# Tracking vortices in superconductors: Extracting singularities from a discretized complex scalar field evolving in time

## Abstract

In type-II superconductors, the dynamics of magnetic flux vortices determine their transport properties. In the Ginzburg-Landau theory, vortices correspond to topological defects in the complex order parameter field. Earlier, in Phillips et al. [Phys. Rev. E 91, 023311 (2015)], we introduced a method for extracting vortices from the discretized complex order parameter field generated by a large-scale simulation of vortex matter. With this method, at a fixed time step, each vortex [simplistically, a one-dimensional (1D) curve in 3D space] can be represented as a connected graph extracted from the discretized field. Here we extend this method as a function of time as well. A vortex now corresponds to a 2D space-time sheet embedded in 4D space time that can be represented as a connected graph extracted from the discretized field over both space and time. Vortices that interact by merging or splitting correspond to disappearance and appearance of holes in the connected graph in the time direction. This method of tracking vortices, which makes no assumptions about the scale or behavior of the vortices, can track the vortices with a resolution as good as the discretization of the temporally evolving complex scalar field. Additionally, even details of the trajectory betweenmore »

- Authors:

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science - Office of Basic Energy Sciences - Materials Sciences and Engineering Division; USDOE Office of Science - Office of Advanced Scientific Computing Research

- OSTI Identifier:
- 1352644

- DOE Contract Number:
- AC02-06CH11357

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review E

- Additional Journal Information:
- Journal Volume: 93; Journal Issue: 2; Journal ID: ISSN 2470-0045

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Phillips, Carolyn L., Guo, Hanqi, Peterka, Tom, Karpeyev, Dmitry, and Glatz, Andreas.
```*Tracking vortices in superconductors: Extracting singularities from a discretized complex scalar field evolving in time*. United States: N. p., 2016.
Web. doi:10.1103/PhysRevE.93.023305.

```
Phillips, Carolyn L., Guo, Hanqi, Peterka, Tom, Karpeyev, Dmitry, & Glatz, Andreas.
```*Tracking vortices in superconductors: Extracting singularities from a discretized complex scalar field evolving in time*. United States. doi:10.1103/PhysRevE.93.023305.

```
Phillips, Carolyn L., Guo, Hanqi, Peterka, Tom, Karpeyev, Dmitry, and Glatz, Andreas. Mon .
"Tracking vortices in superconductors: Extracting singularities from a discretized complex scalar field evolving in time". United States. doi:10.1103/PhysRevE.93.023305.
```

```
@article{osti_1352644,
```

title = {Tracking vortices in superconductors: Extracting singularities from a discretized complex scalar field evolving in time},

author = {Phillips, Carolyn L. and Guo, Hanqi and Peterka, Tom and Karpeyev, Dmitry and Glatz, Andreas},

abstractNote = {In type-II superconductors, the dynamics of magnetic flux vortices determine their transport properties. In the Ginzburg-Landau theory, vortices correspond to topological defects in the complex order parameter field. Earlier, in Phillips et al. [Phys. Rev. E 91, 023311 (2015)], we introduced a method for extracting vortices from the discretized complex order parameter field generated by a large-scale simulation of vortex matter. With this method, at a fixed time step, each vortex [simplistically, a one-dimensional (1D) curve in 3D space] can be represented as a connected graph extracted from the discretized field. Here we extend this method as a function of time as well. A vortex now corresponds to a 2D space-time sheet embedded in 4D space time that can be represented as a connected graph extracted from the discretized field over both space and time. Vortices that interact by merging or splitting correspond to disappearance and appearance of holes in the connected graph in the time direction. This method of tracking vortices, which makes no assumptions about the scale or behavior of the vortices, can track the vortices with a resolution as good as the discretization of the temporally evolving complex scalar field. Additionally, even details of the trajectory between time steps can be reconstructed from the connected graph. With this form of vortex tracking, the details of vortex dynamics in a model of a superconducting materials can be understood in greater detail than previously possible.},

doi = {10.1103/PhysRevE.93.023305},

journal = {Physical Review E},

issn = {2470-0045},

number = 2,

volume = 93,

place = {United States},

year = {2016},

month = {2}

}