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Phase diagram of UCoGe

The temperature-pressure phase diagram of ferromagnetic superconductor UCoGe includes four phase transitions. They are between the paramagnetic and the ferromagnetic states with the subsequent transition in the superconducting ferromagnetic state and between the normal and the superconducting states after which has to occur the transition to the superconducting ferromagnetic state. Here we have developed the Landau theory description of the phase diagram and established the specific ordering arising at each type of transition. The phase transitions to the ferromagnetic superconducting state are inevitably accompanied by the emergency of screening currents. The corresponding magnetostatics considerations allow to establish the significant difference between the transition from the ferromagnetic to the ferromagnetic superconducting state and the transition from the superconducting to the ferromagnetic superconducting state.

The temperature-pressure phase diagram of ferromagnetic superconductor UCoGe includes four phase transitions. They are between the paramagnetic and the ferromagnetic states with the subsequent transition in the superconducting ferromagnetic state and between the normal and the superconducting states after which has to occur the transition to the superconducting ferromagnetic state. Here we have developed the Landau theory description of the phase diagram and established the specific ordering arising at each type of transition.
The phase transitions to the ferromagnetic superconducting state are inevitably accompanied by the emergency of screening currents. The corresponding magnetostatics considerations allow to establish the significant difference between the transition from the ferromagnetic to the ferromagnetic superconducting state and the transition from the superconducting to the ferromagnetic superconducting state.

I. INTRODUCTION
The superconductivity in the uranium ferromagnetic compounds UGe 2 and URhGe discovered more than decade ago [1,2] and more recently in the related compound UCoGe [3] is still the subject of quite active investigations (see recent experimental [4] and theoretical [5] reviews and references therein). The existence of superconducting state at temperatures far below the Curie temperature and very high upper critical field in these materials do not leave doubts that here we deal with the triplet superconductivity like it is in the superfluid 3 He. This is also confirmed by the measurements of the Knight shift on nucleus of 59 Co which is proved unchanged in the superconducting state [6].
One of many peculiar properties of UCoGe is that the ferromagnetism in this compound is suppressed by pressure whereas the superconductivity arising at small pressures inside of the ferromagnetic state continues to exist at high pressures in the paramagnetic state.The pressuretemperature phase diagram shown in Fig.1, has been established first in the paper [7] and then confirmed in many subsequent studies (see f.i. the last one [8]). The phase transition from the paramagnetic to the ferromagnetic state and the following to it the phase transition to the ferromagnetic superconducting state at low pressures and the phase transition from the normal to the superconducting state at high pressures are firmly registered. While the phase transition from the superconducting to the ferromagnetic superconducting state shown on the Fig.1 by the dashed line is still not confirmed experimentally.
A theoretical phase diagram description has been proposed recently by Cheung and Raghu [9]. Making use the numerical calculations applied to the minimal Landau model of neutral ferromagnetic superfluid state with one component order parameter for each spin up-up and spin down-down Cooper pair states they were able to reproduce the general structure of the UCoGe phase di-agram and to predict a first order phase transition near the boundary between the normal phase and the ferromagnetic superconducting phase.
Here I reconsider the same problem making use the analytical calculations applied to the same minimal model for neutral ferromagnetic or nonmagnetic superfluid states. The results of Ref.9 were confirmed.
A phase transition of normal metallic to the superconducting state has its own specific properties different from the properties of a phase transition in the neutral Fermi liquid to the superfluid state. So, in the last part of the paper I will discuss the significant difference between the two transitions, namely, between the phase transition from the ferromagnetic normal state to the ferromagnetic superconducting state and the transition from the superconducting state to the ferromagnetic superconducting state. This difference arises due to the essentially different screening of magnetic moment at these two transitions. In the latter case the screening is complete and instead of a bulk phase transition there is the gradual formation of the Meissner state as it occurs in a superconductor of the second kind under external magnetic field smaller than H c1 .

