We introduce an exactly solvable model of a randomly decorated two-leg spin-ladder that incorporates the topology and main magnetic interactions in CuO ladders found in several cuprate superconducting ceramics. Copper ions at the nodal sites have $S=1/2$. Oxygen ions at the bonds have a random fraction of $S=1/2$ spins due to hole doping. Annealed disorder is assumed to result from the thermal equilibrium with the hole reservoir. Using the transfer matrix technique, we derive the exact thermodynamic behavior and explore its main features. After raising the exact ground-state diagram as a function of the chemical potential and exchange ratio, we explore the thermodynamic signatures of the influence of the model parameters on the entropy and specific heat curves, unveiling competing spin correlations. Finally, we show that distinct hole concentrations in the ladder's legs and rungs are developed, especially when the strength of the Cu-O exchange coupling is larger than the Cu-Cu coupling, as in superconducting cuprate ceramics. We show that for weak Cu-O couplings, thermal fluctuations promote a bias toward a larger hole concentration in the ladder's rungs. For strong Cu-O couplings, magnetic frustration leads to distinct hole concentrations even at zero temperature, with a bias toward the rungs for low doping fractions and the reversed trend toward the ladder's legs at high doping levels.

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• Classical statistical mechanics
• Phase diagrams
• Thermodynamics
• Classical spin models
• Disordered systems
• Equilibrium lattice models
• Exact solutions for many-body systems
• Ising model
• Transfer matrix calculations