Developing techniques for finite density lattice field theory

Quantum Chromodynamics (QCD) is the fundamental field theory of the strong interaction, which is responsible for binding together quarks and gluons, i.e., the strongly interacting particles, into protons and neutrons which make up the atomic nuclei. Although the fundamental equations of QCD are well known, it is a major challenge to calculate physical predictions, since very strong non-linear effects have to be taken into account. A very powerful quantitative method for treating QCD is the formulation of the theory on a 4-dimensional space-time lattice. In this form, the so-called lattice-QCD formulation, the system can be analyzed with so-called Monte Carlo simulations, where one numerically generates configurations of the quark- and gluon fields and computes physical observables from them. Such lattice QCD calculations are the general theme of the SFB/TRR- 55, centered in Regensburg and Wuppertal and Graz as a project partner. Project A15, where Graz is the responsible research group, more specifically focusses on lattice field theory at high densities. Studying QCD at high density is an important topic, since such dense matter states plays a key role in understanding Big-Bang physics and astrophysical objects such as neutron stars. However, when formulating finite density field theories on the lattice a major technical obstacle emerges, the so-called complex action problem: At finite density the Monte Carlo calculation cannot be set up in a straightforward way, since the weight factors used in the numerical treatment have a complex phase. New approaches are necessary for overcoming the complex action problem. In the proposed project we will be dealing with the further development of two such approaches for overcoming the complex action problem: 1) Dual variables, and the 2) density of state approach. 1) In the dual variables approach one searches for an alternative way of writing the mathematical expressions for the observables, such that all complex phases are removed and the Monte Carlo method is again applicable. The Graz group was able to find such dual representations for several lattice field theories, where instead of quantum fields, loops of flux connected by surfaces are the dual degrees of freedom used for formulating the theory. The goal is to develop further these techniques towards a dual formulation free of complex phases for full lattice QCD. 2) In the density of states approach one uses Monte Carlo techniques to count the number of states for a given net particle number. The result is the density of states and physical observables can be computed by integrating this density with the complex phase of the theory. The key challenge is a very accurate determination of the density since the integral with the complex phase is very sensitive to the details of the density. Here we plan to further improve techniques proposed in Graz and to develop our approach into a technique applicable in full QCD simulations.

The project "Developing techniques for finite density lattice field theory" explored new methods for studying quantum systems at finite density. In the lattice approach space-time is replaced by a discrete lattice structure and the quantum mechanical path integral assumes a form where numerical simulations based on stochastic methods are applicable. However, if finite density is considered, the necessary probabilistic interpretation fails and the numerical methods cannot be applied. In the project two different strategies for overcoming this so-called "complex action problem" were developed: In the first approach, so-called "worldline/worldsheet techniques", the system is exactly mapped to new variables, which are worldlines for matter fields and worldsheets for the gauge degrees of freedom. In this form all contributions to physical observables have a probability interpretation such that numerical simulations are again applicable. In the second approach, modern numerical techniques are used to compute an auxiliary quantity, the so-called "density of states", that enumerates quantum mechanical states with very high accuracy. The density of states may then be converted into the physical observables one wants to study at finite density. The new approaches to finite density lattice field theory have a wide portfolio of applications, ranging from elementary particle physics all the way to effective theories in condensed matter theory. In various applications the new methods were used to address open questions in these types of systems, and further publications with physical results for effective field theories of spin systems are in preparation. In a line of research that extends the original project plan, the worldline/worldsheet methods were generalized to theories with topologiocal terms, a very hot topic in modern quantum field theory. Based on these developments, as well as the new density of states techniques, we are currently preparing a follow-up proposal to further develop the successful new techniques.