II. MODEL
The triplet-pairing superconducting state order parameter is given by the complex spin-vector [10] d(k, r) = (1) where ∆ ↑ (k, r), ∆ ↓ (r, k, r), ∆ 0 (k, r) are the amplitudes of spin-up, spin-down and zero-spin of superconducting order parameter depending on the Cooper pair centre of gravity coordinate r and the momentum k of pairing electrons. In the tetragonal ferromagnets with easy axis alongẑ direction there are only two superconducting states A and B with different critical temperature [11]. The general form of the order parameter for the A-state in a two-band spin-up, spin-down superconducting ferromagnet depends from the five complex amplitudes η ↑ x , η ↑ y , η ↓ x , η ↓ y , η 0 z , which obey to coupled Ginzburg-Landau equations, derived in the linear approximation in the papers [5,12].k i = k i /|k|, i = x, y, z are the projections of the unitk vector on the coordinate axis.
The order parameter of the paramagnetic superconducting state [5,13] in a orthorhombic metal looks like the order parameter of superfluid 3 He-B phase [10] ∆ ↑ (k, r) = −k x η x (r) + ik y η y (r), To avoid excessive difficulties the authors of [9] considered the minimal model for the superconducting ferromagnetic state with the order parameter Corresponding simplest order parameter for the paramagnetic superconducting state looks like the order parameter for the discovered recently polar state of superfluid 3 He [14] ∆ ↑ (k, r) = −k x η(r), In neglect of interactions of electron charges with magnetic field created by the magnetization one can write following Ref.9 the gradient independent Landau free energy density as where M is the density of magnetic moment component along the easy axis, T c (P ) is the pressure dependent Curie temperature and T sc0 (P ) is the formal critical temperature of superconducting transition in the single band (say just spin-up) case. The phenomenological treatment does not allow to fix the pressure dependences of these critical temperatures and the other coefficients in Eq. (6). In what follows we shall assume that the pressure dependences T c (P ) and T sc0 (P ) qualitatively correspond to the phase diagram with the intersection of the phase transition lines shown in Fig.1. One can note that the symmetry allows also the following interaction iγ 3 M (η ↑ η ↓ − η ↑ η ↓ ) between the super-conducting and magnetic order parameters [13], but the general enough microscopic calculations [5,12] do not confirm the existence of this term.
In general, the free energy fourth order terms actually have the form different from B(|η 9. If the normal state Green functions are diagonal in the band indices (in our case they are spin-up and spin-down indices ) the Wick's decoupling does not produce any mixing terms between the band order parameters (see f.i. [15]). This case the fourth order terms in respect of the superconducting order parameters are If in the normal state there is the band mixing interaction this leads to emergence of the additional terms One can show that the additional fourth order terms do not introduce a qualitative modification in the phase dia-gram. So, we will work with the same free energy density as in Ref.9

III. PHASE TRANSITIONS IN NEUTRAL FERMI LIQUID
At low pressures the system first passes from the paramagnetic to the ferromagnetic state and then from the ferromagnetic state to the ferromagnetic superconducting state. We begin with consideration of these phase transitions and then discuss the high pressures transitions from the normal to the superconducting state and from the superconducting state to the ferromagnetic superconducting state, and also as well as the transition from the normal to the ferromagnetic superconducting state.
A. Phase transition from the paramagnetic to the ferromagnetic state The second order transition from paramagnetic to ferromagnetic state occurs at T = T Curie (P ). Below this temperature the magnetic moment acquires the finite value and a superconducting ordering is absent B. Phase transition from the ferromagnetic state to the superconducting ferromagnetic state At the subsequent phase transition the superconducting order parameter amplitudes η ↑ , η ↓ appear and the magnetic moment acquires a magnitude M = M 0 + m. Accepting for certainty that coefficient γ 2 = −|γ 2 | is negative we see from Eq. (10) that the phase difference between the superconducting order parameters is absent Here, η 1 and η 2 are the modules of the superconducting order parameters. Thus, one can rewrite the free energy density (10) as The minimization of the free energy density (13) in respect η 1 , η 2 and m yields the equations Here, we have taken into account that M 0 is the minimum of free energy at η 1 = η 2 = 0 and omitted the fourth order terms. The corresponding linear equations for η 1 , η 2 are not coupled with linear equation for m. Equating the determinant of this system to zero and taking into account Eq.(11) we obtain the equation for the temperature T sc of transition to the superconducting ferromagnetic state. We shall not write the explicit formula for T sc in view of its cumbersome shape. Let us only note that according to this equation the pressure decrease of the Curie temperature T c (P ) causes the increase of the superconducting transition temperature T sc (P ), although this is not the only reason for the T sc (P ) pressure dependence. The linear equation in respect of m gives So, m is proved to be of the next order of smallness in comparison with η 1 ∝ η 2 ∝ √ T sc − T . Substitution Eq.(20) to the Eqs. (14) and (15) gives the equations of the third order in respect to the amplitudes η 1 , η 2 . Analytic solution of this system is possible only at negligibly small coefficient |γ 2 |. This case at γ 1 > 0 we obtain This description of the second order phase transition from ferromagnetic to the ferromagnetic superconducting state is valid at the assumption m << M 0 . However, at pressure enhancement the Curie temperature and the critical temperature of superconducting transition (see Fig.1) approach each other, the value of M 0 gets smaller and according to Eq.(20) the value of m increases. One can expect the turning of the second order transition into the first order transition such that the order parameters η 1 , η 2 , m undergo finite jumps from zero to the finite values at temperature larger than the critical temperature given by Eq. (19). Indeed, the this type of behavior was established in Ref. 9 by the numerical solution of nonlinear equations for the order parameter components at close enough values T Curie and T sc .

