Consider QCD with a single generation of massless quarks (u, d). This is probably the simplest variant of QCD which bears some relation to the real world. The theory has the following exact global symmetries:

- U(1) acting on u and d (baryon number)
- Chiral U(1) acting in opposite matter on left-hand and right-handed quarks. It is destroyed by an anomaly so we will not consider it further
- SU(2) isospin rotation mixing u and d
- Chiral SU(2). It is spontaneously broken
- Charge conjugation C
- Parity (spatial reflection) P
- Time reversal T

A thermal equilibrium of the model is characterized by 3 parameters:

- Temperature
- Chemical potential associated to baryon number. Alternatively, baryon number density. We can use C to fix its sign
- Chemical potential associated to isospin. Alternatively, isospin density. It is a vector but using isospin rotation symmetry we can align it along a prescribed axis, so we are left with a positive scalar parameter

Hence, the theory has a 3-dimensional phase diagram

How does the 3D phase diagram look like? Which phases do we have? Which phase transitions? What is the type of each phase transition?

Above a certain temperature T, chiral symmetry is restored. Btw, is this the same phase transition which breaks confinement? Above this T, we can introduce a 4th parameter, namely the chemical potential associated to chiral isospin. I suppose there a certain inequality governing the maximal value of this parameter as a function of the temperature?

How does the 4D phase diagram look like?

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