**7. Criteria for type I and type II spin-triplet ferromagnetic superconductors**

16 Will-be-set-by-IN-TECH

undergo a fluctuation-driven change in the phase transition order from first to second. Such picture is described below, in Sec. 8, and it corresponds to the behavior of real compounds. Our results definitely show that the quantum phase transition near *Pc* is of first order. This is valid for the whole N-FS phase transition below the critical-end point C, as well as the straight line BC. The simultaneous effect of thermal and quantum fluctuations do not change the order of the N-FS transition, and it is quite unlikely to suppose that thermal fluctuations of the superconductivity field *ψ* can ensure a fluctuation-driven change in the order of the FM-FS transition along the line BC. Usually, the fluctuations of *ψ* in low temperature superconductors are small and slightly influence the phase transition in a very narrow critical region in the vicinity of the phase-transition point. This effect is very weak and can hardly be observed in any experiment on low-temperature superconductors. Besides, the fluctuations of the magnetic induction *B* always tend to a fluctuation-induced first-order phase transition rather than to the opposite effect - the generation of magnetic fluctuations with infinite correlation length at the equilibrium phase-transition point and, hence, a second order phase transition [31, 43]. Thus we can quire reliably conclude that the first-order phase transitions at low-temperatures, represented by the lines BC and AC in vicinity of *Pc* do not change their

**Figure 6.** Low-temperature part of the *T* − *P* phase diagram of UGe2, shown in Fig. 5. The points A, B, C are located in the high-pressure part (*P* ∼ *Pc* ∼ 1.6 GPa). The FS phase domain is shaded. The thick solid lines AC and BC show the first-order transitions N-FS, and FM-FS, respectively. Other notations are

Quantum critical behavior for continuous phase transitions in spin-triplet ferromagnetic superconductors with magnetic anisotropy can therefore be observed at other zero-temperature transitions, which may occur in these systems far from the critical pressure *Pc*. This is possible when *TFS*(0) = 0 and the *TFS*(*P*) curve terminates at *T* = 0 at one or two quantum (zero-temperature) critical points: *P*0*<sup>c</sup>* < *Pm* - "lower critical pressure",

<sup>0</sup>*<sup>c</sup>* > *Pm* – "upper critical pressure." In order to obtain these critical pressures one should solve Eq. (17) with respect to *P*, provided *TFS*(*P*) = 0, *Tm* > 0 and *Pm* > 0, namely, when

<sup>0</sup>*<sup>c</sup>* is bounded

the continuous function *TFS*(*P*) exhibits a maximum. The critical pressure *P*�

order as a result of thermal and quantum fluctuation fluctuations.

explained in Figs. 2 and 3.

and *P*�

The analytical calculation of the critical pressures *P*0*<sup>c</sup>* and *P*� <sup>0</sup>*<sup>c</sup>* for the general case of *Ts* �= 0 leads to quite complex conditions for appearance of the second critical field *P*� <sup>0</sup>*c*. The correct treatment of the case *Ts* �= 0 can be performed within the entire two-domain picture for the phase FS (see, also, Ref. [20]). The complete study of this case is beyond our aims but here we will illustrate our arguments by investigation of the conditions, under which the critical pressure *Poc* occurs in systems with *Ts* ≈ 0. Moreover, we will present the general result for *P*0*<sup>c</sup>* ≥ 0 and *P*� <sup>0</sup>*<sup>c</sup>* ≥ 0 in systems where *Ts* �= 0.

**Figure 7.** High-pressure part of the phase diagram of UGe2, shown in Fig. 4. Notations are explained in Figs. 2, 3, 5, and 6.

Setting *TFS*(*P*0*c*) = 0 in Eq. (17) we obtain the following quadratic equation,

$$
\gamma\_1 m\_{0c}^2 - \tilde{\gamma} m\_{0c} - \tilde{T}\_s = 0,\tag{20}
$$

for the reduced magnetization,

$$m\_{0c} = [-t(0, \mathcal{P}\_{oc})]^{1/2} = (1 - \mathcal{P}\_{0c})^{1/2} \tag{21}$$

and, hence, for *P*˜ <sup>0</sup>*c*. For *Ts* �<sup>=</sup> 0, Eqs. (20) and (21) have two solutions with respect to *<sup>P</sup>*˜ <sup>0</sup>*c*. For some sets of material parameters these solutions satisfy the physical requirements for *P*0*<sup>c</sup>* and *P*� <sup>0</sup>*<sup>c</sup>* and can be identified with the critical pressures. The conditions for existence of *P*0*<sup>c</sup>* and *P*� 0*c* can be obtained either by analytical calculations or by numerical analysis for particular values of the material parameters.

