Roots of unity as the system cools

The Bost–Connes system has energy levels log 1, log 2, log 3 and so on, and β is one over the temperature. The faint points on the circle are the roots of unity up to the order the slider allows, coloured by phase; each has a bright point of the same colour, the value the equilibrium state takes there, joined to its root by a thin line. For β up to 1, when the system is hot, there is only one equilibrium state; its values at the roots of unity are real and depend only on a root's order, so the bright points lie on the real axis, and at exactly β = 1 all but the one at 1 sit at the centre. Take β past 1 and there is more than one equilibrium state; the picture follows one of them. The bright points set out from the centre and move outwards, trailing the path they took from β = 1, and the larger β, the colder the system and the closer each point comes to its own root; the mean radius in the corner climbs towards 1. Turn on “Swap every root for its conjugate” and the picture shows another equilibrium state, whose value at each root is the old one's value at the conjugate root. While the system is hot the bright points are on the real axis and nothing changes. Once it cools, every bright point flips across the real axis and the thin lines reach across it: the picture keeps its shape, but its colours swap top for bottom, and the imaginary part of φ_β(e(1/5)) in the corner changes sign. Conjugation is a symmetry of the system; at low temperature it turns the state being followed into a different one, and that state is not symmetric itself: this is how the symmetry is broken.