Update from Nicosia

After barely surviving last week’s heatwave, I’m quite pleased to report things have cooled back down in Nicosia to an almost chilly 37 degrees. In the weeks since my last post, I’ve had the good fortune to discover Nicosia’s wide selection of 24 hour bakeries. I would especially recommend Zorbas for anyone who finds themselves in Nicosia.

My project has been going well. A number of weeks ago, I successfully implemented the MPI read and write functionality for the eigenvectors and eigenvalues. This was an improvement over the previous implementation by a factor of 3. These timings were performed initially on Piz Daint in Switzerland and then on JUQUEEN in Germany, the 6^th and 9^th fastest supercomputers in the world respectively. The 1000 smallest eigenvectors of the Dirac operator of a 48*48*48*96 euclidean lattice were saved on each run, amounting to approximately 1 Terabyte of data per run. The downside to this method is that it outputs a raw binary file with no meta-data or checksumming. In this way, the file could be corrupted without anyone becoming aware.

The solution that was proposed was to use HDF5 to implement a similar method. This was completed last week. HDF5 allows us to store meta-data about the size of the lattice and other properties as well as splitting up the eigenvectors and eigenvalues into different groupings. This facilitates the reading of a subset of the eigenvectors quite easily. In theory, for objects as large as the eigenvectors, the p show nerformance should be close to that of the raw MPI method. The timings for this are almost complete. The write function is slightly slower than the MPI version and a plot of write timings is shown below.

I am currently beginning the next part of the project. This involves the measurement of the topological susceptibility using the stored eigenvalues. The topological susceptibility can be seen as related to the variance of the topological charge of the configuration, which is a generalized version of a winding number. Two systems are said to be topologically equivalent if it is possible to topologically deform one of them continuously into the other without passing through forbidden field configurations which are non-physical.

Above is an example of topologically in-equivalent systems. From the point of view of the green points, it is impossible to smoothly transform one system into the other. Assuming theses curves travel anti-clockwise these systems can be said to have a winding number of 2 and 1 from left to right. Topological charge can be seen as something similar, an integer number, in which systems with the same topological charge are topological equivalent and those with distinct ones are topologically in-equivalent.