For *Ts* = 0, the trivial solution *P*˜ <sup>0</sup>*<sup>c</sup>* = 1 corresponds to *P*0*<sup>c</sup>* = *P*<sup>0</sup> > *PB* and, hence, does not satisfy the physical requirements. The second solution,

$$
\tilde{P}\_{0c} = 1 - \frac{\tilde{\gamma}^2}{\tilde{\gamma}\_1^2} \tag{22}
$$

according to the result for *T*˜

<sup>0</sup>*c*(+) ≥ 0 will exist, if

ferromagnetic superconductors, for example, in UIr.

which generalizes the condition (23).

*TB* < 0, the upper critical pressure *P*�

such maximum (see also Sec. 8).

*P*0*<sup>c</sup>* and *P*�

the upper critical pressure *P*�

obtain that *P*˜

*<sup>m</sup>* in Table 1, this leads to the inequality *T*˜

*κT*˜*<sup>s</sup>*

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 433

<sup>0</sup>*c*. Therefore, for wide variations in the parameters, theory (6)

<sup>0</sup>*<sup>c</sup>* > 0 occurs, whereas the lower critical pressure *P*0*<sup>c</sup>* >

*γ*1 *γ*

≥ 1 +

Now we can identify the pressure *<sup>P</sup>*0*c*(+) with the lower critical pressure *<sup>P</sup>*0*c*, and *<sup>P</sup>*0*c*(−) with

describes a quantum critical point *Poc*, that exists, provided the condition (29) is satisfied. The quantum critical point (*T* = 0, *P*0*c*) exists in UGe2 and, perhaps, in other *p*-wave

Our results predict the appearance of second critical pressure – the upper critical pressure *P*�

that exists under more restricted conditions and, hence, can be observed in more particular systems, where *Ts* < 0. As mentioned in Sec. 5, for very extreme negative values of *Ts*, when

0 does not appear. Bue especially this situation should be investigated in a different way, namely, one should keep *TFS*(0) different from zero in Eq. (17), and consider a form of the

of whether the maximum *Tm* exists or not. In such geometry of the FS phase domain, the

Using criteria like (23) in Sec. 8.4 we classify these superconductors in two types: (i) type I, when the condition (23) is satisfied, and (ii) type II, when the same condition does not hold. As we show in Sec. 8.2, 8.3 and 8.4, the condition (23) is satisfied by UGe2 but the same condition fails for ZrZn2. For this reason the phase diagrams of UGe2 and ZrZn2 exhibit qualitatively different shapes of the curves *TFS*(*P*). For UGe2 the line *TFS*(*P*) has a maximum at some pressure *P* > 0, whereas the line *TFS*(*P*), corresponding to ZrZn2, does not exhibit

The quantum and thermal fluctuation phenomena in the vicinities of the two critical pressures

behavior of the superconducting field *ψ* far below the ferromagnetic phase transitions, where the magnetization *M* does not undergo significant fluctuations and can be considered uniform. The presence of uniform magnetization produces couplings of *M* and *ψ* which are

In order to apply the above displayed theoretical calculations, following from free energy (7), for the outline of *T* − *P* diagram of any material, we need information about the values of *P*0, *Tf* 0, *Ts*, *κ γ*, and *γ*1. The temperature *Tf* <sup>0</sup> can be obtained directly from the experimental phase diagrams. The pressure *P*<sup>0</sup> is either identical or very close to the critical pressure *Pc*, for which the N-FM phase transition line terminates at *T* ∼ 0. The temperature *Ts* of the generic superconducting transition is not available from the experiments because, as mentioned above, pure superconducting phase not coexisting with ferromagnetism has not

<sup>0</sup>*<sup>c</sup>* need a nonstandard RG treatment because they are related with the fluctuation

FS phase domain in which the curve *TFS*(*P*) terminates at *T* = 0 for *P*�

maximum *T*(*Pm*) may exist only in quite unusual cases, if it exists at all.

not present in previous RG studies and need a special analysis.

