Lifshitz-point behaviour of ferromagnetic superconductors

Z. Phys. B - Condensed Matter 46, 149-167 (1982) Condensed Zeitschrift ~ ' t ~ fer Physik B

�9 Springer-Verlag 1982

Lifshitz-Point Behaviour of Ferromagnetic Superconductors

B. Schuh* and N. Grewe*

Institut ffir Theoretische Physik der Universit~it zu K61n, Federal Republic of Germany

Received January 5, 1982

The critical behaviour of the electromagnetically coupled superconductor magnet system is investigated by means of a generalized mean field theory and a renormalization group analysis. We show that in the presence of a genuine anisotropy in systems with an additional pressure-like parameter (like concentration in pseudo-ternary ferromagnetic superconductors (FMS), e.g. Erl_xHo~Rh4B4) the indirect coupling between superconducting and magnetic order parameters (i.e. gauge coupling) can lead to a peculiar kind of critical behaviour characterized by Lifshitz points (LP). These points (quite generally) occur as merging points of three phases: a (magnetically) disordered phase, a homo- geneously ordered phase and a modulated phase. In FMS the latter phase may result from exchange screening by supercurrents. This unusual critical behaviour is found in two varieties:

1. a regular LP which may occur on the lower transition line of a reentrant FMS 2. a similar but slightly different critical point which we term modified Lifshitz point (MLP), and which is to be expected at the merging point of the upper and lower superconducting transition lines with the magnetic order disorder transition lines in the (x, T) phase diagrams of FMS's.

I. Introduction

In the study of the mutual influence between superconducting order and magnetic moments in solids the (pseudo-)ternary compounds, like Erl_xHOxRh4B~, play a particular role [1]. The exchange interaction between electrons producing the magnetic moments localized at the Rare Earth sites and the superconducting 4d-electrons of the metallic complexes seems to be so weak that even a periodic array of moments does not necessarily destroy superconductivity via the pair-breaking mechanism [2]. This has led to a re- vived interest in a Ginzburg Landau theory of magnetic superconductors, first proposed by Krey [3],

* Part of this work was performed during a one year visit of N.G. at the Institute for Theoretical Physics, Santa Barbara and a two year stay of B.S. at the Department of Physics at UCSD, La Jolla. Both authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft and the National Science Foundation (U.S.A.), grant PHY77-27084, as well as the hospi- tality of the departments at Santa Barbara and La Jolla

which stresses the magnetic coupling between the density of magnetic moments and superconducting screening currents. Most of this work has been based on the free energy functional proposed by Blount and Varma [4], which reads in a slightly modified version:

~-~{O,A,a}-Zrd3r[ 1012 +b[Ol4+Yol(F-ieA)Ol 2

+ +41 B2-2 ,-B ] (1) Zero external field is assumed and boundary effects are neglected. All magnetic interactions are contained in the gauge coupling between the vector potential A and the superconducting order parameter 0 (third term) and the last two terms of (1), i.e. the magnetic field energy and the energy of magnetic dipoles, connected with the density of localized spins, in the field B = Fx A.

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150 B. Schuh and N. Grewe: Lifshitz-Point Behaviour of Ferromagnetic Superconductors

The stable states derived from the functional (1) (via the variational procedure) carry a magnetic field B o which is given as usual in terms of the magnetization produced by the spins and the superconducting screening currents:

B o = 47z(M• + Ms). (2)

The transverse fields M• and M s are determined by:

M•177 cVxMs=js . (3)

In principal, there is also a fluctuating part Bfl of the magnetic field

B = B o + Bfl (4)

which is produced by other sources inside the solid. Whether therefore A should be regarded as an actual fluctuating variable of the functional J~ depends on the type of problems one is concerned with. The coefficients c(, fi, G of the pure local moment part in (1) are exclusively determined by exchange (i.e. electrostatic) interactions, which may be of direct or indirect (RKKY) origin. The mean field transition temperature T~o, contained in

~ '= T - Ts , (5)

r ; o

becomes (already in the absence of superconducting order) modified by the presence of the magnetic terms. Observe however that, since F .B=0, this only affects the transverse part ~• of the spin density field, which in the absence of superconductivity and any fluctuations of B (implying B = 4n g6• acquires the mass-coefficient

T - T~o c~• = ~' - 4~z g2 ~ ~0 (6)

Tmo This formally corresponds to an increased mean

4rcg2\ field transition temperature Tmo = T.~, ~ 1 + ~ - ) ,

i ) -

and ~ 0 = ~ + 4 7 z g a. Besides the obvious homogeneous solutions the functional (1) allows for a variety of non-homogeneous states, in which superconducting and modulated ferromagnetic order co- exist, such as domain-like structures [3] (self-induced) vortices [3, 5], spirals [4, 6] and sinusoidally varying linearly polarized states [7], or others [24]. The relative stability of these as well as their stability over the homogeneous states to a high degree depends on the particular values of the coefficients in (1). It seems however that a coexisting spiral phase could actually be stable under realistic conditions at least within mean field theory [7]. Experimentally,

neutron scattering studies of ErRh4B 4 indicate a tendency towards an instability of the pure superconducting state against some kind of order with a characteristic nonzero wave vector slightly above the transition to the pure ferromagnetic state [8]. Such an instability could be brought about by the diamagnetic response of the superconductor to magnetic fluctuations when approaching the (actual) magnetic transition temperature T,, from above, which is the mechanism considered in [4]. This screening of the exchange interactions becomes effective over distances larger than London's penetration depth 2 L which together with the magnetic stiffness ?o sets the scale %1 of the inhomogeneous order in this theory. Thus, in lowering the temperature, one might en- counter three different phases: (t) a superconducting paramagnetic phase, (II) an inhomogeneous coexistent phase of superconductivity and (modulated) ferromagnetism (possibly spiral) with a characteristic wavelength 2~qo 1, and (III) a pure ferromagnetic phase, which most likely is normal conducting. In the following we want to address the question, in which way these phase boundaries could merge as functions of an additional parameter p like pressure or alloy concentration, i.e. what is the nature of the multiple endpoints in the phase diagrams of (pseudo-) ternary componds. This question has been partly addressed in an earlier publication, [9a], where we found that in isotropic systems the fluctuation induced first order nature of the superconducting transition [20] prevails also in the presence of a magnetic order parameter. This fact is a typical feature of field theories with a gauge-type coupling. Here we enlarge our investigation in three direc- tions:

a) we investigate the possible occurance of nonuniform states over the whole phase diagram in the frame of mean field theory and incorporate them into the renormalization group analysis. b) we include the magnetic anisotropies of the systems under consideration (e.g. HoRh4B 4 has an easy axis of magnetization, the pure system ErRh4B 4 an easy plane of magnetization perpendicular to this axis). c) we pay special attention to the critical behaviour in the vicinity of the reentrant transition line where we assume critical fluctuations of the superconducting order parameter to be absent.

It becomes conceivable for example, that with in- creasing p the temperature interval, over which the inhomogeneous phase (I1) is stable, shrinks to a point. Fig. 1 gives a schematic version. Such a triple endpoint, where the above three phases with their

B. Schuh and N. Grewe: Lifshitz-Point Behaviour of Ferromagnetic Superconductors 151

T SC- PM

/

NC - FM

P

Fig. 1. Schematic version of the merging of three phase boundaries in a Lifshitz point (LP). Shaded area (phase II): phase of coexisting superconducting and nonuniform magnetic order

differing magnetic behaviour meet, is called a Lif- shitz point (LP) in the theory of magnetic systems, if q0 there continuously goes to zero from inside phase ( I I ) [10]. It is of special interest because it exhibits a peculiar kind of critical behaviour. In simple magnetic models a LP can be generated by a competition between ferromagnetic and antiferromagnet- ic order [11], or more directly by an externally forced reduction in the magnetic stiffness [12]. In- tuitively, both of these mechanisms are not much different from what may happen in ferromagnetic superconductors. The screening response certainly weakens the (ferro-)magnetic stiffness and imposes a tendency towards net magnetisation zero on large scales. In Sect. II of this paper we proceed by discussing some possibilities for the occurence of a LP in the frame of mean field theory for the functional ~ , i.e. on the basis of London's and Maxwell's equations. In order to be specific, we will refer to the system Erl_xHoxRh4B4, about which much experimental information is available. As mentioned before, it also exhibits an interesting change of the type of ferromagnetic structure with varying alloy parameter x, which could have some bearing on the occurence of LP's. To make sure that the general picture emerging from the mean field treatment is not changed by fluctuation effects, we investigate the influence of fluctuations on the basis of a generalized mean field description for the superconducting and magnetic order parameters in Sect. III. Equipped with some ideas for the results expected we turn to a renormalization group treatment in the following two Sects. IV and V in order to classify the critical behaviour under various circumstances. A resum6 is given in the concluding Sect. VI.

