Schäfer, R.; Chamon, C.; Laumann, C.R.: Hall-on-Toric State: Descendant Laughlin State in the chiral ℤ𝑝 Toric Code. Physical Review Letters 136 (2026), p. 166603/1-7
10.1103/f63q-vqz3
Open Access Version
Abstract:
We demonstrate that the chiral ℤ𝑝 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 ℤ𝑝 charges and increased topological ground-state degeneracy. The Hall-on-Toric phases appear in the vicinity of the transitions between deconfined ℤ𝑝 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.