ALT The phase diagram of a Haldane chain with temperature on the Y-axis and magnetic field on the X-axis. A flattened, upside-down parabola covers the Bose-Einstein condensate phase — the low-temperature area between two critical magnetic fields — the border of which marks a phase transition. Another parabolic shape starts and ends at the same spots on the magnetic field scale, but goes slightly higher in temperature, indicating the Tomonaga-Luttinger liquid phase. A “V” shape originating at the two critical magnetic field points and bordering the Tomonaga-Luttinger liquid phase shows quantum-critical regions. At high temperatures, the system is paramagnetic.