II. Mean Field Characterization of Lifshitz Points

In order to make a close analogy to what is known about Lifshitz-points in magnetic systems, we consider in the following the influence of superconducting order on the magnetic properties of a system described by the functional (1). We also want to be somewhat more general concerning the exchange interaction between localized spins which determines the corresponding quadratic part in ~ and assume:

Eex r = - �89 ~ d 3 r ~ d 3 r' J(r - r') a(r)- a(r'). (7)

It is of little importance for the present qualitative reasoning whether the exchange interaction J is a direct one or is mediated by conduction electrons, i.e. is of the RKKY-type. Were it not for the presence of superconducting order and magnetic interactions this exchange interaction would lead to an order to disorder transition at a temperature T,~ which in mean field theory is determined by the divergence of the susceptibility

%o(q)=C T - J(q) , C=Cur iecons tan t . (8)

This also determines the characteristic wave vectors qo of the instability as the values of q where the Fourier transformed exchange coupling J(q) assumes its maximum:

, C T~,o = ~ S(q0) , S(qo) ~ J(q). (9)

In the most simple ferromagnetic situation J(q) is a continuously decreasing nonnegative function for in- creasing q, so that qo=0 and T~,o--CJ(O)/g 2. The second moment of the function J(r) then is positive. It is a trivial fact that in ~ the coefficient of the term quadratic in the field a (apart from a factor g2) coincides with the inverse susceptibility, provided superconducting order and magnetic interactions (except with an external field) are neglected: In the simple isotropic case assumed in (1) this allows the coefficients ~' and ~ to be identified as:

2 c(=~(T-T, ' ,o) , i.e. ct;=J(0);

- - 1 f d3r r 2 J(r) > 0. (10)

More generally, and to be consistent with (7), one should use the full nonlocal expression g 2 z 0 ( q ) - t ~ r ( q ) - 6 ( - - q ) . The next goat now is to implement the change in the susceptibility which is caused by the magnetic interactions among the ions alone and by the diamagnetic response of the super-


conductor to the magnetisation field ga• of the spins. The former interactions cause the coefficient c( for the transverse field as to be lowered by 4ng 2, and the corresponding critical temperature to be raised to Tmo, as discussed in Sect. II (see (6)). Diamagnetic effects may be included as a reduction of the exchange interaction J in (8). For this purpose we express the field 4nMs, which is produced by the superconducting current is, in terms of the spin density field a. The effective exchange field H'exc(q) which is produced by the genuine contribution Hexc(q)=J(q)a(q)/g and the screening field 4nMs then defines an effective J ' via:

H'~,r - Hexo(q) + 4 n M,(q) = a'(q) a(q)/g. (11)

In order to establish a connection between M~ and one can use the Ginzburg-Landau equation derived from the functional (1) by variation with respect to the vector potential A [13]:

c 4n V• ( B ~

=ec?o{ l [o*(VO)-O(V~t ,* ) ] -e lOl2Ao} . (12)

The r.h.s, of (12) is readily identified as the superconducting current density j, and the transverse part B 0 - 4 n g a • of the field on the 1.h.s. therefore is the screening field 4nM, . Using a polar representation for the order parameter O=Oo eie, (12) may be expressed as

Vx 4riMs= -~{2(Ao--e-1 V(p), (13)

where the London screening length ;t L =(4r~70e2~,2o) -1/2 has been introduced. For a general state, 2 L is a position dependent function through the varying modulus ~0. Below the normal to superconducting transition temperature T~ however, large (relative) fluctuations in qJo become unimportant 1-14], so that 2 L may be assumed to be constant with ~o replaced by its mean value. Then (13) fouriertrans- formed reads:

4;re )~2 q2 Ms(q ) = _ iq x Ao(q) = - 4n(Ms(q) + ga•

(14)

where we have used Maxwell's equation (2), and the effective exchange constant splits into a longitudinal and a transversal part:

4rig 2 J(l(q)=J(q), J~(q)=J(q) l+(2Lq) 2" (15)

The additional contribution in J',(q) can shift the maximum from its position at q = 0 to a value with q =qo>O.

In the isotropic case j(q)__+j(q2) the wave vector qo is determined by

dJ 2 4ng 2~2 (16) dq2 (qo) + (1 + (J~Lq0)2) 2 =- 0,

which after approximating J(q)~,O:'o-y,~q 2 gives explicitely:

12q = Here we have introduced the magnetic length

l - (?J (4n g2))l/z (18)

which is a measure for the spin aligning tendency of the exchange interactions relative to the strength with which the spins couple to the magnetic field. This interaction length has to be smaller than the penetration depth ;~L if J' is to develop a maximum at a finite qo, as is obvious from (17). This maximum signifies an instability towards an inhomogeneous state of coexisting superconducting and modulated magnetic order. In order to determine the temperature Tmo a at which the instability occurs, we have to insert the screened exchange interaction J; into the transverse spin susceptibility (8) which also has to be corrected for the effect of magnetic interactions discussed in connection with (5) and (6). From the result:

4n z• ~ = Zo(q)- ~ - 4n -} 1 + (,,~L q) 2 (19)

one gets, after inserting (8), from z• =0:

C [j(qo2)+4ng 2 (2Lqo) a ] T~od = ~- 1-7 (2~%)2J' (20)

Again using the expansion J(q)=c%-3~q 2 (20) gives a simplified expression for Tmo a which can easily be compared to the other two characteristic magnetic temperatures T,~ ~ and Tmo

( mod T~,o<_ T,~ o 1 +--C(o =

=T,,o(1 4rig2 1 2L _ (21)

It is clear from (21) that the transition into the coexistent phase cannot occur above Tmo although the tendency towards the instability develops continuously with the onset of superconductivity at T, (211=t=0). As for the real system ErRh4B 4 it is not clear whether such a phase ever becomes thermodynamically stable at all. There are general argu-


ments [15] and numerical calculations E9b] that a proper treatment of fluctuations based on the isotropic functional (1) renormalizes the transition temperature Tmo d to - o o . We shall review this point in the next section. The argument breaks down, however, in the presence of anisotropies, as they are encountered in e.g. ErRh4B 4 (easy plane of magnetization) or Eq _xHo~Rh,B4, x ~> 0.3 (easy axis). This may also have a bearing on the nature of the coexistent state; in the last case for example a linearly polarized state may be preferred to the spiral state. We shall simply refer to "phase II" in the following, regardless of the specific form of the coexistent phase since we are only interested in the possible merging points of this state with the other phases. So far we have shown that within mean field theory for the superconducting order parameter ~9 and the magnetic induction B the free energy functional (1) may be replaced by the "magnetic" functional:

~ ' { 1 2 - 1 "~4"m 0"}=22{g Zo(q) all(q)'all(q) q

+g2z . (q ) - ~ a.(q).a,(-q)}+�89 a. (22)

The dynamics of the transverse field a• is now determined by the "screened" susceptibility:

1 l z,(q) --- ~ - f ( q ) ,

[4~@ (i/~L)2 ]-1 f ( x ) = +x24 ( i / ~ x 2 ] (23)

(a, is given by (6)). For l < 2 L # oO this function exhibits a maximum at a nonzero wavevector x 2 =12q 2 whose location varies with the parameter

