# Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid

## Abstract

Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and classify level crossings, aka. singularities in the spectrum of a family of Hamiltonians. The approach we use is singularity theory. Using this language, we obtain a characterization of Dirac points and also show that the branching behavior of the level crossings is given by an unfolding of A{sub n} type singularities. Which type of singularity occurs can be read off a characteristic region inside the miniversal unfolding of an A{sub k} singularity. We then apply these methods in the setting of families of graph Hamiltonians, such as those for wire networks. In the particular case of the Double Gyroid we analytically classify its singularities and show that it has Dirac points. This indicates that nanowire systems of this type should have very special physical properties. - Highlights: Black-Right-Pointing-Pointer New method for analytically finding Dirac points. Black-Right-Pointing-Pointer Novel relation of level crossings to singularity theory. Black-Right-Pointing-Pointer More precise version of the von-Neumann-Wigner theorem for arbitrary smooth families of Hamiltonians of fixed size. Black-Right-Pointing-Pointer Analytical proof of the existence of Dirac points for the Gyroid wire network.

- Authors:

- Department of Mathematics, Purdue University, West Lafayette, IN 47907 (United States)
- Department of Physics, Purdue University, West Lafayette, IN 47907 (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 22157013

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York)

- Additional Journal Information:
- Journal Volume: 327; Journal Issue: 11; Other Information: Copyright (c) 2012 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 77 NANOSCIENCE AND NANOTECHNOLOGY; BRILLOUIN ZONES; DIAGRAMS; DISPERSION RELATIONS; ELECTRONS; GRAPH THEORY; HAMILTONIANS; HOLES; QUANTUM WIRES; SINGULARITY

### Citation Formats

```
Kaufmann, Ralph M., E-mail: rkaufman@math.purdue.edu, Khlebnikov, Sergei, E-mail: skhleb@physics.purdue.edu, Wehefritz-Kaufmann, Birgit, E-mail: ebkaufma@math.purdue.edu, and Department of Physics, Purdue University, West Lafayette, IN 47907.
```*Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid*. United States: N. p., 2012.
Web. doi:10.1016/J.AOP.2012.08.001.

```
Kaufmann, Ralph M., E-mail: rkaufman@math.purdue.edu, Khlebnikov, Sergei, E-mail: skhleb@physics.purdue.edu, Wehefritz-Kaufmann, Birgit, E-mail: ebkaufma@math.purdue.edu, & Department of Physics, Purdue University, West Lafayette, IN 47907.
```*Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid*. United States. doi:10.1016/J.AOP.2012.08.001.

```
Kaufmann, Ralph M., E-mail: rkaufman@math.purdue.edu, Khlebnikov, Sergei, E-mail: skhleb@physics.purdue.edu, Wehefritz-Kaufmann, Birgit, E-mail: ebkaufma@math.purdue.edu, and Department of Physics, Purdue University, West Lafayette, IN 47907. Thu .
"Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid". United States. doi:10.1016/J.AOP.2012.08.001.
```

```
@article{osti_22157013,
```

title = {Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid},

author = {Kaufmann, Ralph M., E-mail: rkaufman@math.purdue.edu and Khlebnikov, Sergei, E-mail: skhleb@physics.purdue.edu and Wehefritz-Kaufmann, Birgit, E-mail: ebkaufma@math.purdue.edu and Department of Physics, Purdue University, West Lafayette, IN 47907},

abstractNote = {Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and classify level crossings, aka. singularities in the spectrum of a family of Hamiltonians. The approach we use is singularity theory. Using this language, we obtain a characterization of Dirac points and also show that the branching behavior of the level crossings is given by an unfolding of A{sub n} type singularities. Which type of singularity occurs can be read off a characteristic region inside the miniversal unfolding of an A{sub k} singularity. We then apply these methods in the setting of families of graph Hamiltonians, such as those for wire networks. In the particular case of the Double Gyroid we analytically classify its singularities and show that it has Dirac points. This indicates that nanowire systems of this type should have very special physical properties. - Highlights: Black-Right-Pointing-Pointer New method for analytically finding Dirac points. Black-Right-Pointing-Pointer Novel relation of level crossings to singularity theory. Black-Right-Pointing-Pointer More precise version of the von-Neumann-Wigner theorem for arbitrary smooth families of Hamiltonians of fixed size. Black-Right-Pointing-Pointer Analytical proof of the existence of Dirac points for the Gyroid wire network.},

doi = {10.1016/J.AOP.2012.08.001},

journal = {Annals of Physics (New York)},

issn = {0003-4916},

number = 11,

volume = 327,

place = {United States},

year = {2012},

month = {11}

}