**8. Application to metallic compounds**

**8.1. Theoretical outline of the phase diagram**

*<sup>s</sup>* <sup>&</sup>gt; <sup>−</sup>*γ*2/4*κγ*1. So, we

<sup>0</sup>*<sup>c</sup>* > 0, irrespective

*oc*

*<sup>γ</sup>* , (29)

is positive for

$$\frac{\gamma\_1}{\gamma} \ge 1\tag{23}$$

and, as shown below, it gives the location of the quantum critical point (*T* = 0, *P*0*<sup>c</sup>* < *Pm*). At this quantum critical point, the equilibrium magnetization *m*0*<sup>c</sup>* is given by *m*0*<sup>c</sup>* = *γ*/*γ*<sup>1</sup> and is twice bigger that the magnetization *mm* = *γ*/2*γ*<sup>1</sup> ([20]) at the maximum of the curve *TFS*(*P*).

To complete the analysis we must show that the solution (22) satisfies the condition *P*0*<sup>c</sup>* < *P*˜ *m*. By taking *P*˜ *<sup>m</sup>* from Table 1, we can show that solution (22) satisfies the condition *P*0*<sup>c</sup>* < *P*˜ *m* for *n* = 1, if

$$
\gamma\_1 < \mathfrak{K}\_\prime \tag{24}
$$

and for *n* = 2 (SFT case), when

$$
\gamma < 2\sqrt{3}\kappa.\tag{25}
$$

Finally, we determine the conditions under which the maximum *Tm* of the curve *TFS*(*P*) occurs at non-negative pressures. For *n* = 1, we obtain that *Pm* ≥ 0 for *n* = 1, if

$$\frac{\gamma\_1}{\gamma} \ge \frac{1}{2} \left( 1 + \frac{\gamma\_1}{\kappa} \right)^{1/2},\tag{26}$$

whereas for *n* = 2, the condition is

$$\frac{\gamma\_1}{\gamma} \ge \frac{1}{2} \left( 1 + \frac{\gamma^2}{4\kappa^2} \right)^{1/2}. \tag{27}$$

Obviously, the conditions (23)-(27) are compatible with one another. The condition (26) is weaker than the condition Eq. (23), provided the inequality (24) is satisfied. The same is valid for the condition (27) if the inequality (25) is valid. In Sec. 8 we will show that these theoretical predictions are confirmed by the experimental data.

Doing in the same way the analysis of Eq. (17), some results may easily obtained for *Ts* �= 0. In this more general case the Eq. (17) has two nontrivial solutions, which yield two possible values of the critical pressure

$$\tilde{P}\_{0c(\pm)} = 1 - \frac{\gamma^2}{4\gamma\_1^2} \left[ 1 \pm \left( 1 + \frac{4\tilde{T}\_s \kappa \gamma\_1}{\gamma^2} \right)^{1/2} \right]^2. \tag{28}$$

The relation *P*˜ <sup>0</sup>*c*(−) <sup>≥</sup> *<sup>P</sup>*˜ <sup>0</sup>*c*(+) is always true. Therefore, to have both *<sup>P</sup>*˜ <sup>0</sup>*c*(±) ≥ 0, it is enough to require *P*˜ <sup>0</sup>*c*(+) <sup>≥</sup> 0. Having in mind that for the phase diagram shape, we study *<sup>T</sup>*˜ *<sup>m</sup>* > 0, and according to the result for *T*˜ *<sup>m</sup>* in Table 1, this leads to the inequality *T*˜ *<sup>s</sup>* <sup>&</sup>gt; <sup>−</sup>*γ*2/4*κγ*1. So, we obtain that *P*˜ <sup>0</sup>*c*(+) ≥ 0 will exist, if

$$\frac{\gamma\_1}{\gamma} \ge 1 + \frac{\kappa \mathcal{T}\_s}{\gamma} \,\tag{29}$$

which generalizes the condition (23).