0 <= p - I/}% ==_ 1 (24)

according to (17). The situation is schematically depicted in Fig. 2. The maximum value of Z, tends to infinity for T"~Tmo a given by (21) indicating the temperature where phase II "locks in". Regardless of the relative stability of this phase we may now ask the question under what conditions its characteristic wave vector x o =lqo vanishes, a necessary condition for the presence of a Lifshitz point. From (17) it follows that there are two distinct ways to achieve qo ~ 0 if we regard p as an independent parameter. The first possibility is p---~pLp=l with TLp=Tmod( p = 1)= T" ~ as the corresponding critical temperature (from (21)). From an expansion of f - 1 (i.e. Z• 1) in (23) around x =0:

f (x) -~ = 0~• g2) -t - 1 -t- X2(1 -p-2)+p-4x4+O(x6)

(25)

~ V1 ( ~ , P=0

f(x)

{ ~-~z +1) -1

x 2

Fig. 2. Qualitative plot of the transverse susceptibility z • (see (23)) for various values of p=l/),L, decreasing from p=0 (top curve) to p = 1 (bottom curve)

we can see that the reduced stiffness vanishes in this case and the fourth order term dominates. This is exactly the behaviour the susceptibility displays at an ordinary Lifsfiitz point (LP). The second possibility to achieve qo~0, namely: p-*PMLp=0, rMgp=rmod(O)=rmo is slightly different due to the nonanalyticity of Z, as a function of (p,x) in the neighbourhood of (0, 0). As depicted in Fig. 2 f develops its maximum very sharply in the limit p ~ 0 and the maximum value approaches 4ng2/a , instead of (c~,/(4ng2)+l) -1 as in the former case. We shall call the corresponding critical point "modified Lifshitz point" (MLP) from now on. Strictly speaking (25) must not be used now for x>=p. We can nevertheless try to characterize the MLP in terms of (25), i.e. the behaviour of the effective stiffness 7eff and the fourth order term which we call F from now on. In such a characterization the MLP is associated with

~eff----~ - - (30, F---~ o% 7ef f /F~0- (26)

Up to now we have ignored an important fact, however: p=I /2 L is not an independent parameter but depends on temperature via the superconducting order parameter 00 contained in 2 L. (See the line following (13).) Thus at T=Tmod, which in turn is given through p, the operating value of p (or Tmoa) has to be determined selfconsistently from the implicit equation:

p = 1/,)cL(Tmod(P) ). (27)

In the presence of an additional parameter x, like e.g. concentration in the case of Erl_xHoxRh4B 4 (27) determines the possible values of p as a function of x. One needs to investigate now whether the LP- and MLP-values p(xc)=0 , 1 are possible solutions of


(27) for some concentration x~, thus determining the location of Lifshitz points (x~, Tmoa(P(Xc) ) in the (x, T)-plane. In order to get a rough idea of such a procedure we may use the mean field result

a a 0 T~- T = ( 4 x 70 e2)-i 2{ z (28) r 2b 2b T s

in (27). Inserting Tmod(p) from (21) then leads to a quadratic equation for p which is solved by

p+ =(I +pTJT,,o) -1

�9 (1 + { 1 - (1 - Ts/T,,o) ~(1 + p TJTmo)} ~/2). (29)

The dimensionless constants

T T sc- pt~

Tm 4b

would - - - . . . . . . . mo be LP

Fig. 3. Merging of phase boundaries if the actual transition temperature T,, for the SC-PM~NC-FM transition lies above the mean field phase boundary T/,o(x )

p=-o:ob/(2~?oy,,eZao), v=-~:o/(4rcg2)> l (30)

contain specific information about the material. They are weakly concentration dependent. The main concentration dependence may be thought of as being given by Ts(x ) and Tmo(X ). Let us now turn to the question whether (29) permits LP and MLP solutions, i.e. p+ = 1 or 0. In the first case only the + - b r a n c h allows for p + = l solutions. The solubility condition relates TJT,, o to v and p via:

TJTmo =(v- 1)/(v-p) (31) if l < p < v or p < l < v . In all other cases TJTmo would be negative. In principle (31) then determines the critical concentration. From (21), however, it became clear already that the corresponding temperature would be Tmod( p = 1)= T,~o. Consequently, even if p = 1 is admitted by (27) for some concentration x, the states discussed here could already be metastable or unstable if the free energy of the system could be lowered by entering the normal ferromagnetic phase at a temperature T m higher than T~0, i.e. if T/,o<Tm<Tmo. This would provide an explanation for a region in the phase diagram, into which phase II is confined such that the characteristic wave vector qo nowhere goes to zero, as sketched in Fig. 3. There are experimental hints that such metastability phenomena, involving phase II, exist 1-17]. If a point p = l lies in the stable region, it follows from (17) that the Lifshitz phenomenon q0~0 should be observable there for phase trajectories coming out of region II. Moreover, the critical exponent connected with the vanishing of qo is of the mean field type, since in this case qo ~(P-Pc) 1/2. On the basis of the ansatz (28), however, it seems unlikely that systems like Erl_xHOxRh4B4 possess even the metastable LP, since for realistic values of the parameters: v = 1 + ~/(47c gZ) > 1, p = v22/I 2. (T~ -T)/T~> 1 so that (31) leads to the unphysical condition TJT,,o < O.

T ~ ~ . ~ NC-PM

\ . •

Fig. 4. Location of phase II (shaded area) in the vicinity of the MLP according to the ansatz (28) for the temperature dependence of the superconducting order parameter

This deficiency is probably due to the extension of the temperature dependence of r to temperatures well below T s. It possibly is partly removed in a more re f ined treatment of this temperature dependence which we shall introduce shortly. First we discuss the MLP-solutions of (27), i.e. p--,0. Since we exclude / = 0 this can only happen if AL~OO , or r Clearly only the negative branch p_ in (29) permits such a solution. The correpsond- ing critical temperature according to (21) and (28) has to fulfill: TML p = Tmoa( p = 0) = Tmo = T s. Therefore the only candidate for a MLP is the triple endpoint of the lines T~, T2 (the para-to ferromagnetic transition line) and T~ (the transition line between uniform ferromagnetic order and phase II). (see Fig. 4). However, again the ansatz (28) leads to TIT,, ~ < 1 (which is a necessary condition for p_ >0) such that phase II would have to merge with the other phases in the MLP as sketched in Fig. 4. It is very unlikely that phase II would be a stable phase in this location (The MLP could be observable as such nevertheless). The outcome is due to the crude ansatz made in (28) for 2 L2 as a function of temperature. In reality one has to take into account that the super-


conducting order parameter is reduced by magnetic fluctuations when approaching the reentrant transition temperature T R < T~; this is found in thermal conductivity measurements on ErRhcBr [25]. It is therefore more realistic to use instead of (28) the ansatz:

;~2=11_ 2 T~-T TIT~ (32) #~ rs r~

with a new phenomenological temperature r R = TR(x ). The implicit equation (27) now leads to a quartic equation for p:

(v- lp2-- 2 v - lp + 1- - t+ )2= t 2_-(t2+ --t2_) p v - l p 2

t + -- (T~ + TR)/(2 Tmo ). (33)

We shall not attempt to discuss this equation in detail since too many assumptions about the material parameters need to be made to come to definite conclusions. For our purposes it suffices to note that (33) permits a solution p"~ 0 if

- ~ v = o + 2 V - 1

~rR ,). �9 (d@ (~-o) p= o + 2 v - (34)

Thus Ts/Tmo>l >TR/Tmo is permissible now in the neighborhood of the MLP such that phase II may well be stable and located as indicated in Fig. 5. We summarize some of the possibilities for the existence of LP's in this figure. Since p(x) as it is determined by the selfconsistency equation (27) need not be a single valued function of x in the whole intervall O<_x<_x c (where x c is the critical concentration for the occurrence of superconductivity) there is an almost unlimited variety of possibilities. From the class of solutions for which p(x) is a single-valued function we select four representative ex- amples, a) p < l for all concentrations, b) p mo- notonically decreasing from a value po > 1, c) an intermediate region of concentrations where p > 1, d) same as c) with an additional increase of p at small concentrations. In all cases a MLP is present at the triple endpoint of T~, R, T~ >, T < . Its associated modulated phase (shaded) may be thermodynamically stable or not. In any case it should show up experimentally as an instability in the susceptibility. For a real material like Erl_xHOxRh4B 4 not only the x-dependence of the fundamental parameters de-

.termining p through (27) has to be known but also the influence of crystal anisotropies must be esti- mated. Roughly speaking, the spiral state seems to

