Zak Phase Dislocations in Trimer Lattices

TL;DR

This work reveals that in off-diagonal trimer lattices, screw-type dislocations of the Zak phase occur in the three-dimensional parameter space, with the degeneracy axis forming the organizing center for phase winding. By treating a cyclic adiabatic modulation as a synthetic dimension, the authors show that the Chern number of a pumping loop equals the negative Zak-phase winding around the dislocation, enabling precise control of edge-state sequences and pumped charge. The key results include the relation C_μ = -l_μ, explicit winding numbers for the three bands, and constructive prescriptions for designing adiabatic pumps with large, tunable Chern numbers, along with a bulk–edge correspondence for evolving finite lattices. The findings provide a framework for engineering flat bands in synthetic dimensions and robust topological pumping in photonic and related platforms. They also generalize to negative hopping regimes and higher Chern numbers, offering a versatile toolbox for topological transport in one- and quasi-two-dimensional systems.

Abstract

Wave propagation in periodic media is governed by energy-momentum relation 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 pump, corresponding closed trajectory in parameter space is characterized by a Chern number equal the negative total winding number of Zak phase dislocations enclosed by the trajectory. We discuss correspondence between bulk Chern numbers and the edge-states in a finite system evolving along various pumping cycles.

Figures (6)

• Figure S1: Illustration of a trimer lattice in photonics Alex. Plotted is the refractive index $n_0(x,z)$ for two unit cells, numbered $n$ and $n+1$, of a trimer lattice with the intra-cell coupling strengths $J_{1,2}(z)$ 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.
• Figure S2: ( a) Parameter domain $J_{1,2,3}\ge0$ on the 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 (\ref{['disp']}) 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 $J_0$).
• Figure S3: Family of states along the dashed curve $p=0.1$ in Figure \ref{['Fig1']}. ( a) Hopping parameters $J_m(\phi)$ help an eye to connect the dashed grid lines at $\phi= \pi s/3$ ($s=1,2,4,5$) with noncentered inversion symmetries $J_3=J_1$ and $J_3=J_2$inversionPRB19. ( 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 (\ref{['Zak']}) are plotted mod$\mathopen{}\mathclose{\left[-\pi,\pi \right]$; note that $Z_\mu\equiv0$ for $J_1=J_2$ at $\phi=0,\pi$.
• Figure S4: Zak phases Equation (\ref{['Zak']}): ( 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 \ref{['Fig2']}c. 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.
• Figure S5: 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 $\varphi =\tan^{-1}(J_2/J_1)$. In ( a,b) the white domains of "positive" dislocations $l_\mu=\{1,-2,1\}$, as in Figure \ref{['Fig3']} 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 \ref{['Fig1']}a. The solid red line in all panels, passing through 3 negative dislocations, is the AAH critical trajectory Equation (\ref{['J']}) with $\lambda=4$ChaoYuri. Other trajectories indicated in ( b,d) with black and blue lines are discussed in the text and Figure \ref{['Fig5']}.