Hall-on-Toric: Descendant Laughlin state in the chiral $\mathbb{Z}_p$ toric code

TL;DR

This paper uncovers emergent topological phases called Hall-on-Toric states in the chiral $ ext{Z}_p$ toric code, characterized by fractional charges, increased degeneracy, and robustness without $U(1)$ symmetry, confirmed via iDMRG simulations.

Contribution

It introduces Hall-on-Toric states as descendant fractional quantum Hall-like phases in the chiral $ ext{Z}_p$ toric code, expanding understanding of topological order and excitations.

Findings

Hall-on-Toric states exhibit fractionalized $ ext{Z}_p$ charges.

These states show increased topological ground-state degeneracy.

Hall-on-Toric phases are robust even without $U(1)$ symmetry.

Abstract

We demonstrate that the chiral $\mathbb{Z}_p$ toric code -- the quintessential model of topological order -- hosts additional, emergent topological phases when perturbed: descendant fractional quantum Hall-like states, which we term \textit{Hall-on-Toric}. These hierarchical states feature fractionalized $\mathbb{Z}_p$ charges and increased topological ground-state degeneracy. The Hall-on-Toric phases appear in the vicinity of the transitions between deconfined $\mathbb{Z}_p$ phases with different background charge per unit cell, in a fixed non-trivial 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.