NC-PM

/'Tm'o

NCiF I l ,

Xc a concentration

Xc

P ~ NC-PM

SC-PM

eL Tmo

b XL XC concentration

PX 1

Xt. X c X

~ ~ NC-PM

E

I I I I t

XL2 XL! XC C concentration

~<L2 XLI X c X

1 ~ NC- PM

S P

I I

I I [ I I L

d XL 3 XL2 XL1 X C

Fig. 5. Various possibilities for the location of Lifshitz points (dots) and modified Lifshitz points (dots labeled MLP) in the temperature concentration plane of ferromagnetic superconductor alloys. For explanation see Sect. III

[k A

XL. 3 XL2 XL1X C X


have a lower free energy than the linearly polarized one in a nearly isotropic situation [7], mostly since the former does not involve amplitude variations. Therefore, if a spiral phase II were stable in ErRhgB 4 (x=0), which seems favored because there is an easy plane of magnetisation, it could terminate at a concentration x where the easy axis structure of HoRh4B , becomes effective, simply because then neither a spiral nor a linearly polarized phase II is stable any more. Then one would expect a LP near the minimum of the Tin-curve, which has been in- terpreted [9c] to reflect the competition in the magnetic order. As has been extensively studied in magnetic systems [16], crystal anisotropy does change the nature of the inhomogeneous state, some details of the phase boundaries (such as common tangents etc.) [18] and possibly the order of transitions but not the essential structures of the phase diagram. Since the foregoing considerations did not depend on details like how much phase space and what kind of final states are available for the instability to occur, conclusions like the existence of Lifshitz points as endpoints of inhomogeneous phase regions are unchanged in the presence of such anisotropies.

III. Fluctuation Effects

We want to study in greater detail now the influence of fluctuations near a Lifshitz point or a modified Lifshitz point. Since the functional (1) is quadratic in A(B), fluctuations in the gauge field can be treated exactly. Fluctuations in the superconducting modulus and the magnetic order parameter are included in a generalized mean field theory of the kind used in [9b] already. In a polar representation of 4 = 0 o e~e the functional (1) may be written:

J~{0o, q), 0-, A} =�89 g + 7o(V0o) 2 + bO{}

+ ~o{(P, 0-, A} ( , ' 3 t 2 (VxA)2 ~,~ r Yo0o(Vq~-eA) 2+l~4zt

- 2g 0- - ( g • A) + ~0-2 at- fi(0-2)2 -I- 7~r( [7o 0-)2 } . (35)

We absorb the phase cp into the longitudinal part of A and then employ a selfconsistent molecular field approximation for the first term in ~ �9 ~0"

02A z = (02 - (0o2)) (A 2 - (A 2 )) - (0o 2) (A 2 )

+O 2 (A 2 ) + A 2 (0o 2)

~1/t2 ( A 2 ) + A 2 (0o2) + const. (36)

where the bar stands for the spatial average: --

_ 1 j.d3r.., and ( ) denotes the thermal average Vol

with the new effective functional:

O~eff{O0,0-, A } =�89 ~ d3r {arenO2-l-'y,p( V00)2 + b004

+/zA 2 +r( Vx A)2 -2gA .( Vx 0-)

+ ~0-2 + fi(0-2)2 + 7~( Vo 0")2}. (37)

Here we have introduced the renormalized mass

are n = a -q- ~),p e 2 (A 2 ) (38)

and the abbreviations

e 2 < >, r = 1/4 . (39)

This description is appropriate in the neighbourhood of the MLP where there are fluctuations in the superconducting order parameter. In regions where its modulus is a constant to a good approximation we need not make the molecular field approximation (36). In this case we simply drop the

(g0o)Z-term in (37) and replace (02) by 0~ in (39). Probably such a description is good in the neighbourhood of the reentrant transition line T m ~ T R ~ T,, i.e. for the description of ordinary Lif- shitz points. Both cases are covered by the same notation. The part of ~eff quadratic in A and a reads after Fourier transformation:

~fz) {a, A} =�89 Z {#All (q) All(-- q) q

+ (# + rq 2) A• A• - q) + (~' + ?~q2) 0-(q). a( - q)

- 2 ig(q • 0-(q)) �9 A ( - q)}. (40)

For q q= 0 it may be diagonalized with respect to the vector potential by referring to its mean field value A 0 [28] :

A=A0+Af1• Ao(q)=igqx0-(q)/(Iz+rq2). (41)

The result is:

~ee (2) {0-, A} = ~sing -}- �89 2 (~ -}- rq2) Aflz(q)" At1• - q) ff q=~0

+�89 ~ {(C(+ 7~q2) 0- ii (q) 0- ir ( -- q)

( Iz+rq ] q)}. + r 0-•177 (42)

Here O~s~ng suppresses states with nonzero uniform superconducting order parameter Oo and magnetic induction B o (MeiBner effect). Depending on the importance one attributes to independent fluc-


tuations of A the second term on the r.h.s. (42) is either zero (no fluctuations) or may be integrated out in the partition function leading to

A O~fl = 2 ln(# + rq2). (43) q

This contribution is the first important source for fluctuation effects. For small # it contributes a term proportional to _#3/2 to the free energy which is clear from a dimensional analysis of (43). If # is identified with ~2 o (39) this means a negative third order term in the superconducting order parameter indicating that the superconducting transition even in zero field is (weakly) of first order, due to the fluctuations of the gauge field. This fact is well known, and can be made rigorous [20, 22]; it also holds in the coupled superconductor magnet system [9a]. Note, however, that the crude argument based on (43) breaks down in the presence of anisotropies if these also confine the fluctuating inner fields to a lower dimensional subspace. As we shall see shortly also the more rigorous RG treatment in this case avoids the runaway behaviour typical of first order transitions. Another effect of the elimination of the gauge field is the screening of the magnetic interactions, which we have discussed in the foregoing chapter already. There we were lead to the screened susceptibility (23) for the transverse components of the aa-field. Recalling that

~' = c~• + 4~ g2 = ~• -t- g2/r (44)

and that # corresponds to rJ~/72, we can see now from (42) that elimination of the A• of free- dom leads to the identical result for )C,. It is also instructive to write down the A-susceptibility (which is identical to the B-susceptibility up to a factor q2). From (41) and (42) we get:

Za(q) = <W(q)A~(- q)>~eff g2 v~q4 ]_1

= [ # + ( r - ~ ) q2-t o( c(+7oq2[

= # - ~ / ~ + r q 2 - t 7~ O:'q-Yaq 2] " (45a)

Each of the two different forms for ZA is appropriate for one of the two different Lifshitz points we consider. We may also rewrite the 0.-susceptibility 7~ in the corresponding forms:

z• • •

= [~'+ @-7)q2+ e # + r q e j g2 ge # -1

= [ c ( - ~ - + , o q 2 q - r # + r q 2 ] . (45b)

Table 1. Behaviour of Ginzburg Landau coefficients and suscepti- bilities at Lifshitz point (LP) and modified Lifshitz point (MLP) respectively

LP MLP

F ~)eff = 7~ - - g 2 / / / ~ 0 Fef f = r - - g2/6{! = ~ C(• ~ 0

Z - 1 4, ___~0 ~ • ~q tq ) X~l~q4(q ~0)

The symmetry (#,r,A) (e,7~,0"• is apparent. More importantly, this symmetry is supplemented by the correspondence MLP~-+LP. To see this we recall that near the LP e ' ~ 0 whereas the condition l/)~L~ 1 reads 7~-g2/# ~0 since in the absence of 0- fluctuations we may identify # with (4~/~L)2--1 ~ i//02 :t=0. Thus the first form of )/• in (45b) displays the typical LP behaviour with vanishing stiffness and dominating fourth order term. Exactly the same behaviour is exhibited by ZA at the MLP since there the transition temperature is close to Tmo , or equivalently c q = ( r - g 2 / c ( ) . c ( / r ~ O , and # ~ ( t p 2 ) ~ 0 , see the first form in (45a). Conversely, the behaviour which X• displays at the MLP corresponds to the behaviour ZA shows at the LP, see the second form of (45a, b). The correspondence is summarized in Table 1. This close analogy may justify the use of the term "modified" Lifshitz point. Having assured that we can relate to the same mean field picture as outlined in Sect. II we can now proceed to discuss the effect of 0.-fluctuations. It is well known that they drastically affect the stability of a modulated phase [15, 9b] in an isotropic system. The argument probably dates back to Brazovskii [26] and was raised in connection with the phase transition in cholesteric liquid crystals. For clarity we will rephrase the reasoning given in [9b]. We concentrate on the transverse part of the field 0., since it contains the dominant fluctuations, and de- compose it according to:

