Constraints on the merging of the transition lines at the tricritical point in a wing-structure phase diagram

We consider the phase diagram of a ferromagnetic system driven to a quantum phase transition with a tuning parameter $p$. Before being suppressed, the transition becomes of the first order at a tricritical point, from which wings emerge under application of the magnetic field $H$ in the $T-p-H$ phase diagram. We show that the edge of the wings merge with tangent slopes at the tricritical point.

Figure 1

Schematic temperature-pressure-magnetic field ($T-p-H$) phase diagram. $H$ is applied along the easy axis of magnetization. The regions of the ferromagnetic (FM) and paramagnetic (PM) phases are shown. The diagram also indicates the tricritical point (TCP), the quantum phase transition lines (QPT), as well as two quantum wing critical points (QWCPs). Solid lines are second-order transitions. Dashed lines and blue planes are first-order transitions.

Figure 2

Schematic projections of the temperature-pressure-magnetic field ($T-p-H$) phase diagram in the $T-p$ ($H=0$), $H-p$ ($T=0$), and $T-H$ ($p=0$) planes.