C. Phase transitions from the normal to the ferromagnetic superconducting state
To establish the whole phase diagram one must consider the phase transition from the normal nonmagnetic state to the superconducting state. The free energy density (13) minimization in respect η 1 , η 2 , M yields At α > 0 there are two type solutions of these equations such that and In the first case the transition to the ferromagnetic superconducting state occurs by means of two consecutive phase transitions: the phase transition from the normal state to the nonmagnetic superconducting state followed at lower temperature by the transition to the ferromagnetic superconducting state. In the second case the phase transition to the ferromagnetic superconducting state occurs directly from the normal state. We consider these situations separately.

Two consecutive phase transitions from the normal to the ferromagnetic superconducting state
The solution (26) is realized at large enough positive α when formation of a ferromagnetic state is not energetically profitable. This case the common magnitude of the superconducting amplitudes is At positive sum 2β 1 + β 2 > 0 this phase transition is of the second order and occurs at that coincides with Eq.(19) at M 0 = 0. To pass in the ferromagnetic superconducting state the system must undergo one more phase transition. At this transition the magnetization M spontaneously appears and the superconducting order parameter amplitudes acquire the deviations from the value given by Eq. (28) The free energy acquires the following form Here we have taken into account that η is the minimum of free energy at M = δ 1 = δ 2 = 0 and omitted the zero order terms in respect of M, δ 1 , δ 2 . The order parameters are determined from the conditions of the free energy minimum One can easily check that in linear approximation the equations for (δ 1 − δ 2 ) and M are decoupled from the equation for (δ 1 + δ 2 ). Hence, the latter combination is of the next order of smallness in comparison with Here T scM is the critical temperature of transition from the superconducting to the superconducting ferromagnetic state which is determined from the equation given by the equality to zero of the determinant of the system (33) 2. Direct phase transition from the normal to the ferromagnetic superconducting state The second type solution (27) is realized at small enough positive α. The analytical treatment is possible in neglect of third order term 4βM 3 in Eq.(25), then .
Passing to the sum and the difference of Eqs.(23) and (24) and using the Eq. (36) we come to the equations The solution of these equations at η = 0, δ = 0 is Thus, at direct transition from the normal to the ferromagnetic superconducting state the order parameter components M, η 1 , η 2 undergo the finite jumps. This is the phase transition of the first order. At a phase transition the free energy is not changed, that gives the equation for the phase transition temperature We solve this equation in the assumption that the coefficient |γ 2 | is negligibly small. Then at small enough values of α one can use the approximate expressions Substituting them to the Eq.(42) we obtain So, at small enough positive α the phase transition temperature is given by that exceeds the critical temperature of the second order transition given by Eq. (29).

D. Phase diagram
The analytic derivation made at several not strongly restrictive assumptions leads to the conclusion that the direct phase transition from the normal to the ferromagnetic superconducting state is of the first order. This confirms the statement numerically established in Ref. 9. On the other hand, as we have pointed out in the section IIIB., when the temperatures of phase transitions to the normal ferromagnetic and to the superconducting ferromagnetic state are close to each other, the transition from the ferromagnetic to the ferromagnetic superconducting state is of the first order.
Thus, the simple phase diagram with intersection of ferromagnetic and superconducting phase transition lines like it is drown in Fig.1 cannot be realized at least in frame of the model under consideration. Near the intersection of the critical lines of the ferromagnetic and the superconducting transitions there is a piece of the transition of the first order as this is shown in Fig.2 by the thick line. The pressure interval where this transition takes place can be quite small. The presence of the first order transition seems to be in correspondence with the sharp drop of resistivity at superconducting phase transition in this pressure interval found in the paper [8].

