Motivated by the order fractionalization in the Kitaev-Kondo model, we propose an exactly solvable spin-charge ladder model to study order fractionalization with discrete symmetry. The spin-charge ladder is composed of a spin chain and a superconducting wire coupled via an Ising-type interaction, and we obtain the exact solution in the flat band limit. The exact solution reveals the ${\mathbb{Z}}_2$ order fractionalization with dual symmetry breaking and intertwined order parameters. We investigate the topological phase transition of the spin-charge ladder via the spectral chiral index, and we identify the correlated topological superconductor phase with gapped ${\mathbb{Z}}_2$ Kondo flux excitations. We demonstrate that Majorana spinons generate odd frequency pairing in the superconducting wire. We also discuss the order fractionalization from the perspective of ${\mathbb{Z}}_2$ lattice gauge theory.

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• Fractionalization
• Quantum spin models
• Strongly correlated systems
• Exact solutions for many-body systems
• Spin chains
• Spin ladders