical results of Ref. [14]. Topological nature of Chern numbers is explicit here as $C_{\mu }$ do not depend on a particular trajectory as long as it encircles degeneracy axis $J_0$ once in the counter-clockwise direction.

The next open question is to understand the topological transition in AAH model with

$$J_m\left(\varphi \right)=\tilde{J}\left\{1-\lambda cos\left(2\pi m/3-\varphi \right)\right\},$$

(10)

predicted to occur for modulation parameter $\lambda >4$ [14]. The factor $\tilde{J}=J\sqrt{2/3(2+{\lambda }^2)}$ scales hopping parameters to the sphere J. For small modulation, $\lambda \ll 1$, Equation (10) describes a circle around $J_0$ axis, similar to $p=const$ loops close to $J_0$ in Figure 2. At $\lambda =1$ the circle touches the boundary $J_m=0$ of the spherical triangle in Figure 2a, which corresponds to the anti-continuum limit $p=0$ with collapsing bands, ${\omega }_{\mu }=J\{-1,0,1\}$, see Appendix B.2.

Although hopping parameters for cold atoms [24] and coupled waveguides [26] take positive values, other photonic systems offer the possibility of controlling hopping phases [52,53,54]. Therefore, we extend our analysis to the complete parameter space including negative hopping parameters and distinguish “positive” and “negative” Zak-phase dislocations in Figure 5a,b. The critical parameter of topological transition $\lambda =4$ in Equation (10) corresponds to a circle in Figure 5 (red line) passing trough three negative dislocations. Increasing $\lambda >4$ includes four dislocations within the circle, one positive and three negative; the corresponding Zak winding numbers change from $l_{\mu }=\{1,-2,1\}$ to $l_{\mu }=\{-2,4,-2\}$; and we recover the numerical result of Ref. [14], $C_{\mu }=\{2,-4,2\}$.

Figure 5. Full parameter space on the sphere J (a,c) and corresponding Mercator projection in (b,d) with $\zeta ={tanh}^{-1}(J_3/J)$ and spherical azimuth $\phi ={tan}^{-1}(J_2/J_1)$. In (a,b) the white domains of “positive” dislocations $l_{\mu }=\{1,-2,1\}$, as in Figure 4 with $p>0$, and the gray domains with “negative” dislocations with $p<0$ and opposite winding numbers, $l_{\mu }=\{-1,2,-1\}$. Positive and negative dislocations are marked in (b,d) by open red and blue circles, respectively. In (c,d) the shading of different domains indicates the number of edge states; see the legend in panel (d) and Figure 2a. The solid red line in all panels, passing through 3 negative dislocations, is the AAH critical trajectory Equation (10) with $\lambda =4$ [14]. Other trajectories indicated in (b,d) with black and blue lines are discussed in the text and Figure 6.

Mapping of Zak-phase dislocations in Figure 5 offers a general way to design pumping protocols with desired Chern numbers by choosing appropriate trajectory in parameter space. For example, using the same Chern numbers as above, $C_{\mu }=\{2,-4,2\}$, we obtain an elliptic loop encircling two negative dislocations in Figure 5b. The maximal Chern numbers with a single, simply connected loop are $C_{\mu }=\{-4,8,-4\}$ for the zig-zag-shaped contour drawn with a black line in Figure 5b, cf. Refs. [55,56,57]. Furthermore, the adiabatic trajectory is not limited by simple loops, e.g., the self-intersecting figure-eight cycle in Figure 5b has a well-defined winding and Chern numbers $\{0,0,0\}$. The question we address below is how such a variety of Chern numbers relates to the adiabatic pump dynamics; namely, what is the bulk–boundary correspondence in our system?

Thouless pumping in Fermionic systems is characterized by the total Chern number of the filled bands below the gap with Fermi level [23]. In our case $C=C_1$ for the bottom gap, $C=C_1+C_2$ for the top gap, and $C=\sum C_{\mu }=0$ when all bands are full. Photonic lattices realize filled bands with excitation of a single lattice site [14], while they also allow precise excitation of selected modes and adiabatic pumping with edge states [26]. Therefore, in the following we discuss the bottom gap states and $C=C_1$.

