# Zero-Energy Modes, Fractional Fermion Numbers and The Index Theorem in a Vortex-Dirac Fermion System

## Abstract

**:**

## 1. Introduction

## 2. Fermion Zero-Energy Modes And Solitons

#### 2.1. A Vortex-Dirac Fermion Model

#### 2.2. Effect of the Chemical Potential

#### 2.3. Dirac Fermions and Soliton Fields

## 3. Dirac Operator and Fractional Fermion Number

#### 3.1. Index of the Dirac Operator

#### 3.2. Fractional Fermion Number

#### 3.3. Fractional Vortex and Dirac Index

#### 3.4. Fermion Number and Kinks

## 4. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | two-dimensional |

TI | topological insulator |

## References

- Qi, X.L.; Hughes, T.L.; Zhang, S.C. Topological field theory of time-reversal invariant insulators. Phys. Rev. B
**2008**, 78, 195424. [Google Scholar] [CrossRef] [Green Version] - Atiyah, M.; Patodi, V.; Singer, I. Spectral asymmetry and Riemannian geometry. Bull. Lond. Philso. Soc.
**1973**, 5, 229. [Google Scholar] [CrossRef] - Atiyah, M.; Patodi, V.; Singer, I. Spectral asymmetry and Riemannian geometry I. Proc. Camb. Philos. Soc.
**1975**, 77, 42. [Google Scholar] [CrossRef] - Atiyah, M.; Patodi, V.; Singer, I. Spectral asymmetry and Riemannian geometry II. Proc. Camb. Philos. Soc.
**1975**, 78, 405. [Google Scholar] [CrossRef] - Atiyah, M.; Patodi, V.; Singer, I. Spectral asymmetry and Riemannian geometry III. Proc. Camb. Philos. Soc.
**1976**, 79, 71. [Google Scholar] [CrossRef] - Jackiw, R.; Rossi, P. Zero modes of the vortex-fermion system. Nucl. Phys. B
**1981**, 190, 681. [Google Scholar] [CrossRef] - Callan, C.G.; Harvey, J.A. Anomalies and fermion zero modes on strings and domain walls. Nucl. Phys. B
**1985**, 250, 427. [Google Scholar] [CrossRef] - Weinberg, E.J. Classical Solutions in Quantum Field Theory; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Manton, N.S.; Sutcliffe, P. Topological Solitons; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Rajaraman, R. Solitons and Instantons; North-Holland: Amsterdam, The Netherlands, 1982. [Google Scholar]
- Yanagisawa, T.; Hase, I.; Tanaka, Y. Massless and quantized modes of kinks in the phase space of superconducting gaps. Phys. Lett. A
**2018**, 382, 3483. [Google Scholar] [CrossRef] [Green Version] - Jackiw, R.; Rebbi, C. Solitons with fermion number 1/2. Phys. Rev. D
**1976**, 13, 3398. [Google Scholar] [CrossRef] - Su, W.P.; Schrieffer, R.; Heeger, A.J. Solitons in polyacetylene. Phys. Rev. Lett.
**1979**, 42, 1698. [Google Scholar] [CrossRef] - Su, W.P.; Schrieffer, R.; Heeger, A.J. Soliton excitations in polyacetylene. Phys. Rev. B
**1980**, 22, 2099. [Google Scholar] [CrossRef] - Goldstone, J.; Wilczek, F. Fractional quantum numbers on solitons. Phys. Rev. Lett.
**1981**, 47, 986. [Google Scholar] [CrossRef] [Green Version] - Hosur, P.; Ghaemi, P.; Mong, R.S.K.; Vishwanath, A. Majorana modes at the ends of superconductor vortices in doped topological insulators. Phys. Rev. Lett.
**2011**, 107, 097001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jiang, K.; Dai, X.; Wang, Z. Quantum anomalous vortex and Majorana zero modes in iron-based superconductor Fe(Te, Se). Phys. Rev. X
**2019**, 9, 011033. [Google Scholar] [CrossRef] [Green Version] - Tanaka, Y.; Yamamori, H.; Yanagisawa, T.; Nishio, T.; Arisawa, S. Experimental formation of a fractional vortex in a superconducting bi-layer. Physica C
**2018**, 548, 44. [Google Scholar] [CrossRef] - Kong, L.; Zhu, S.; Papaj, M.; Chen, H.; Cao, L.; Isobe, H.; Xing, Y.; Liu, W.; Wang, D.; Fan, P.; et al. Half-integer level shift of vortex bound states in an iron-based superconductor. Nat. Phys.
**2019**, 15, 1181. [Google Scholar] [CrossRef] [Green Version] - Iyo, A.; Kawashima, K.; Kinjo, T.; Nishio, T.; Ishida, S.; Fujihisa, H.; Gotoh, Y.; Kihou, K.; Eisaki, H.; Yoshida, Y. New-structure-type Fe-based superconductors: CaAFe
_{4}As_{4}(A= K, Rb, Cs) and SrAFe_{4}As_{4}(A= Rb, Cs). J. Am. Chem. Soc.**2016**, 138, 3410. [Google Scholar] [CrossRef] - Liu, W.; Cao, L.; Zhu, S.; Kong, L.; Wang, G.; Papaj, M.; Zhang, P.; Liu, Y.; Chen, H.; Li, G.; et al. A new Majorana platform in an Fe-As superconductor. arXiv
**2019**, arXiv:1907.00904. [Google Scholar] - Yanagisawa, T. Fermion zero-energy modes and fractional fermion numbers in a fractional vortex-fermion model. In Proceedings of the 32nd International Symposium on Superconductivity, Kyoto, Japan, 3–5 December 2019; J. Phys. Conf. Ser.. 2020. [Google Scholar]
- Pontryagin, L.S. Ordinary Differential Equations; Pergamon: New York, NY, USA, 1962. [Google Scholar]
- Coddington, E.A.; Levinson, N. Theory of Ordinary Differential Equations; McGraw-Hill Education: New York, NY, USA, 1984. [Google Scholar]
- Yanagisawa, T.; Higashi, Y.; Hase, I. Fractional skyrmion and absence of low-lying Andreev bound states in a micro fractional-flux quantum vortex. J. Phys. Soc. Jpn.
**2019**, 88, 104704. [Google Scholar] [CrossRef] - Beenakker, C.W. Specular Andreev reflection in graphene. Phys. Rev. Lett.
**2006**, 97, 067007. [Google Scholar] [CrossRef] [Green Version] - Jackiw, R.; Pi, S.Y. Persistence of zero modes in a gauged Dirac model for bilayer graphene. Phys. Rev. B
**2008**, 78, 132104. [Google Scholar] [CrossRef] [Green Version] - Khaymovich, I.M.; Kopnin, N.B.; Mel’nikov, A.S.; Shereshevskii, I.A. Vortex core states in superconducting graphene. Phys. Rev. B
**2009**, 79, 224506. [Google Scholar] [CrossRef] [Green Version] - Niemi, A.J. Topological solitons in a hot and dense Fermi gas. Nucl. Phys. B
**1985**, 253, 14. [Google Scholar] [CrossRef] - Niemi, A.J.; Semenoff, G.W. Index theorems on open infinite manifolds. Nucl. Phys. B
**1986**, 269, 131. [Google Scholar] [CrossRef] - Yanagisawa, T.; Tanaka, Y.; Hase, I.; Yamaji, K. Vortices and chirality in multi-band superconductors. J. Phys. Soc. Jpn.
**2012**, 81, 024712. [Google Scholar] [CrossRef] [Green Version] - McClure, J.W. Diamagnetism of graphite. Phys. Rev.
**1956**, 104, 666. [Google Scholar] [CrossRef] - Slonczewski, J.C.; Weiss, P.R. Band structure of graphite. Phys. Rev.
**1958**, 109, 272. [Google Scholar] [CrossRef] - Ando, T. Theory of electronic states and transport in carbon nanotubes. J. Phys. Soc. Jpn.
**2005**, 74, 777. [Google Scholar] [CrossRef] [Green Version] - Yanagisawa, T. Kondo effect in the presence of spin-orbit coupling. J. Phys. Soc. Jpn.
**2012**, 81, 094713. [Google Scholar] [CrossRef] [Green Version] - Yanagisawa, T. Kondo effect in Dirac systems. J. Phys. Soc. Jpn.
**2015**, 84, 074705. [Google Scholar] [CrossRef] [Green Version] - Yanagisawa, T. Dirac fermions and Kondo effect. J. Phys. Conf. Ser.
**2015**, 603, 012014. [Google Scholar] [CrossRef] [Green Version]