18 Will-be-set-by-IN-TECH

<sup>0</sup>*<sup>c</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>γ</sup>*˜ <sup>2</sup>

*γ*1 *γ*

*γ* < 2 √

at non-negative pressures. For *n* = 1, we obtain that *Pm* ≥ 0 for *n* = 1, if

*γ*1 *γ* ≥ 1 2 <sup>1</sup> <sup>+</sup> *<sup>γ</sup>*<sup>1</sup> *κ*

*γ*1 *γ* ≥ 1 2 <sup>1</sup> <sup>+</sup> *<sup>γ</sup>*<sup>2</sup> 4*κ*<sup>2</sup>

Finally, we determine the conditions under which the maximum *Tm* of the curve *TFS*(*P*) occurs

Obviously, the conditions (23)-(27) are compatible with one another. The condition (26) is weaker than the condition Eq. (23), provided the inequality (24) is satisfied. The same is valid for the condition (27) if the inequality (25) is valid. In Sec. 8 we will show that these theoretical

Doing in the same way the analysis of Eq. (17), some results may easily obtained for *Ts* �= 0. In this more general case the Eq. (17) has two nontrivial solutions, which yield two possible

<sup>0</sup>*c*(+) is always true. Therefore, to have both *<sup>P</sup>*˜

<sup>0</sup>*c*(+) <sup>≥</sup> 0. Having in mind that for the phase diagram shape, we study *<sup>T</sup>*˜

1/2

1/2

1/2<sup>2</sup>

and, as shown below, it gives the location of the quantum critical point (*T* = 0, *P*0*<sup>c</sup>* < *Pm*). At this quantum critical point, the equilibrium magnetization *m*0*<sup>c</sup>* is given by *m*0*<sup>c</sup>* = *γ*/*γ*<sup>1</sup> and is twice bigger that the magnetization *mm* = *γ*/2*γ*<sup>1</sup> ([20]) at the maximum of the curve *TFS*(*P*). To complete the analysis we must show that the solution (22) satisfies the condition *P*0*<sup>c</sup>* < *P*˜

*<sup>m</sup>* from Table 1, we can show that solution (22) satisfies the condition *P*0*<sup>c</sup>* < *P*˜

*γ*˜ 2 1

*P*˜

<sup>0</sup>*<sup>c</sup>* = 1 corresponds to *P*0*<sup>c</sup>* = *P*<sup>0</sup> > *PB* and, hence, does not

≥ 1 (23)

*γ*<sup>1</sup> < 3*κ*, (24)

3*κ*. (25)

, (26)

. (27)

. (28)

<sup>0</sup>*c*(±) ≥ 0, it is enough to

*<sup>m</sup>* > 0, and

(22)

*m*.

*m* for

For *Ts* = 0, the trivial solution *P*˜

and for *n* = 2 (SFT case), when

whereas for *n* = 2, the condition is

values of the critical pressure

<sup>0</sup>*c*(−) <sup>≥</sup> *<sup>P</sup>*˜

The relation *P*˜

require *P*˜

predictions are confirmed by the experimental data.

*P*˜

<sup>0</sup>*c*(±) <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup>

4*γ*<sup>2</sup> 1

 1 ± 1 + 4*T*˜ *<sup>s</sup>κγ*<sup>1</sup> *γ*2

is positive for

By taking *P*˜

*n* = 1, if

satisfy the physical requirements. The second solution,

Now we can identify the pressure *<sup>P</sup>*0*c*(+) with the lower critical pressure *<sup>P</sup>*0*c*, and *<sup>P</sup>*0*c*(−) with the upper critical pressure *P*� <sup>0</sup>*c*. Therefore, for wide variations in the parameters, theory (6) describes a quantum critical point *Poc*, that exists, provided the condition (29) is satisfied. The quantum critical point (*T* = 0, *P*0*c*) exists in UGe2 and, perhaps, in other *p*-wave ferromagnetic superconductors, for example, in UIr.