0-=0-0+ ~ 0-• (46) q ~ 0

Next we decouple the quartic term in 0. in (37) in a self consistent fashion, similar to (36):

(0"2) 2 ~ 2 < 0 " 2 > 0 -2 - - < 0 " 2 > 2. ( 4 7 )

The effect we are interested in is the renormalization of Z21 by the first summand of (47); it amounts to the replacement

c~' --+ c~ = e ~ ' + 2 fl ( 0"g + A ) - do --T - Tm eff (48) r m e f f


in (42). A has to be calculated selfconsistently from

1 ~ ( ~ • 1 7 7 A=V~ol q ,o

1 [ g2q2 1-1 - V o l q~0 d+yaq2 ]A+rq 2] " (49)

The point is that ( a •177 becomes singular on a whole spherical shell given by (17) at the effective "spiral" temperature (see (21)):

Tm elf -- Tmeff (1 + (1 - ] ~ a / g 2 ) 2 g2/(r do) ). o d - -

As a consequence the integral on the r.h.s, of (49) diverges like eff -- 1/2 [T-T~oa] for T close to Teff This ~mod" is inconsistent with (48), unless Tm e l f = - - 0 0 . Thus in the isotropic system the stability of phase II is suppressed by fluctuations. Beside the corrections of this result due to the self-consistency condition (47), anisotropies would change it qualitatively. This can easily be understood by noticing that the integral in (49) would be finite if the region in q-space where the singularity occurs, shrinks to a point. This is expected for example for a spiral phase II in a material with an easy plane of magnetisation which fixes the direction of the spiral energetically and thus does not allow for the long ranged directional fluctuations leading to the singular behaviour. A qualitatively similar effect could result from boundary conditions, if the material exists in form of small crystallites (powder). This concludes our investigation of fluctuation effects. We have seen that for anisotropic systems fluctuations do not alter drastically the mean field picture outlined in Sect. II. In a RG treatment of the functional (1) we may therefore expect the existence of fixed points corresponding to the LP and the MLP. This expectation is confirmed in the following two sections.

IV. RG Analysis for the LP

We now turn to the critical properties at the possible Lifshitz-points. For this purpose a renormalization group analysis with e-expansion can be used. However, certain changes in the standard procedure [21] are necessary in order to account for the inhomogeneous nature of the states involved [10]. Ad- ditionally, the gauge coupling, if not treated proper- ly, could cause problems [9b, 22]. In the following we give a short summary of the technique used. Whereas in the present section scaling arguments and some knowledge about the interactions generated in the course of renormalization will be sufficient

to map the situation onto the standard one found in magnetic systems [12], a more detailed study of the renormalization group equations is necessary in case of the modified LP, which we will give in Sect. V. As a guide for a general formulation we use a particular situation which may occur in d=3 dimensions, e.g. in the case of ErRh4B 4. There the easy plane of magnetisation introduces one definite direction, say %, in which the instability can take place, i.e. %11%. All wave vectors q which are parallel to e x span a re=l-dimensional space No, the "space of instable wavevectors". The ( d - m = 2)-dimensional or- thogonal complement of N o is called Nb- In a hy- pothetical spiral coexistent phase for example Nb would contain the vectors of magnetic induction B and spin density a. It turns out to be convenient to use a nonstandard gauge condition for the vector potential A adapted to the geometry under consideration. If P ~ x denotes the projector onto the subspace No, then one demands:

(_po. V ) A - Va.A=0. (50)

As a consequence, the Fourier-transformed field can be expressed in terms of the magnetic induction in a simple way:

qO x B(q) (51) A(q) = i qO2

It is obvious from (51) that the gauge transformation (50) is singular on the manifold r i.e. for qa=0. The corresponding states however have zero weight in the partition function. The generalisation to more dimensions d and to different decompositions lRe=~a | of the wave vector space into an m-dimensional subspace of instable wave vectors and its (d-m)-dimensional ortho- gonal complement is straightforward. One keeps (50) with pa being the projector on ~a and defines the vector product in (51) only in the sense that a unique prescription is given how contractions have to be performed in the renormalization procedure [9a]. If the complex superconducting order parameter ~ is written as a p-dimensional real vector O=(ReO, Im ~, ...) (p=2 will always be kept, although genera- lizations to p > 2 may be of interest), the Fourier- transformed functional (1) may be written as [-9a]

~ {~ ,B , a} =~o + V~A+ V~2A2 + V~4+ V~4

where:

(52)

1 T 1 ~o = ~ ~ {O(q) Xo(q)- ~k( - q) + a(q)rX~(q)-i a( _ q) q

+ B(q)r X~(q)-~ B( - q) - 2 g ~(q)B( - q)

B. Schuh and N. Grewe: Lifshitz-Point Behaviour of Ferromagnetic Superconductors

W + ~ (q- B(q))(q. B( - q))} (53 a)

= 1 y, {qt(q)rX0(q)~O( -- q) + a(q)TX ~(q) ~( -- q) q

+ l~(q) ~ X: ~(q)B( - q)}, (53 b)

qa V~A=y0e y , (q ,_ �89 x B ( - - q )

q,q' qa2

�9 O(q - q,)r[ _ ia , ] O(q'), (54a)

V 1 2 qa x B(q) qa, x B(q') 0 2 A Z = 2 ~ o e E qa,2

q,q,,q,, q az

�9 ~9(q). ~O(-q - q ' - q"), (54b)

Vq,,=�89 ~ 0 ( q ) ' 0 ( q ' ) 0 ( q ' ) ' 0 ( - q - q ' - q " ) , (54c) q,q ' ,q"

V~=�89 ~, t r ( q ) . g ( q ' ) ~ ( q " ) . ~ ( - q - q ' - q ' ) . (54d) q' ,q ,q"

In a graphical language the different per turbat ions

-1o) are as shown in Fig. 6. We use - i ~ = and

have in t roduced an addit ional term into ~o which guarantees that B is a transversal field if the limit w ~ o o is performed. The coefficient functions in ~o are given in their general form by: (Xi(q) -1 =- zi(q)-i 1, i = O, ~, B)

a a 2 b b 2 a 4 zo(q)-~=a+voq +Toq +Foq (55a) za(q)-i , a a 2 b b 2 a 4 =c~+7~q +7~q +F~q (55b)

x~(q)-t = r + a a 2 7nq + ]:bBqba-'[-FBqa4. (55C)

In diagonalizing ~o (see (53b)) the field B has been t ransformed according to l ~ ( q ) = B ( q ) - g [ z ~ l ( q ) l +wPq]-1 . a(q) and the corresponding new coefficien~ functions are XT~(q) - l = z ~ l ( q ) l + w _ P q and ~ ( q ) - i =z~(q)-~l_g2~7~(q) . pq__qoq/~2 i s the projector onto the direct ion of q . A renormal izat ion step con- sists as usual of an integrat ion of certain contracted

a) b)

d~

Fig. 6. Graphical representation of interaction terms of the functional (54) which are treated as perturbations

159

terms in a per turbat ion expansion of the par t i t ion function

Z = S 9 0 j" ~ B S ~ a exp[ - (~o + V)] (56)

over a shell A)t21<lq"l<A, A2yl<[qb[<A (A is a cutoff and will be set equal to 1), an enlargement of the remaining sphere of q-vectors to the old size, and a rescaling of all fields q0(q~,qb)=r AS a consequence of the inhomogeneous si tuation considered, there are different scaling factors 2~, 2 b in the subspaces N a and N b. For Lifshitz-points, where the stiffness 7 ~ along subspace N o vanishes, one has to fix the coefficients yb and F of the qb2 and qa4-parts in J~o both to order 1 which implies the following scaling:

qa__>qa,=)~aqa ' qb___~qb,=)%qb, ,~b = ,~a2-}-O(~). (57)

