**Principal Investigator:**

Dénes Sexty

**Affiliation:**

Heisenberg Fellow, Bergische Universität Wuppertal (Germany)

**Local Project ID:**

pr84to

**HPC Platform used:**

SuperMUC of LRZ

**Date published:**

**The simulation of nuclear matter at nonzero baryon density presents a notoriously hard problem in lattice QCD. The usual simulation strategies depend on the exploration of the configuration space by interpreting the weight of each configuration in an average as a probability, which is however not valid here as the weight is non positive. This is called the ‘sign problem’ of nonzero density QCD. In this project the researchers compare two different simulation strategies which evade the sign problem with very different methods.**

The different phases of water (liquid water, vapour and ice) are well known to everybody, and almost everyone knows even the temperature at which the water changes its phase i.e. it gets frozen or boils to vapour. Systematic studies yield the phase diagram of the water as a function of the temperature and the pressure, well known to both experimental and theoretical physics.

One of the current outstanding problems of theoretical physics is to produce such a phase diagram for QCD as a function of the temperature and the baryonic density. Currently very little is known about the phase diagram of QCD, but effective models give some guidelines as illustrated above. The simulation of nuclear matter at small densities is a well established part of theoretical physics since decades. There are very sophisticated algorithms based on the so called ‘importance sampling’ which uses the fact that the system can be described with an ensemble of configurations which describe an instance of gluon and quark fields. Each configuration has a weight which is interpreted as a probability depending on the temperature, quark masses and so on. This probabilistic description however breaks down as soon as there is a nonzero baryonic density in the system, because the weight of each configuration can now become negative. This is the infamous ‘sign problem’, a long standing challenge in the field of lattice QCD as well as other fields such as condensed matter physics, non-equilibrium physics, etc.

In this project the researchers test two methods designed to evade the sign problem. The first is called ‘reweighting’ as it changes the weights of the configurations such that they are positive again, and in turn mimics the nonzero baryon density by changing the physical quantities to be measured in an appropriate fashion. This allows to explore into the region of non zero densities, however the method becomes unreliable if we go too far from the original zero density phase, moreover its cost increases very rapidly as the system size is increased.

The second method is based on the so called Langevin equation which describes a random walk in configuration space, and can be used as a simulation method also at zero densities. This method however does not require the interpretation of weights as probabilities, and therefore is generalizable to non-positive weights as well. This generalization is based on the structure of the complex numbers, hence it has the name ‘Complex Langevin equation (CLE)’ . It was invented more than 30 years ago [1], however its status was unclear as sometimes this method gives unreliable results. In recent years however important results have clarified what conditions must be satisfied and what technical improvements are needed to make the method and its results trustworthy [2].

The researchers have determined that there is a good agreement between reweighting and Complex Langevin results where both are applicable. The range of applicability is however quite different. Reweighting performs well at small baryon densities while Complex Langevin works at small lattice spacings (high temperatures). Their reliabilities can be assessed independently of the other method.

See for illustration the figures below where the ‘Polyakov loop’ (a quantity connected to the energy of free quark) is plotted as a function of the chemical potential and the ‘beta’ parameter controlling the lattice spacing. These findings led to further studies which are carried out to simulate the physics at high temperature and to determine whether the applicability of the Complex Langevin method can be extended to lower temperatures.

**References:**

[1] G. Parisi, Phys. Lett. 131B (1983) 393; J. R. Klauder, Acta Phys. Austriaca Suppl. 25 (1983) 251.

[2] D.Sexty PoS LATTICE2014 (2014) 016; G. Aarts, E. Seiler, D. Sexty, I.-O. Stamatescu JHEP 1705 (2017) 044

**Scientific Contact:**

Dr. Dénes Sexty

Institut für Theoretische Teilchenphysik

Fakultät für Mathematik und Naturwissenschaften

Bergische Universität Wuppertal, D-42097 Wuppertal (Germany)

e-mail: sexty [at] uni-wuppertal.de