# Emergent Commensurability from Hilbert Space Truncation in Fractional Quantum Hall Fluids

## TL;DR

This paper introduces a universal scheme based on Hilbert space truncation and translational invariance to determine fractional quantum Hall states, revealing emergent properties and predicting realizability without microscopic Hamiltonians.

## Contribution

It proposes the concept of emergent commensurability as a fundamental property of FQH states, enabling universal identification of topological features without detailed Hamiltonian models.

## Key findings

- Unique determination of FQH model states via Hilbert space truncation.
- Identification of filling factors, shifts, and pairing unambiguously.
- Prediction of state realizability based on emergent commensurability.

## Abstract

We show that model states of fractional quantum Hall fluids at all experimentally detected plateau can be uniquely determined by imposing translational invariance with a particular scheme of Hilbert space truncation motivated from physical local measurements. The scheme allows us to identify filling factors, topological shifts and pairing/clustering of topological quantum fluids unambiguously in a universal way without resorting to microscopic Hamiltonians. This prompts us to propose the notion of emergent commensurability as a fundamental property for at least most of the known FQH states, which allows us to predict if a particular FQH state conforming to a set of paradigms can be realised \emph{in principle}. We also discuss the implications of certain missing states proposed from other phenomenological approaches, and suggest that the physics of fractional quantum Hall physics could fundamentally arise from the algebra of the Hilbert space in a single Landau level.