# Evolution of quantum entanglement with disorder in fractional quantum Hall liquids

## TL;DR

This paper investigates how quantum entanglement in disordered fractional quantum Hall liquids signals phase transitions from topological phases to insulators across various filling fractions using exact diagonalization.

## Contribution

It provides detailed analysis of entanglement entropy behavior at different filling fractions, identifying potential disorder-driven phase transitions in fractional quantum Hall systems.

## Key findings

- Minimum in entropy derivative indicates phase transition at certain fillings.
- Entropy derivative diverges in the thermodynamic limit at specific fillings.
- Finite-size effects vary significantly across different filling fractions.

## Abstract

We present a detailed study of the ground-state entanglement in disordered fractional quantum Hall liquids. We consider electrons at various filling fractions $f$ in the lowest Landau level, with Coulomb interactions. At $f=1/3,1/5$ and $2/5$ where an incompressible ground-state manifold exists at zero disorder, we observe a pronounced minimum in the derivative of entanglement entropy with respect to disorder. At each filling, the position of this minimum is stable against increasing system size, but its magnitude grows monotonically and appears to diverge in the thermodynamic limit. We consider this behaviour of the entropy derivative as a compelling signal of the expected disorder-driven phase transition from a topological fractional quantum Hall phase to a trivial insulating phase. On the contrary, at $f=1/2$ where a compressible composite fermion sea is present at zero disorder, the entropy derivative exhibits much greater, almost chaotic, finite-size effects, without a clear phase transition signal for system sizes within our exact diagonalization limit. However, the dependence of entanglement entropy with system size changes with increasing disorder, consistent with the expectation of a phase transition from a composite fermion sea to an insulator. Finally, we consider $f=1/7$ where compressible Wigner crystals are quite competitive at zero disorder, and analyze the level statistics of entanglement spectrum at $f=1/3$.