Figure 5c,d maps the domains with different number of edge states with the same coloring as in Figure 2a; it allows us to predict the bifurcation of each gap edge state, e.g., in the gray parameter areas, the gaps are empty. In Figure 6 we plot the total intensity of edge states in the bottom gap, $|{\psi }_L{|}^2+{\left|{\psi }_R\right|}^2$, along different trajectories. The states on two edges overlap only in the b sublattice because the left-edge state has sublattice c empty, ${\psi }_L={\{a_L,b_L,0\}}^T$, while the right-edge state has sublattice a empty, ${\psi }_R={\{0,b_R,c_R\}}^T$. Figure 6a corresponds to the dashed loop $p=0.1$ in Figure 2 with a sequence of “right-both-left” edge states along $\varphi$, cf. Figure 3b. This sequence corresponds to the Thouless pumping of a single charge from the right to the left edge [26], i.e., negative current with the total Chern number $C=C_1=-1$. We will count the pair of edge states in the sequence RL (right–left) as a negative “Thouless pair”, in contrast to the positive pair LR (left–right). Examples below establish the correspondence: the Chern number equals the total number of (signed) edge state pairs.

Figure 6. Total intensity of two edge states in the bottom gap, $|{\psi }_R{|}^2+{\left|{\psi }_L\right|}^2$, along different loops for an open chain of $N=7$ unit cells. (a) The loop $p=0.1$ in Figure 2 (dashed contour) and Figure 3. The oval (b) and figure-eight (c) trajectories shown as blue curves in Figure 5b,d. (d) The zig-zag-shaped contour shown with a black line in Figure 5b,d. Note that the length scale in (a–c) is the same, which allows better comparison of the edge states and emphasizes the different lengths of the loops in parameter space, important for experimental realization. The contour in (d) is much longer, and this panel length is scaled down.

Indeed, the elliptic loop in Figure 5b corresponds to the sequence of two positive edge-state pairs LR–LR in Figure 6b; thus, $C=2$ (similar to the circle Equation (10) with $\lambda >4$). The figure-eight loop in Figure 5b demonstrates the sequence LR–RL in Figure 6c, where the two opposite currents cancel each other within one cycle; thus, $C=0$. The zig-zag loop with $C=-4$ has four consecutive negative pairs RL-RL-RL-RL, see Figure 6d, and it realizes the maximal Chern numbers in our system, $C_{\mu }=\{-4,8,-4\}$.

A final note on the design of adiabatic pumps is that the full three-dimensional parameter space can be used, e.g., the self-crossing of the figure-eight trajectory in Figure 5b and Figure 6c, and can be avoided if the coupling strength J varies along the cycle. In other words, the trajectories are not limited to the sphere considered above, and they can form three-dimensional loops winding around degeneracy axes. The Chern numbers are exactly the same and the edge states follow the same bifurcation sequences as their counterparts along the loops projected on a sphere. This degree of freedom might be useful in experimental designs.

4. Conclusions

In conclusion, we consider trimer lattices and show that the gap-closing degeneracies, embedded in three-dimensional parameter space, generate screw-type dislocations in geometric phase [18]. The Dirac-type spectral degeneracies allow for classification of Floquet–Bloch systems [58]; here we provide the route to design these structures based on geometric phase dislocations. This procedure allows us to construct flat bands in synthetic dimensions as well as engineer the Chern numbers and edge states in adiabatic pumps. It remains to be explored how our findings could be applied in closely related and rapidly developing topological systems, such as two-dimensional photonic crystals [59,60,61], non-hermitian lattices [62,63,64,65], plasmonics [66], metamaterials [67], and acoustics [68,69]. Perhaps more generally, spectral singularities [70] and topological invariants [71,72] map to real space phase defects; we expect similar phenomena with Zak-phase dislocations. It will also be of interest to study the role that geometric phase dislocations may play in topological transitions induced by on-site potentials [14] and nonlinear interactions [73,74,75].