Here e = d c - d is the deviat ion from the upper critical dimension de, which depends on the parameter m :

dc(m ) = 4 + m/2 (58)

[10]. This is an immediate consequence of the Ginz- burg criterion [27] and the form of the a-cor- relat ion function at the LP: X2*~qb2+qQ Strictly speaking, a dimensional expansion can be made a round any point (too, d~(mo) ) on the line (58) in the two parameters e, = m o - m and e b = d~(mo) - m o - (d -m) , but to first order in ea, eb only the com- binat ion e = eo/2 + eb appears [-12]. The contract ions are expectat ion values (with regard to Yo) of pro- ducts of fields, which are of the form [23]:

QP(q) ~ ~P'(q'))Jo = (2z0 d 3(q + q')I~; ~ < ]qal < 1

" Ix~-i < ]qb I < a G,q,,(q). (59)

Here I . . . is the characteristic function for the interval indicated and the matr ix functions G for the interesting cases are in the limit w ~ oo:

Goq,(q) = zo(q) lp (60 a)

z B ( q ) - i Goo(q) = z~(q)-i zB(q)-~ _g2 ( 1 - Pq) + z~(q)Pq (60b)

g (1 -Pq) (60c) G B * ( q ) = Z , , ( q ) - i Z B ( q ) - I _ g2 _ _

z~(q)-i G~B(q) = z~(q) -~ zB(q)-I _ g2 (1 -_Pq). (60d)

We note in passing that according to (60b) only the transverse components of a are affected by the presence of the (transverse) field B. We encountered this fact in the last section already.


Up to now the considerations have been quite general. We now turn our attention specifically to the LP. According to the discussion in the foregoing sections we may have the following expectations: (1) the critical temperature is in the neighbourhood of T,~o, i.e. we expect the fixed point value of c( to be of order e. Furthermore, the critical susceptibility will be dominated by the qb2 and q"%terms. We therefore expect the corresponding coupling constants to be of order 1. (2) We are far from the superconducting transition temperature T,, which means we may neglect critical fluctuations in ~ and expect a * = O(1). (3) As a consequence of (2) the vector potential acquires a mass # ~ 2 , which can be considered to be a manifestation of the Meigner effect. Point (1) now suggests the following choice of rescaling factors for the fields a and B:

( 2 _ _ ] m ] d - - m } 2 - - ~ l ~ ff2 ) m ) d - - m ] 2 - - r l B (61) (7 - - "~a "~b "~b b B = "~a "~b "~b

where the choice for (B is motivated by the intimate connection between the two fields. Point (2) leads to:

2 - - ,]m,]d --m ]. --tb# (62) ~O - - *~a "~b "~a

so that the mass a is of order 1 and the interaction Vo, irrelevant. The fixed-point propagator of the field ~ then is:

_ 1 1 G~o(q) - ~ =p = O(1). (63)

From the gauge couplings ViA and V~,2A~ there are two contributions to first order in e which renormal- ize the part of the functional ~ which is quadratic in the magnetic induction B. These are shown in Figs. 7a and 7b. After an appropriate mass renormalization [22, 9a] the two contributions cancel up to a term of order a*.~Z~ (see also (75) in the following section). In contrast to the situation in the pure superconductor (or more general in the vicinity of the T~-transition line, where a*=O(e)) we now have (63) due to the absence of critical fluctuations in ~b. Therefore a vector potential mass is im- mediately generated in the course of the renormalization according to:

# ' = 2 ~ ( 22-''2B , b a+m,t#+ Cue2)=26+~ C~, e2) (64)

a) r . . . . 7 b)

x_ .... j j

A 2 A 2

Fig. 7. The two graphs contributing to a vector potential mass

where Cu is a positive constant [29] and (61), (62) have been used in the second equality sign. Initially C,=#0 and # = 0 , eventually C ~ 0 (since it is proportional to 7~) but # + m. In fact # * = oQ is the only stable fixed point of (64). The mass-term for the vector-potential changes the propagator of the B- field, since one has to replace

zB(q)-1 __+zB(q)- 1 + ~ ( 1 - Pqa) (65)

and therefore

~/~(q) -- 1 = 0~B(q) -- 1 ~- q~--~-) (_1 - - g )

+ (w + zB(q)'l)Pqa + (w + zB(q)-l +q~--j ) Pqb.

In the limit w ~ o e one now obtains instead of (60):

zB(q)-i + #/q~ G~(q) = z~(q)-i (z,(q)-i + #/q~:) _ g2 (1_ --P_q)

+ z~(q)Pq (66a)

g Gs~(q)=z~(q)_l(zB(q)_l+#/q~)_g 2 (_1 -- Pq) (66b)

z~(q) -1 GBB(q) =z~(q)-l(z~(q)_~ + #/qa2)_g2 (1 -- Pq). (66c)

In order to determine the form of these progagators at criticality, it is necessary to inspect the behaviour of the constants #, g and those contained in Z~, XB near the corresponding fixed point. From (64) it is clear that # ~ 2 6 ' where t is the number of iterations. The coefficient r of the magnetic field energy renormalizes according to

t 2 - -m - - d + m F = ( B ) L a Ab (r+C)=2~+~ ( 6 7 )

where initially C = 0 and therefore r ~24t at the stable fixed point [23]. The higher coefficients in 2171, if generated initially, scale in a similar way, but with even lower powers of 2 a. Since there are no contributions to 7~, 7b,, F~ from the four-vertices to first order in e, one finds:

a, - 2 y 2 } - m ~ - d + m , ~ , a _ _ ) 2 + O ( s ) , ~ , a (68a)

yb, _ ~o(e) @ , = )o(~) F - - ' - a --a, /'~ ..~ _~. (68b)

a 2t b* This means 7 ~ 2 a and 7~ , F~*=O(1). Finally, the nondiagonal coefficient g obeys a RG-equation of the form

g'=(B(a2am2~d+m(g+D)=A4a+O(~)(g+D), (69)

where initially D=0. Hence g~2~ behaves like r [23].


As a result, the asymptotic form of the functions appearing in (66) at the fixed point is:

1 # -1 qaZ 0~,(q)- + ~ ) ~ - ( 1 - ~ q a 2 ) , (70a)

~ a a 2 b b 2 ~ a 4 ~7~q +~'~q +F~q . (70b)

The new constants are given by:

~a = ya __ g2/#, (71 a)

/~ -- F~ + rg2/# 2 , (71b)

and are readily identified as the renormalized coun- terparts of the mean field coefficients emerging from (25). Comparing directly with an expansion of (23) in powers of q one sees that (71) defines a renormalized screening length 2/~ via #-1=4rc2~. The condition for the Lifshitz-point is

c,'* = 0(~), ~a. =0, (72)

which sorts out one (or possibly several) isolated points on the curve Tm(X ). Notice that g Z / # ~ 2 2 t 7a and rg2/# 2 ~ 2 ~ ~F~ asymptotically at the fixed point, so that a compensation of the two terms in ~a is possible. A consequence of (65) and (67) is that the fixed point propagators come out to be:

1 G*~, (q) = )~o(q) Pq = ~ Pq = 0 (73 a)

1 - 1 . G*~(q) = z~(q) -~ - gZ()~,(q)-~ + #/q.~)-~

1-e~ b, b~ +F,qa" . (73b)

7~ q

G~(q) = 0 = G~B(q). (73 c)

This result has a clear physical interpretation: The longitudinal component of the spin density field has no reduced stiffness and cannot take part in the phase transition to the inhomogeneous state. There- fore it exhibits no critical fluctuations at the fixed point considered and its propagator vanishes. Quite on the contrary, the transversal part contains all the physics considered and has a propagator of order 1. The fact that the mass coefficients # and r of the vector potential and the magnetic induction diverge, or likewise that the B-field does not propagate at the fixed point according to (73c) is a consequence of the Meil3ner effect which is present for ~bo>0. It suppresses all long ranged variations of the vector potential and thereby fixes (in the gauge used here)

its fixed point value A*=0. The influence of the magnetic interactions on the spin density field at criticality - as is apparent from (73) - is solely to provide the instability in the transverse part. The picture emerging out of the foregoing discussion is that the subsystem of localized spins which in- teract via the exchange interaction only (the magnetic interactions are screened) can be treated sep- arately at the Lifshitz-point. There is, however, a shift in the transition temperature (corresponding to the shift from Tmo to T,~ o in mean field theory) caused by the Meissner-effect, the instability occurs in the transverse component of o- only (for q6N,), and the critical behaviour is possibly modified through couplings of the type ~r3B, a2B 2, ..., which are mediated by the field ~ in the early steps of the RG-procedure [23]. In principle, one can now proceed in the framework of conventional Lifshitz- point-theory for magnetic systems and transfer known results E12] from there.

