Topological defects in two-dimensional (2D) and three-dimensional topological materials can induce exotic phenomena, such as protected localized modes. The introduction of synthetic dimensions further enables the exploration of such effects in higher-dimensional systems. Recent studies reveal that a 2D photonic crystal with 2D synthetic translation space ($\mathrm{\Delta }x,\mathrm{\Delta }y$) supports gapless dislocation modes, protected by the second Chern number of the four-dimensional pseudomomentum space $(k_x,\mathrm{\Delta }x,k_y,\mathrm{\Delta }y)$. However, the complexity of the dislocation lattice and the unclear mode-formation mechanism hinder applications in wave devices. To simplify the structure and clarify the physics, we discretize the lattice into four sectors. So that the sectored dislocation lattice can be viewed as the interface between two one-dimensional (1D) edge systems with edge states. The $(k_y,\mathrm{\Delta }y)$ subspace topology staggers the edge bands of two 1D edge systems, yielding a common edge gap with differing band counts below it, while the $(k_x,\mathrm{\Delta }x)$ subspace topology ensures a first Chern number of $-1$ to each band. Their joint contribution guarantees a gapless dislocation band spanning the common edge gap, protected by the second Chern number. Based on this mechanism, we propose an easy-to-integrate topological wavelength-selective component using sectored translation scheme. This work elucidates the formation mechanism of gapless dislocation modes and expands the design paradigm for topological functional systems in synthetic translation dimensions.

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• Crystal defects
• Disclinations & dislocations
• Photonics
• Topological defects
• Topological effects in photonic systems
• Metamaterials
• Photonic crystals