Tricritical Behavior in Itinerant Quantum Ferromagnets

It is shown that the peculiar features observed in the low-temperature phase diagrams of ${\mathrm{ZrZn}}_2$, ${\mathrm{UGe}}_2$, and MnSi can be understood in terms of a simple mean-field theory. The nature of the ferromagnetic transition changes from second order to first order at a tricritical point, and in a small external magnetic field surfaces of first-order transitions emerge which terminate in quantum critical points. This field dependence of the phase diagram follows directly from the existence of the tricritical point. The quantum critical behavior in a nonzero field is calculated exactly.

Figure 1

Schematic phase diagram in the temperature-pressure-magnetic-field ($T-p-h$) space. Shown are the ferromagnetic (FM, dark shaded) and paramagnetic (PM) phases at $h=0$, the tricritical point (TCP), and the two quantum critical points (QCP). Also shown are various lines of first-order (dashed lines) and second-order (solid lines) phase transitions, and the “wing” surfaces of first-order transitions (light shaded).

Figure 2

The free energy at $T=0$ for $h=0$, $t=t_1$ (solid curve) and $h=0.02$, $t=0.1878$ (dashed curve), respectively. In both cases, $u=v=m_0=1$ and $T=0$.

Figure 3

The surface of first-order transitions in the space spanned by $T$, $t$, and $h$. It is bounded by lines of first-order transitions in the $T=0$ and $h=0$ planes, respectively, and by the line of second-order transitions discussed after Eqs. (4). A symmetric surface extends into the region where $h<0$.