V. RG Analysis for the MLP

For the MLP there are two important differences to the situation at the LP. We recall these from Ta- ble 1. First the transition temperature in mean field theory is T,,o instead of T,~o. As a consequence we have e• and we expect e' to behave like (see (44)):

Or' = 0(~) + g2 /r. (74)

Secondly we have fluctuations in the superconducting order parameter, since the transition temperature is also close to T~. We must therefore expect a* =O(e) at least. Another consequence is that the vector potential mass # vanishes at the MLP. This is clear from (39) already and follows directly from its recursion relation (64), since C, ~ a. To see this consider the contribution of the two graphs in Fig. 7 to the q-dependent mass of the vector potential for an isotropic system in d dimensions; they add up to

1 {1 2 Top 2 (~rnA(q)=lyoeZ. 2 ~ p a+70p 2 \ d a+T0lp+q] 2]

2 a+7op 2 ] =�89 a+~,p2 (1 d a+,ol~+~q,2]

2 1 1 + 7oeZ~a~ �9 . (75)

a+?op 2 a+yolp+q[ 2

The first term for q = 0 is cancelled b y the mass renormalization which becomes necessary since the finite cutoff formulation used here is incompatible


0-2 ~2

a)

o..2A 2 0-2 B 2

o-3B

Fig. 9. a Graph contributing to the renormalization of 70. b Equa- tion determining the four-vertex-coupling Y0 e2

b) c)

B 2 a'B

Fig. 8. Course of the generation of couplings of the type a2B 2 and a. B during the RG iteration

with gauge invariance [9 a, 22]. The remaining terms contribute to the q-dependence of Zff ~. In particular the quantity C u in (64) is given by the second term for q = 0 if 70 is replaced by 7~, d by m and p2 by p~, as will become clear shortly. As indicated before it is of order eZ.a and may be neglected if a = o ( ~ ) .

Thus we have two critical masses at the MLP, namely y (or equivalently a) and c~• Before we proceed to explore the consequences of this fact it will be help- ful to point out how we can expect the MLP to show up in a RG-treatment at all. For this reason we go back to our mean field treatment and the a~- correlation function, (23). Apparently an essential feature of the MLP is the nonanalyticity of Z~ as a function of (I/,~L) 2 and q in the neighbourhood of q =0 and (1/2• 2=0. We must not expect such non- analytical behaviour to show up in a perturbational treatment like the e-expansion. In other words, to grasp the essential feature of the MLP we would have to sum up all orders in e 2, since (l/2L)Z~e2r which in turn is proportional to the critical mass y. What we may expect to identify in the RG-treatment is the behaviour of Z• in two limiting cases: (a) first g ~ 0 (i.e. 2L~O0), then q~0 , (b) first q~0 , then y ~ 0 . In case (a) the a-correlation function displays the behaviour characteristic of a usual critical point with the critical temperature given by e• In case (b) the behaviour given in (26) is to be expected, i.e. the effective coupling constant of the q2-term diverges to - o % whereas the effective coupling constant of the q4_ term diverges to + oo.

Let us further study the consequences of the fact that r is critical at the MLP. Clearly the choice (62) for the rescaling factor ~0 is not possible in this case. On the other hand, the standard choice [31]

~2 - - q t n~d -m~ 2 -r/q, (76) 0--'~'~b "~

leads to a runaway behaviour of 7~ whose recursion relation is given by

7~' = 2 ; 2 ~ 2;'n 2b -a +m 7;(1 -- 4 C~ 7; e2). (77)

Here the second term arises from the contribution of the graph shown in Fig. 9a, and Cr is given explicitely by CT=Y'Z~(p).trG*B(p)/p "~. Since there is

p no external parameter left to fix 7~ =0 which would be a possible solution of (77), ?~--.o�9 if the choice (76) is made. Correspondingly the gauge coupling perturbations VjA and VO2A2 would diverge and r would not exhibit critical fluctuations since Goo~0 at criticality. We interpret this situation as being reminiscent of the first order nature of the transition which one encounters in the isotropic uniform case. However, the fact that we take into account nonuniform states by means of the scaling-ansatz (57) pro- vides us with a possibility to avoid the critical behaviour following from (76). Namely we can choose

~ 2 __ ] m ] d - - m )2--qq,, (78) 0 -- "~a "~b "~a '

With this choice (77) has a solution ?~=1, thus b fixing t/q, to order e 2. Furthermore (78) renders 70

and F~ irrelevant. The fixed point propagator is then given by

1 1 1 U 5.

Since the recursion relation for b is of the form

(79)

br __ ~'4~-- 3m]-- 3(d--m)[l, ~ ~_ . -- bO"~a "~b ku ~ . .)

2 ( 4 + m / 2 - - d ) - - 4 + O(~:) =2 . (b+.. .) ,

B. Schuh and N. Orewe: Lifshitz-Point Behaviour of Ferromagnetic Superconductors 163

b is also rendered irrelevant by the choice (78), [29]. This does, however, not affect the criticality of ~b since a 0-mass renormalization is still achieved by the Voza~ vertex. Explicitely we have

a , = 2 2 rhp(a_l_ . 2 C~y~e ) (80)

(where C ~ = ~ t r * ~ GBB(p)/p ) with the fixed point so- p

lution (to order e 2)

42 Ca e 2 . (81)

We supplement the choice of scaling factors (,, (B in (61) by

2 2 - , _ ~ 4 - , ' b --'~ (11: G, 11B) (82)

which is the usual refinement of the scaling ansatz (57) and introduces four critical indices since each field q?=a, B has different correlations Qp(q)qff-q)) depending on whether the field components have a q-vector belonging to the subspace N, or Nb' We can now proceed to a discussion of the remaining recursion relations, and in particular the fixed point form of the propagators given by (66). The recursion relation for the charge e 2 c a n be deduced from the recursion relation for the coupling constant of the four vertex V~,~ao~: c5- " 2 ?oe illus- trated in Fig. 9b:

6' = Z 2p- ~* 6(1 - 4 C, 6) (83)

and the one for y~,, (77). The result is

(e2) '= 22Ve 2, (84a)

with p given by

p = 4 + m/2 - d + (m - 8)11B/4 - (m - 4) t/~/8 (84 b)

[30]. Since the mean field transition for the superconducting order parameter is shifted due to A- fluctuations, see (81), we must seek solutions 0 < e 2 * < ~ . Thus we can deduce from (84) d~=4 +m/2 and fix 11B, 11~ by (p=0):

= (m - 4)11~/8 - ( m - 8)11s/4 + 0(82). (85)

Since n o w [ ? = 0 0 3 2 ) the solution of (84a) including higher order terms in e 2 would yield a fixed point solution

e 2 . = 0 (82) . (86)

We may now safely ignore terms of order e 2 in our O(e)-calculation. In particular the contribution of the ViA-vertex to Z~ -1 mentioned in (75) may be neglect-

ed and the set of RG-equations for the couplings defined in (55b, c) reads:

~);' 2-t/• a ,gb' ~_,,~-2t/,# ~;b = h a 70 ' - e ' a ; o '

F~=22"; g (q0 = a,B), (87a)

/ _ _ 4--rtl 3 r - 2 , r, (87b)

(~')' = 24-'a(e ' + 2(n + 2)/n Ca,. fi), (87 c)

where n is the number of a-components and C=, = ~ tr G.r The equation for the a-B-coupling reads

P

g' =22+ 'B- '~ g. (88)

