First principles Quantum Fields Theory for cold and dense matter

First principles Quantum Fields Theory for cold and dense matter is one of the grand challenges since more then 20 years. The reason is that the computer simulations are hampered by the notorious "sign problem". The recent years have seen some remarkable successes to tackle this issue. These include the Complex Langevin Simulations, the Taylor expansion method, dualisation techniques and one approach developed by the supervisor of the project - the density-of-states method (Phys.Rev.Lett. 109 (2012) 111601). The thesis will critical review the current state of affairs and will then focus on the advance of the density-of-states method. The foundations have been thoroughly investigated (see Eur.Phys.J. C76 (2016) no.6, 306). The PhD project will start with a complementing study of the performance of the method using Wilson loop expectation values in SU(2) Yang-Mills theory, which are known for a very poor signal-to-noise ratio. Also, the potential of the Polykov loop gauge potential will be calculated. The latter result will feed into an ongoing collaboration with scientists from the University of Heidelberg. Main focus will then be a continuation of the study of the cold and dense matter in QCD, the theory of strong interactions. The project will also address the simulation of Graphene. This research here will contribute to a successfully running project with colleagues at the University of Giessen, Germany

Markov chain Monte-Carlo (MCMC) simulations face two limitations: due to ergodicity problems (making sure the distribution converges to the invariant distribution regardless of our choice of starting point) and in cases of a non-positive Gibbs factor, that is the inverse temperature. Instead of MCMC simulations, we can consider a new class of non-Markovian Random Walk simulation, which do not rely on an update of the configurations according to Importance Sampling with respect to a positive Gibbs factor. One particular set-up that is especially relevant for QFTs with our limitations uses the inverse density-of-states as a measure for updating configuration. This measure is semi-positive definite by definition and aims to generate a random walk-in configuration space even in 'deprived' regions with low probabilistic measure. This approach should reduce the negative affect of the 'sign problem' (numerical methods failing due to the cancellation of the positive and negative contributions to the integral).
The key to density of states is to estimate the derivative of the logarithm of the density at any given magnetisation value. Thanks to the universities condor system, I can acquire this derivative at multiple magnetisation values, which I can then numerically integrate. This has been done for a 100x100 lattice, and I am working on altering the code to work with different lattice sizes, with a personal aspiration to investigate how this would work with a 3-dimensional lattice (e.g. 100*100*100)