**9. Final remarks**

22 Will-be-set-by-IN-TECH

with *PB* and *PC*). It is possible to achieve a lower *Pm* value (while leaving *Tm* unchanged), but this has the undesirable effect of modifying *Pc*<sup>0</sup> to a value that disagrees with experiment. In SFT (*n* = 2) the multi-critical points are located at slightly higher *P* (by about 0.01 GPa), as for ZrZn2. Therefore, the results from the SFT theory are slightly worse than the results produced

The estimates for UGe2 imply *γ*1*κ* ≈ 1.9, so the condition for *TFS*(*P*) to have a maximum found from Eq. (17) is satisfied. As we discussed for ZrZn2, the location of this maximum can be hard to fix accurately in experiments. However, *Pc*<sup>0</sup> can be more easily distinguished, as in the UGe2 case. Then we have a well-established quantum (zero-temperature) phase transition of second order, i.e., a quantum critical point at some critical pressure *P*0*<sup>c</sup>* ≥ 0. As shown in Sec. 6, under special conditions the quantum critical points could be two: at the lower

in systems with *Ts* = 0 (as UGe2) occurs when the criterion (23) is satisfied. Such systems (which we label as U-type) are essentially different from those such as ZrZn2 where *γ*<sup>1</sup> < *γ* and hence *TFS*(0) > 0. In this latter case (Zr-type compounds) a maximum *Tm* > 0 may sometimes occur, as discussed earlier. We note that the ratio *γ*/*γ*<sup>1</sup> reflects a balance effect between the two *ψ*-*M* interactions. When the trigger interaction (typified by *γ*) prevails, the Zr-type behavior is found where superconductivity exists at *P* = 0. The same ratio can be expressed as *γ*0/*δ*0*M*0, which emphasizes that the ground state value of the magnetization at *P* = 0 is also relevant. Alternatively, one may refer to these two basic types of spin-triplet ferromagnetic superconductors as "type I" (for example, for the "Zr-type compounds), and

As we see from this classification, the two types of spin-triplet ferromagnetic superconductors have quite different phase diagram topologies although some fragments have common features. The same classification can include systems with *Ts* �= 0 but in this case one should

In URhGe, *Tf*(0) ∼ 9.5 K and *TFS*(0) = 0.25 K and, therefore, as in ZrZn2, here the spin-triplet superconductivity appears at ambient pressure deeply in the ferromagnetic phase domain [6–8]. Although some similar structural and magnetic features are found in UGe2 the results in Ref. [8] of measurements under high pressure show that, unlike the behavior of ZrZn2 and UGe2, the ferromagnetic phase transition temperature *TF*(*P*) ∼ *Tf*(*P*) has a slow linear increase up to 140 kbar without any experimental indications that the N-FM transition line may change its behavior at higher pressures and show a negative slope in direction of low temperature up to a quantum critical point *TF* = 0 at some critical pressure *Pc*. Such a behavior of the generic ferromagnetic phase transition temperature cannot be explained by our initial assumption for the function *Tf*(*P*) which was intended to explain phase diagrams where the ferromagnetic order is depressed by the pressure and vanishes at *T* = 0 at some critical pressure *Pc*. The *TFS*(*P*) line of URhGe shows a clear monotonic negative slope to *T* = 0 at pressures above 15 kbar and the extrapolation [8] of the experimental curve

<sup>0</sup>*<sup>c</sup>* < *Pm*. This type of behavior

*oc*) = 0 at *P*0*<sup>c</sup>* ∼ 25 − 30 kbar. Within the

**8.4. Two types of ferromagnetic superconductors with spin-triplet electron**

by the usual linear approximation (*n* = 1) for the parameter *t*.

critical pressure *P*0*<sup>c</sup>* < *Pm* and the upper critical pressure *P*�

"type II" – for the U-type compounds.

use the more general criterion (29).

*TFS*(*P*) tends a quantum critical point *TFS*(*P*�

**8.5. Other compounds**

**pairing**

Finally, even in its simplified form, this theory has been shown to be capable of accounting for a wide variety of experimental behavior. A natural extension to the theory is to add a *M*<sup>6</sup> term which provides a formalism to investigate possible metamagnetic phase transitions [45] and extend some first order phase transition lines. Another modification of this theory, with regard to applications to other compounds, is to include a *P* dependence for some of the other GL parameters. The fluctuation and quantum correlation effects can be considered by the respective field-theoretical action of the system, where the order parameters *ψ* and *M* are not uniform but rather space and time dependent. The vortex (spatially non-uniform) phase due to the spontaneous magnetization *M* is another phenomenon which can be investigated by a generalization of the theory by considering nonuniform order parameter fields *ψ* and *M* (see, e.g., Ref. [28]). Note that such theoretical treatments are quite complex and require a number of approximations. As already noted in this paper the magnetic fluctuations stimulate first order phase transitions for both finite and zero phase-transition temperatures.
