# Sequencing the Entangled DNA of Fractional Quantum Hall Fluids

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Statement of the Problem

**Theorem 1.**

## 3. Some Basics Applications

#### 3.1. Laughlin State

#### 3.2. Jain-2/5 State

#### 3.3. Tao–Thouless State

#### 3.4. Thin Cylinder

## 4. Proof

**Proof.**

**Lemma 1.**

**Proof.**

**Corollary 1.**

**Proof.**

**Lemma 2.**

**Proof.**

**Corollary 2.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Half infinite ladder of single particle quantum Hall states with simple $j\ge 0$ boundary condition, as appropriate, e.g., for a single Landau level in the disk geometry with an additional spin or valley index, or multiple Landau levels in the half-infinite cylinder.

**Figure 2.**Half infinite ladder of quantum Hall states for a system of multiple Landau levels in the disk geometry, subject to the constraint $r\ge -j$.

**Figure 3.**Graphical representation of patterns/partitions (bottom) associated to configurations (top). The pattern/partition is obtained by dropping the vertical index (r) present in the configuration, and representing each horizontal index (j) by a single orbital. These single orbitals are then occupied by identical particles that may multiply occupy each orbital, resulting in what we refer to as a pattern of occupancies ${n}_{j}$ (see text). (If the original particles were bosons, multiple occupancies are already possible for each state represented by a horizontal line in the configuration on top.) On patterns, one may define inward squeezing processes as indicated by the arrows and as explained in the text.

**Figure 4.**Root state, or DNA, of the Jain-2/5 wave function. Ovals represent singlets. This configuration manifestly occupies 2/5 of all available hall states, and satisfies the entangled Pauli principle that two particle entangled states must occur within a distance of 2.

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**MDPI and ACS Style**

Cruise, J.R.; Seidel, A.
Sequencing the Entangled DNA of Fractional Quantum Hall Fluids. *Symmetry* **2023**, *15*, 303.
https://doi.org/10.3390/sym15020303

**AMA Style**

Cruise JR, Seidel A.
Sequencing the Entangled DNA of Fractional Quantum Hall Fluids. *Symmetry*. 2023; 15(2):303.
https://doi.org/10.3390/sym15020303

**Chicago/Turabian Style**

Cruise, Joseph R., and Alexander Seidel.
2023. "Sequencing the Entangled DNA of Fractional Quantum Hall Fluids" *Symmetry* 15, no. 2: 303.
https://doi.org/10.3390/sym15020303