Phases of cold and dense quark matter

Depending on conditions like temperature or pressure, a given substance can appear in different phases, such as solid, liquid or gaseous. This project is devoted to the investigation of phases of matter under extreme conditions, addressing two main questions. What happens to matter when compressed to extremely high density? The answer to this question lies in the phase diagram of the theory of the strong nuclear force, Quantum Chromo- Dynamics (QCD), yet in a region accessible neither to current experiments, nor to theoretical computations. The goal of this project is to obtain, for the first time, model-independent constraints on the phase diagram using a combination of indirect input from numerical simulations and general rigorous arguments. I will study properties of dense matter in theories similar to QCD which are amenable to computer simulations due to their "positivity". With the help of already existing numerical data, I will extract information about how the dense medium affects the nuclear force, in particular its ability to "confine" the fundamental constituents of matter, the quarks. Moreover, I will make model predictions that will facilitate the comparison to future simulations, thus preparing ground for further improvement of the expected results. Why are some systems in equilibrium homogeneous, while others develop order? This second question is more general and although seemingly unrelated, it is in fact deeply connected to the first one. According to a little-known conjecture made a decade ago, positivity of a theory implies that its equilibrium state must be homogeneous. No counterexample has been found so far, but a definitive proof is still missing. I will investigate this intriguing link between positivity and homogeneity and strive to either prove or disprove the hypothesis. In any case, the work will provide further rigorous constraints on the structure of matter at high density. In a complementary study, I will develop a model-independent, effective description of systems with spontaneous order, valid at low energy and temperature. While this research is directly motivated by the desire to understand the behavior of dense nuclear matter, the results will be of much wider relevance, with applications for instance to condensed-matter or atomic physics. To summarize, in this project I put forward a novel approach to matter under extreme conditions, based on exploiting available numerical data for QCD-like theories, and on the systematic use of symmetry. Whereas it does not pretend to answer all questions at once, it does intend to show an alternative way to tackle some longstanding problems.

What happens to matter when compressed to extremely high density? The present project was concerned with the development of mathematical tools that might push us a bit closer to the ultimate answer to this question. I followed two different paths, independent in aims and scope but related by the motivation.At extremely high densities such as in the cores of neutron stars, all fundamental interactions of nature but one become negligible: the strong nuclear interaction. We have had a successful theory of the strong interaction, called quantum chromodynamics (QCD), for four decades. Large-scale numerical simulation of QCD is by now an established theoretical tool, yet it is notoriously difficult to apply to the conditions prevalent in neutron stars. On the other hand, a number of simplified model approaches exists that can be applied to such situations, yet we do not know the input parameters of these models precisely due to the lack of experimental data. In the first part of the project, I tried to bridge the gap between the direct numerical simulation of QCD and the simplified models. I used the fact that there are mathematical theories similar to QCD for which numerical simulation at high matter densities is feasible. Exploiting the data from these simulations, I was able to constrain the parameters of the phenomenological models in a way that will, in the future, allow more precise calculations of the structure of neutron stars.When perturbed slightly, matter tends to respond by collective oscillations of its constituents: waves. In the second part of the project, I studied general properties of such matter waves. Many of their properties can be understood based on symmetry using the so-called effective field theory framework. This has the advantage of being independent of the microscopic structure of matter so that, for instance, superfluid Helium and certain superfluid phases in dense nuclear matter can be described by the same mathematical theory, although they appear under vastly different physical conditions. The response of matter to stress gives us information about its elastic properties and the propagation of sound waves. Together with my collaborators, we used the fact that the effect of external stress is mathematically equivalent to placing the material in a curved space. We therefore worked out a general formalism for a description of quantum matter in curved spacetime. We now have a complete classification of matter waves based on symmetry as well as a model-independent mathematical framework to describe them. In the future, I plan to use this to complement the above-mentioned model study of extremely dense matter in neutron stars.