IV. MAGNETOSTATICS
The authors of Ref.9 discussed the phase transition between the SC state and FM+SC state in the neutral Fermi liquid. The situation is changed in a charged Fermi liquid because the magnetization in the superconducting state is inevitably accompanied by the screening currents. The emergency of superconducting state in the ferromagnet state of UCoGe takes place at finite magnetization. Whereas the arising of ferromagnetic state in the superconducting state of UCoGe accompanied by the smooth increasing of magnetization from zero to the finite value. This determines the difference between the transition from the ferromagnetic to the superconducting ferromagnetic state and the transition from the superconducting to the superconducting ferromagnetic state which we discuss here.
Let us consider a cylindrical sample of radius R with axis parallel to the easy magnetization axis. A phase transition to the superconducting ferromagnetic state is accompanied by the appearance of super-currents. The corresponding London equation for magnetic induction is where δ is the London penetration depth. The contribution to the current due to the term c curlM is nonvanishing only in the surface layer with thickness of the order of coherence length ξ << δ << R [10], hence, the small enough magnetic field has to decay in the sample volume: where the surface magnetic field is determined by the magnetic moment created by the super-currents flowing in the surface layer In UCoGe according to the phase diagram drown in Fig.2 the formation of ferromagnetic superconducting state occurs (i) from the normal ferromagnetic state, (ii) from the nonmagnetic normal state either through the two consecutive phase transitions of the second order N → SC and then SC → F M + SC or directly by means of the first order transition.
A. Magnetostatics below transition from the ferromagnetic to the ferromagnetic superconducting state Just below the temperature of the phase transition from the ferromagnetic to the ferromagnetic superconducting state discussed in the section III.B the field at surface is According to the experimental results reported in [16][17][18] this field at ambient pressure is larger than the lower critical field field H c1 in UCoGe. So, the complete field screening is not realised, and the phase transition occurs directly to the superconducting mixed state. The vortex cores occupy the small part of the sample volume, and almost whole volume is in the superconducting state with the order parameter given by Eqs.
Here, we have neglected the temperature dependence of M 0 (T ), just taking it as the constant M 0 = M 0 (T sc ). The specific heat jump at this transition has been registered [4]. Another situation takes place at the transition from the superconducting to the ferromagnetic superconducting state. This case discussed in the section III.C the surface field is determined by the small magnetic moment arising at transition in magnetic superconducting state This field is certainly smaller than the lower critical field in well developed superconducting state with superconducting density n s ∝ η 2 ∝ (T sc − T ) The transition to the ferromagnetic superconducting state is characterized by the emergency of magnetic moment M and the magnetic part of superconducting ordering ∼ (δ 1 − δ 2 ). But the super-currents completely screen this magnetism in the bulk of material. The gradual increase of magnetization from zero to some finite value is not accompanied by a bulk phase transition as it is in the process of the Meissner state formation in a superconductor of the second kind under external magnetic field smaller than H c1 . The pressure decrease stimulates ferromagnetism. Hence, at low temperatures and low pressures the magnetization will exceed the lower critical field and a sample passes to the ferromagnetic superconducting mixed state. So, instead a phase transition between the superconducting and the ferromagnetic superconducting state one can expect just the transition between the Meissner and the mixed superconducting states.
Thus, there is no bulk phase transition at all. The ferromagnetic superconducting Meissner state exists in the region between two dashed lines shown in Fig.3. The actual position of H c1 line is subject to experimental dedetermination. It can be in principle much more to the left, than it is drawn in Fig.3, and also as well as more to the right such that near the first order transition the two dashed lines can be merged to one line.

V. CONCLUSION
The Landau theory allows to establish specific properties of phase transformations in anisotropic ferromagnetic superconducting material UCoGe. There was found that the phase transition from the ferromagnetic to the ferromagnetic superconducting state at ambient pressure is characterized by the appearance of superconducting part of the order parameter whereas the ferromagnetic component does undergo insignificant changes. However, at higher pressures this transition can turn to the transition of the first order.
There was proven that the direct phase transition from the nonmagnetic normal state to the ferromagnetic superconducting state is of the first order and exists in small pressure interval. Out of this interval the transition to the ferromagnetic superconducting state occurs by means of two consecutive phase transitions of the second order from normal to nonmagnetic superconducting state and then from the nonmagnetic superconducting state to ferromagnetic superconducting state. The model phase diagram in neutral Fermi liquid acquires the shape shown in Fig.2.
The phase diagram modification introduced by screen-ing currents in charged Fermi liquid is shown in Fig.3.
The magnetic moment at phase transition from the ferromagnetic to the ferromagnetic superconducting state is just partially screened by the superconducting currents. Whereas at phase transition from the superconducting state to the ferromagnetic superconducting state this screening is complete what shades the manifestations of a bulk phase transition. The ferromagnetic mixed superconducting state occurs only at lower pressures where the spontaneous magnetic moment exceeds the lower critical field. The position of H c1 line is subject to experimental determination.