# Statistical Topology—Distribution and Density Correlations of Winding Numbers in Chiral Systems

## Abstract

**:**

## 1. Introductory Remarks

## 2. Winding Numbers and Chirality

#### 2.1. A Simple Topological Invariant in Complex Analysis

#### 2.2. Kitaev Chain and Winding Numbers

#### 2.3. Chirality, Random Winding Numbers and Modelling Aspects

#### 2.4. Connections to Quantum Chromodynamics and Condensed Matter Physics

## 3. Formulation of the Problem and Mathematical Setup

#### 3.1. Chiral Random Matrix Ensembles with Parametric Dependence

#### 3.2. Statistical Quantities Considered

#### 3.3. Mapping a Topological to a Spectral Problem

## 4. Results

#### 4.1. Winding Number Correlators in the Chiral Unitary Case

#### 4.2. Winding Number Distribution

#### 4.3. Aspects of Universality

#### 4.4. Generators in the Chiral Unitary and Symplectic Cases

## 5. Discussion and Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RMT | Random Matrix Theory |

QCD | Quantum Chromodynamics |

chGUE | chiral Gaussian Unitary Ensemble |

chGSE | chiral Gaussian Symplectic Ensemble |

${\mathbb{S}}_{N}$ | permutation group of N objects |

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**Figure 1.**

**Left**: Three points ${z}_{i},\phantom{\rule{4pt}{0ex}}i=1,2,3$ in the complex plane $\mathbb{C}$ and a closed contour $\gamma $.

**Right**: A closed contour $\Gamma $ encircling zeros and poles of a meromorphic function $f\left(z\right)$.

**Figure 2.**Kitaev chain, electrons as larger open circles (red), Majorana fermions as small dots (green) with the pairing indicated by connecting lines (green). Top: All Majorana fermions are paired, normal or trivial superconducting phase. Bottom: Unpaired Majorana fermions at the ends of the chain, topological superconducting phase.

**Figure 3.**Ellipses described by $\overrightarrow{d}\left(k\right)$ (

**left**) and corresponding dispersion relations $E\left(k\right)$ (

**right**).

**Top**: $t=0.25$, normal superconducting phase, $W=0$.

**Center**: $t=0.5$, phase transition point.

**Bottom**: $t=1$, topological superconducting phase, $W=1$. Courtesy of Nico Hahn.

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**MDPI and ACS Style**

Guhr, T.
Statistical Topology—Distribution and Density Correlations of Winding Numbers in Chiral Systems. *Entropy* **2023**, *25*, 383.
https://doi.org/10.3390/e25020383

**AMA Style**

Guhr T.
Statistical Topology—Distribution and Density Correlations of Winding Numbers in Chiral Systems. *Entropy*. 2023; 25(2):383.
https://doi.org/10.3390/e25020383

**Chicago/Turabian Style**

Guhr, Thomas.
2023. "Statistical Topology—Distribution and Density Correlations of Winding Numbers in Chiral Systems" *Entropy* 25, no. 2: 383.
https://doi.org/10.3390/e25020383