Our results predict the appearance of second critical pressure – the upper critical pressure *P*� *oc* that exists under more restricted conditions and, hence, can be observed in more particular systems, where *Ts* < 0. As mentioned in Sec. 5, for very extreme negative values of *Ts*, when *TB* < 0, the upper critical pressure *P*� <sup>0</sup>*<sup>c</sup>* > 0 occurs, whereas the lower critical pressure *P*0*<sup>c</sup>* > 0 does not appear. Bue especially this situation should be investigated in a different way, namely, one should keep *TFS*(0) different from zero in Eq. (17), and consider a form of the FS phase domain in which the curve *TFS*(*P*) terminates at *T* = 0 for *P*� <sup>0</sup>*<sup>c</sup>* > 0, irrespective of whether the maximum *Tm* exists or not. In such geometry of the FS phase domain, the maximum *T*(*Pm*) may exist only in quite unusual cases, if it exists at all.

Using criteria like (23) in Sec. 8.4 we classify these superconductors in two types: (i) type I, when the condition (23) is satisfied, and (ii) type II, when the same condition does not hold. As we show in Sec. 8.2, 8.3 and 8.4, the condition (23) is satisfied by UGe2 but the same condition fails for ZrZn2. For this reason the phase diagrams of UGe2 and ZrZn2 exhibit qualitatively different shapes of the curves *TFS*(*P*). For UGe2 the line *TFS*(*P*) has a maximum at some pressure *P* > 0, whereas the line *TFS*(*P*), corresponding to ZrZn2, does not exhibit such maximum (see also Sec. 8).

The quantum and thermal fluctuation phenomena in the vicinities of the two critical pressures *P*0*<sup>c</sup>* and *P*� <sup>0</sup>*<sup>c</sup>* need a nonstandard RG treatment because they are related with the fluctuation behavior of the superconducting field *ψ* far below the ferromagnetic phase transitions, where the magnetization *M* does not undergo significant fluctuations and can be considered uniform. The presence of uniform magnetization produces couplings of *M* and *ψ* which are not present in previous RG studies and need a special analysis.

### **8. Application to metallic compounds**

### **8.1. Theoretical outline of the phase diagram**

In order to apply the above displayed theoretical calculations, following from free energy (7), for the outline of *T* − *P* diagram of any material, we need information about the values of *P*0, *Tf* 0, *Ts*, *κ γ*, and *γ*1. The temperature *Tf* <sup>0</sup> can be obtained directly from the experimental phase diagrams. The pressure *P*<sup>0</sup> is either identical or very close to the critical pressure *Pc*, for which the N-FM phase transition line terminates at *T* ∼ 0. The temperature *Ts* of the generic superconducting transition is not available from the experiments because, as mentioned above, pure superconducting phase not coexisting with ferromagnetism has not been observed. This can be considered as an indication that *Ts* is very small and does not produce a measurable effect. So the generic superconducting temperature will be estimated on the basis of the following arguments. For *Tf*(*P*) > *Ts* we must have *Ts*(*P*) = 0 at *P* ≥ *Pc*, where *Tf*(*P*) ≤ 0, and for 0 ≤ *P* ≤ *P*0, *Ts* < *TC*. Therefore for materials where *TC* is too small to be observed experimentally, *Ts* can be ignored.

Although the expanded temperature scale in Fig. 3, the difference [*Tm* − *TFS*(0)] = 5 mK is hard to see. To locate the point *max* exactly at *P* = 0 one must work with values of *γ*˜ and *γ*˜1 of accuracy up to 10−4. So, the location of the *max* for parameters corresponding to ZrZn2 is very sensitive to small variations of *γ*˜ and *γ*˜1 around the values 0.2 and 0.1, respectively. Our initial idea was to present a diagram with *Tm* = *TFS*(0) = 0.29 K and *ρ*<sup>0</sup> = 0, namely, *max* exactly located at *P* = 0, but the final phase diagram slightly departs from this picture because of the mentioned sensitivity of the result on the values of the interaction parameters *γ* and *γ*1. The theoretical phase diagram of ZrZn2 can be deduced in the same way for *ρ*<sup>0</sup> = 0.003 and this yields *Tm* = 0.301 K at *Pm* = 6.915 kbar for initial values of *γ*˜ and *γ*˜1 which differs from *<sup>γ</sup>*˜ <sup>=</sup> <sup>2</sup>*γ*˜1 <sup>=</sup> 0.2 only by numbers of order 10−<sup>3</sup> <sup>−</sup> <sup>10</sup>−<sup>4</sup> [18]. This result confirms the mentioned sensitivity of the location of the maximum *Tm* towards slight variations of the material parameters. Experimental investigations of this low-temperature/low-pressure region with higher accuracy may help in locating this maximum with better precision.