Appendix B.1. Zak-Phase Vanishes in the Inversion-Symmetric Lattice $J_1=J_2$

Although directly evident from Equation (6) that Berry connection vanishes for $J_1^2=J_2^2$, we also notice that the eigenvector $\psi ={\left\{a,b,c\right\}}^T$ in Equation (4) gains additional symmetry, $c=\pm a^{*}$ for $J_2=\pm J_1$. We can generalize and consider a vector $\psi ={\left\{a,b,\pm a^{*}\right\}}^T$ with real $b=b^{*}=\sqrt{1-2{\left|a\right|}^2}$. It is straightforward to derive $⟨\psi |{\partial }_q\psi ⟩\equiv 0$ for such an eigenstate.

Furthermore, the eigenstate amplitude ${\Psi }_{\mu }$ in Equations (4) and (6) diverges in this case, which may complicate the direct numerical integration of Equations (5) and (7). The dispersion relation Equation (3) for $q=0,\pi$, plotted in Figure 5b,c, gives the following results:

For	$J_3>J_1=J_2$,	$\varphi =\pi$,	$\left\{\begin{array}{c}{\omega }_3\left(0\right)=J_3\hfill \\ {\omega }_1\left(\pi \right)=-J_3\hfill \end{array}\right.$
For	$J_3<J_1=J_2$,	$\varphi =0$,	$\left\{\begin{array}{c}{\omega }_2\left(0\right)=J_3\hfill \\ {\omega }_2\left(\pi \right)=-J_3\hfill \end{array}\right.$

so that the factor $1/({\omega }_{\mu }^2-J_3^2)$ diverges for respective bands. Similarly, the factor $1/(3{\omega }_{\mu }^2-J^2)$ diverges at the degeneracy $J_0$, i.e., $J_{1,2,3}=J/\sqrt{3}$, where a pair of bands touch and form Dirac cones in Figure 2b,c, namely, ${\omega }_2\left(0\right)={\omega }_3\left(0\right)=J/\sqrt{3}$ and ${\omega }_1\left(\pi \right)={\omega }_2\left(\pi \right)=-J/\sqrt{3}$. These poles are not essential; they appear as a consequence of band folding.

Appendix B.2. Band Collapse and Zak Phase in the Anti-Continuum Limit

Along the boundary of the spherical triangle in Figure 2a, one of the hopping parameters vanishes. This corresponds to the fragmented (anti-continuum) lattice and leads to band collapse. Indeed, because parameter $p=0$, the dispersion relation Equation (3) gives the triplet ${\omega }_{1,2,3}=J\{-1,0,1\}$, independent of the location at the boundary. We were able to derive Zak phase in analytical form in this limit using expressions for eigenvectors Equation (4) and Berry connection Equation (6). We obtain the following results, referring to the edges of the spherical triangle in Figure 2a:

Right edge:	$J_1=0$,	$\pi /3<\varphi <\pi$,	$\left\{\begin{array}{c}{Z}_{1,3}=\pi J_3^2/J^2\hfill \\ Z_2=2\pi J_2^2/J^2\hfill \end{array}\right.$
Left edge:	$J_2=0$,	$-\pi <\varphi <-\pi /3$,	$\left\{\begin{array}{c}{Z}_{1,3}=-\pi J_3^2/J^2\hfill \\ Z_2=-2\pi J_1^2/J^2\hfill \end{array}\right.$
Bottom edge:	$J_3=0$,	$-\pi /3<\varphi <\pi /3$,	$Z_{1,2,3}=0$

These results are fully consistent with those numerically obtained in Figure 4, namely, the winding numbers of Zak-phase dislocations $l_{1,2,3}=\left\{1,-2,1\right\}$. It is noteworthy that such a simple analysis allows us to identify the presence of a dislocation and its winding number. In particular, it offers a quick proof that the winding numbers change sign when one of the hopping parameters above changes sign, as shown in Figure 5.