**Table 1.**Allowed values of ℓ for positive integers Q. ℓ takes half-integers when Q is an even integer. m indicates a power of ${\chi}_{1\ell}$ for small $r\sim 0$.

Q | ℓ | $\mathit{m}\equiv (\mathit{Q}-1)/2+\mathit{\ell}$ |
---|---|---|

1 | 0 | 0 |

2 | $-1/2$, $1/2$ | 0, 1 |

3 | $-1$, 0, 1 | 0, 1, 2 |

4 | $-3/2$, $-1/2$, 1/2, 3/2 | 0, 1, 2, 3 |

5 | $-2$, $-1$, 0, 1, 2 | 0, 1, 2, 3, 4 |

**Table 2.**Allowed values of ℓ for positive odd half-integers Q. $2\ell $ takes half-integers in this case.

Q | ℓ | $\mathit{m}\equiv (\mathit{Q}-1)/2+\mathit{\ell}$ |
---|---|---|

$\frac{1}{2}$ | No solutions | No solutions |

$\frac{3}{2}$ | $-\frac{1}{4}$, $\frac{1}{4}$ | 0, $\frac{1}{2}$ |

$\frac{5}{2}$ | $-\frac{3}{4}$, $-\frac{1}{4}$, $\frac{1}{4}$, $\frac{3}{4}$ | 0, $\frac{1}{2}$, 1, $\frac{3}{2}$ |

$\frac{7}{2}$ | $-\frac{5}{4}$, $-\frac{3}{4}$, $-\frac{1}{4}$, $\frac{1}{4}$, $\frac{3}{4}$, $\frac{5}{4}$ | 0, $\frac{1}{2}$, 1, $\frac{3}{2}$, 2, $\frac{5}{2}$ |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yanagisawa, T.
Zero-Energy Modes, Fractional Fermion Numbers and The Index Theorem in a Vortex-Dirac Fermion System. *Symmetry* **2020**, *12*, 373.
https://doi.org/10.3390/sym12030373

**AMA Style**

Yanagisawa T.
Zero-Energy Modes, Fractional Fermion Numbers and The Index Theorem in a Vortex-Dirac Fermion System. *Symmetry*. 2020; 12(3):373.
https://doi.org/10.3390/sym12030373

**Chicago/Turabian Style**

Yanagisawa, Takashi.
2020. "Zero-Energy Modes, Fractional Fermion Numbers and The Index Theorem in a Vortex-Dirac Fermion System" *Symmetry* 12, no. 3: 373.
https://doi.org/10.3390/sym12030373