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 435

Fig. 4 shows the high-pressure part of the same phase diagram in more details. In this figure the first order phase transitions (solid lines BC and AC) are clearly seen. In fact the line AC is quite flat but not straight as the line BC. The quite interesting topology of the phase diagram of ZrZn2 in the high-pressure domain (*PB* < *P* < *PA*) is not seen in the experimental phase diagram [13] because of the restricted accuracy of the experiment in this range of temperatures

These results account well for the main features of the experimental behavior [13], including the claimed change in the order of the FM-FS phase transition at relatively high *P*. Within the present model the N-FM transition is of second order up to *PC* ∼ *Pc*. Moreover, if the experiments are reliable in their indication of a first order N-FM transition at much lower *P* values, the theory can accommodate this by a change of sign of *b <sup>f</sup>* , leading to a new tricritical point located at a distinct *Ptr* < *PC* on the N-FM transition line. Since *TC* > 0 a direct N-FS phase transition of first order is predicted in accord with conclusions from de Haas–van Alphen experiments [44] and some theoretical studies [40]. Such a transition may not occur in other cases where *TC* = 0. In SFT (*n* = 2) the diagram topology remains the same but points

The experimental data for UGe2 indicate *Tf* <sup>0</sup> = 52 K, *Pc* = 1.6 GPa (≡ 16 kbar), *Tm* = 0.75 K, *Pm* ≈ 1.15 GPa, and *P*0*<sup>c</sup>* ≈ 1.05 GPa [2–5]. Using again the variant *n* = 1 for *Tf*(*P*) and the above values for *Tm* and *P*0*<sup>c</sup>* we obtain *γ*˜ ≈ 0.0984 and *γ*˜1 ≈ 0.1678. The temperature *TC* ∼ 0.1

Using these initial parameters, together with *Ts* = 0, leads to the *T* − *P* diagram of UGE2 shown in Fig. 5. We obtain *TA* = 0 K, *PA* = 1.723 GPa, *TB* = 0.481 K, *PB* = 1.563 GPa, *TC* = 0.301 K, and *PC* = 1.591 GPa. Figs. 6 and 7 show the low-temperature and the high-pressure parts of this phase diagram, respectively. There is agreement with the main experimental findings, although *Pm* corresponding to the maximum (found at ∼ 1.44 GPa in Fig. 5) is about 0.3 GPa higher than suggested experimentally [4, 5]. If the experimental plots are accurate in this respect, this difference may be attributable to the so-called (*Tx*) meta-magnetic phase transition in UGe2, which is related to an abrupt change of the magnetization in the vicinity of *Pm*. Thus, one may suppose that the meta-magnetic effects, which are outside the scope of our current model, significantly affect the shape of the *TFS*(*P*) curve by lowering *Pm* (along

B and C are slightly shifted to higher *P* (typically by about 0.01 − −0.001 kbar).

and pressures.

**8.3. UGe**<sup>2</sup>

K corresponds to *κ* ∼ 4.

As far as the shape of FM-FS transition line is well described by Eq. (17), we will make use of additional data from available experimental phase diagrams for ferroelectric superconductors. For example, in ZrZn2 these are the observed values of *TFS*(0) and the slope *ρ*<sup>0</sup> ≡ [*∂TFS*(*P*)/*∂P*]<sup>0</sup> = (*Tf* 0/*P*0)*ρ*˜0 at *P* = 0; see Eq. (17). For UGe2, where a maximum (*T*˜*m*) is observed on the phase-transition line, we can use the experimental values of *Tm*, *Pm*, and *P*0*c*. The interaction parameters *γ*˜ and *γ*˜1 are derived using Eq. (17), and the expressions for *T*˜ *m*, *P*˜ *<sup>m</sup>*, and *ρ*˜0, see Table 1. The parameter *κ* is chosen by fitting the expression for the critical-end point *TC*.
