**Phase Transitions in Neutron Stars Fiorella Burgio**

The QCD phase transition in extremely dense nuclear matter plays an important role in the compact star phenomenology. The aspect of finite temperatures is essential for EOS models applicable to supernova simulations and neutron star mergers. Another aspect corresponds to the excitation of hyperons in neutron star interiors which has led to the “hyperon puzzle” that finds a solution in the sufficiently early phase transition to deconfined quark matter. A key question is whether the deconfinement transition occurs in supernova explosions and neutron star interiors or not.

**JWST, Primordial Black Holes and Cosmological Phase Transitions Günther Hasinger**

The QCD phase transition is important in a cosmological context where it has recently been found to play a crucial role in explaining the formation of primordial black holes and their mass spectrum. It also suggests a natural interpretation of the recent astonishing results of the James-Webb Space Telescope (JWST) about the very early formation of galaxies and quasars, i.e. of supermassive primordial black holes as their seeds and as dark matter candidates.

**Critical Point and Dynamics of a Phase Transition in Heavy Ion Collisions Joseph Kapusta**

These lectures introduce the solid basis of the methods of finite temperature quantum field theory and guides towards the active research works that are currently performed on phase transition constructions that are being developed in the community. One of them uses a “switch function” between phases and allows a flexible modeling of the character of the phase transition in the QCD phase diagram, including a critical point. The corresponding EOS is the key ingredient for hydrodynamical simulations of the heavy-ion collisions and the diagnostics of critical parameters of the QCD phase transition. It is fascinating that the theory of nucleation can not only be applied to describe the growth of droplets of the new phase in the dynamical situation of a heavy-ion collision, but also to the nucleation of black holes, a process which potentially plays a key role in explaining the formation of galactic structures in the early Universe.

**Quark deconfinement in supernova explosions: How to probe it ? Takami Kuroda **

While for isolated neutron stars this question is under debate, it has been shown in that quark deconfinement can be an engine for driving a supernova explosion of very massive blue supergiant stars with 50 solar masses which would not explode within the standard neutrino-driven explosionmechanism without a phase transition. The lecture introduces a novel generalization of the one-dimensional model calculation to the multidimensional case and reports the first 2D fully general-relativistic radiation hydrodynamics simulations showing that the QCD phase transition induced explosion does take place albeit depending upon the progenitors' mass. Also the detectability of neutrino and gravitational-wavesignals is outlined.

**Warm Dense Plasmas from Inertial Fusion to Planetary Interiors Burkhard Militzer**

In planetary science, computer simulations aid the interpretation of spacecraft measurements of gravitational and magnetic fields. When laboratory measurements probe yet more extreme states of warm dense matter, computer simulations are employed to plan experiments, interpret them, and to derive properties that cannot yet be measured. In the last ten years, novel computer simulation techniques have been developed that are, e. g., based on Feynman's path integral formalism but these methods are not described in books nor in review articles and are thus not accessible to a broad audience. Furthermore, machine learning algorithms are currently designed to first learn the forces between the nuclei and then accelerate their simulations. The details of such methods are not available anywhere. The lectures will fill this gap and provide a forum for such advanced computer simulation techniques to be explained, shared, and scrutinized.

**Topological Phases of Matter in and out of Equilibrium Roderich Moessner**

Topological Phases of Matter are an exceptionally dynamic field of research: several of the most exciting recent experimental discoveries and conceptual advances in modern physics have originated in this field. These have generated new, topological, notions of order, interactions and excitations. Among the systems are topological insulators, magnets, semimetals, and superconductors. The lectures on topological phases of matter in and out of equilibrium will provide an accessible, unified and comprehensive introduction to the phenomena surrounding topological matter, with detailed expositions of the underlying theoretical tools and conceptual framework, alongside accounts of the central experimental breakthroughs. Topics to be covered concern: signatures of topological phases; transitions between topological phases; detecting microscopic physics differing between topologically identical systems; topological electrostatics.

**The complexity of neutron star matter: from the liquid gas phase transition to chiral symmetry breaking and restoration Constança Providência **

One key aspect of phases and phase transitions concerns symmetries and their dynamical breaking due to interactions and emergence of collective phenomena. Under extreme conditions of high temperatures and densities broken symmetries can be restored. A similar effect of breaking and restoration of symmetries is well-known from spin systems and the response of their magnetization to strong external fields. In particle physics, it is crucial to understand the mechanisms of fermion mass generation by dynamical chiral symmetry breaking. A paradigmatic model was developed by Nambu in analogy to the Bardeen-Cooper-Schrieffer model for superconductivity and earned him the Nobel prize. Dynamical chiral symmetry restoration is a key aspect of the phase transition in quantum chromodynamics (QCD) from a hadron resonance gas (confined quarks with large constituent masses) to the quark-gluon plasma (deconfined, almost massless quarks). At subnuclear densities, nuclear matter transitions from a nuclear liquid to a gas of nuclides (clusters) via a mixed phase with structures called “pasta” phases that play a role in neutron stars at the crust-core transition. These two aspects of phase transitions in strongly interacting matter play a key role for the investigation of applications in heavy-ion collisions and compact star astrophysics as well as cosmology. The specifics of the hadron-to-quark matter transition under neutron star constraints corresponds to charge neutrality and beta equilibrium which led to the creation of the CompOSE repository of equations of state for application in astrophysical simulations.

**Correlations, cluster formation and phase transitions in dense fermion systems ****Gerd Röpke**

Electron-ion plasmas, nuclear matter and quark-gluon plasmas are examples of interacting fermion systems in which the formation of bound states (atoms, nuclei, hadrons) is possible. The thermodynamic and transport properties are strongly influenced by correlations that determine the composition of the system. Quantum statistical methods (Green functions and Feynman diagrams) and numerical simulations (density functional theory, molecular dynamics, path integral Monte Carlo) are used to calculate these properties.Various approximations (resonance gas, Beth-Uhlenbeck formula and virial expansions, in-medium wave equation) are discussed. Interesting effects are self-energy, shielding and Pauli blocking, which lead to the dissolution of bound states (so-called Mott effect). Of particular interest are quantum phase transitions (Bose-Einstein condensation and Cooper pairing, quartetting) as well as liquid-gas-like phase transitions. Various examples are presented: the thermodynamics of the uniform electron gas, the electrical conductivity of hydrogen plasmas, the composition of nuclear matter, heavy ion collisions, the fission of actinides, the hadron-quark transition.

**The Universality of Critical Behaviour Helmut Satz**

Despite the diversity of thermodynamical conditions for the phase transitions and their character, it turned out that their critical behavior has universal properties. In recent years, critical behavior was found to be among the most ubiquitous phenomena in nature. From color confinement/ deconfinement in statistical QCD to the hadronisation transition in early universe cosmology, critical behavior takes place on a variety of scales in the universe. In addition, the study of self-organized criticality has come into play, dealing with such diverse phenomena as avalanches, earth quakes, word frequencies, and swarm structure. All occur essentially at a critical point, although the reason for such behavior is not yet understood.