Hall-on-Toric State: Descendant Laughlin State in the Chiral Z_{p} Toric Code

We demonstrate that the chiral Z_{p} toric code-the quintessential model of topological order-hosts additional, emergent topological phases when perturbed: descendant fractional quantum Hall-like states, which we term Hall-on-Toric. These hierarchical states feature fractionalized Z_{p} charges and increased topological ground-state degeneracy. The Hall-on-Toric phases appear in the vicinity of the transitions between deconfined Z_{p} phases with different background charge per unit cell, in a fixed nontrivial flux background. We confirm their existence through extensive infinite density matrix renormalization group (iDMRG) simulations, analyzing the topological entanglement entropy, entanglement spectra, and a generalized Hall conductance. Remarkably, the Hall-on-Toric states remain robust even in the absence of U(1) symmetry. Our findings reinforce the foundational interpretation of star and plaquette defects as magnetic and electric excitations, and reveal that this perspective extends to a much deeper level.