Finally the four vertex coupling fl obeys

fl' = 22p fi(1 - 2 C~/3) (89 a)

with Ce = ~ {4(tr G=o~) 2 + n(n + 4) tr G~o}/n z and P

p = 4 + m / 2 - d + (m - 8) 11o/4 - m/7"/8. (89 b)

First we note that the ratio rig may be fixed (to unity say) since

(r/g)' = 2, "B- "" r/g

leading to G=11B to order e 2. Then, fi'om (82), the /

same follows for 11'o and///3. This was to be expected from the likeness of the fields a and B. Next we notice that r, g and c~' tend to infinity with the same power of 2, so that the divergencies can cancel in the combination % = c ( - g 2 / r ; explicitely, from (87b, c) and (88) we get

~, _ 2 4 - .ag~, + 2 ( n + 2)/n C a, fl) ( 9 0 )

with the fixed point solution

2~ 2 n + 2 (91)

The fixed point fl* is extracted from (89a). As usual there are two solutions, a Gaussian and a Heisen- berg fixed point given by

fl* = 0, (92 a)

fi. lu2, p = ~ - . (92b)

The stability of the two fixed points is governed by p: if p > 0 the Heisenberg fixed point is stable, if p < 0 the Gaussian fixed point is stable. Let us now investigate whether the fixed point characterized by (81), (91), (86) and (92) can be associated with the MLP introduced in mean field theory in


section II. For this purpose we have to investigate the behaviour of the correlations functions given by (66). The explicit form of )~l=tr{(l_-Pq)Go~}/(n-1) is after using (55):

Z 21( q~, qb) = c( + y; q~ + Tb qb2 + Fo q~4

--g2/(r + #/q~+ y~q~ + 7~qb~ + FBq~4).

In complete analogy to the mean field form (45) this can be written as

Z2 1 = ~, _ g 2 / r + 7; q"~ + 7~ q bz -}- f f a q~"

g2 i/q_ 7~ q~ + 7~ q~ qb~ , (93)

r # + r q~

where we have used the fact that 7~/r, 7~/r, FB/r~O , g /r=l in the course of renormalization. We have not yet used #* =0, however, in order to keep track of the nonanalytic behaviour of Z• as discussed before. Expanding (93) further for small q~ we get

Z2 ~ = c( + (7; - gZ/#) q~ + 7~ qb~

+ (r~ + g~ 7;/(~#) - g~r/# ~) q~ +. . . a 2 b b 2 a 4

=~'q-Yeffq +7~q + ~ f f q " (94)

At the fixed point #* =0 the effective couplings 7~ff, F~ff equal - c o , + co respectively, with (Te,/F~f0* =0. This is exactly the behaviour described in (26) which we associate with the MLP. Formally, we can deduce the LP values for the coupling ]~ from (89a) by absorbing the (divergent) coupling F~ff into a redefinition of ft. Since C a in (89a) behaves like

'X 2 c p ~ j • 2+~-~)/2

we define fl=-fl.F~ 2+~-~)/2 and deduce the recursion relation

/~'= 2~ "-/~(1 - 2 C~-/~) (95)

r _ -n~- from (89a) and F~fr-2 . F~ff.

C~= lim {Fe(~+"-e)/2Cp} feff ~ 00

now is a nonzero regular function of c~'/Fef f. For the exponent/~ one obtains

= 4 + m / 2 - d + ( m - 8) t/J4. (96)

If t/~ = O(e2), which follows from 7b~ = 1, we get

} = 4 + m / 2 - d = e . (97)

The fixed points of (95) are given by (92) with /3 replaced by fl, Cr by Cp-and p by }. Thus the Lifshitz Heisenberg fixed point is stable for d<d~ and the Gauss• fixed point for d>dc, and the critical dimension is d~ = 4 + m/2.

Going back to the susceptibility (93) we can discuss the other limiting case #* =0. Z2 ~ then reduces to'

Z~ 1 = 6• -]- (7 a + g27~/r 2) q,2 + (7] + g27~/r 2) qb 2

+ (F~ + F B g2/r2) qa4. (98)

We can simplify (98) with g / r= 1, ~"b*--F*-- ~ ~ = F~* = 7~* = 0, which are the initial values, to get:

Z x- 1 = c~• + 7~, q,2 + 7~qb b 2 �9 (99)

From the recurs• relation for 7] it is clear that 7~-'co since initially 7~ This is exactly the behaviour corresponding to usual critical points in this description [10]. Again, formally we can see this by absorbing (7;) -a+(d-")/2 into the definition of/~. The recurs• relation for /~ remains unchanged, but for the exponent p we obtain instead of (97)"

2 } = 4 - d + ( m - 8 ) qo /2 - (dc -4 -m/2 ) t f J2 . (100)

Also the charge e z has to be redefined since the B- correlation function zB=trGBB/(n--1) at # * = 0 behaves like (see (66c))

z ; ~ = 6. r/~' + (7~ + g~ 7~/~' ~) q"~ + (7~ + 7~ g2/c(2) qb~ (101)

a __ a with a diverging coupling constant 7Beff=7~ -I- 7 a g2/~ , 2 Defining ~2 ~ 22(7~ e f f ) ( d - m - 2)/2 one gets

~2,~___ 22(/3+ 1) ~2 (102)

with exactly the same exponent } as given by (100). Thus at kt*=0 the theory describes a usual critical point with critical dimensionality d~=4 for the a- field and e2~o% indicating the importance of fluctuations in the vector potential which lead to the first order nature of the superconducting transition. Thus we arrive at the following picture. The MLP cannot be described as a usual critical point in a unique fashion by means of an expansion around its mean field behaviour. The physical reason for this is the presence of two simultaneously critical fields or, in other words, the presence of two competing correlation lengths. The ambiguity is apparent in the mean field picture already where different critical behaviour shows up depending on the choice of scale (set by the correlation lengths) on which fluctuations are considered. This corresponds to approaching the MLP from different regions in the phase diagram. Coming from a region where the intrinsic magnetic correlation length (corresponding to the critical mass 6• is much larger than the correlation length set by the superconducting fluctuations the behaviour displayed is essentially of


Lifshitz character. In the RG analysis this show up as a (to order e) decoupled fixed point (since e 2 = O(e2)) where the magnetic subsystem has the usual Lifshitz point FP (95), (97) with corresponding critical indices [10], and the superconducting order parameter displays mean field behaviour to order e, see (81). Correlations in the transverse a-field components display Lifshitz point behaviour (94). On the other hand, approaching the MLP from a region where the superconducting correlation length is comparatively large the magnetic response is of the usual ferromagnetic nature, the correlation function of the transverse field components behaves effec- tively like (99). The RG analysis is also complicated by the fact that in the absence of a magnetic subsystem the superconducting transition is weakly first order [-201, and also in the presence of a uniform magnetic order parameter [9a]. In the foregoing analysis the 22- divergence of the charge (102), (100) is thereby initiated.

VI. Summary

In this section we summarize our results and some predictions which emerge from this investigation. Our main concern are the possible equilibrium states and the critical behaviour of systems which can be described by a Ginzburg-Landau functional like (1). The essence of this functional is the simula- taneous presence of a superconducting order parameter ~ and a magnetic order parameter a coupled indirectly by electromagnetic interactions (a~--~B~--~A~--~p). It is widely believed that this free energy is of some relevance for the description of ferromagnetic superconductors. We shall not discuss the validity of this assumption here. We quote Er 1 _xHOxRh4B4 as a representative example for pseudoternary compounds which are both superconductors and ferromagnets, though in different parameter regions (temperature, concentration). Both parameters may be thought of as being comprised in the GL-coefficients a, 7 etc. Ad- ditional coexisting phases of superconducting and a modulated magnetic order are probably present in some of these compounds. This may well be understood on the basis of the electromagnetic coupling contained in (1): the presence of supercurrents leads to a screening of the exchange interaction between the localized moments on a length scale roughly given by London's penetration depth 2 r. This weakening of the magnetic interactions results in an instability of the spin system against fluctuations with a finite wavelength, and thus to the condensa-

tion of a spatially inhomogeneous magnetic order in the presence of superconductivity (phase II). A simple derivation of this statement on the basis of London's and Maxwell's equation is given in Sect. II. A number of possible states of the functional (1) are: NC-PM a normal conducting paramagnetic phase, SC-PM a superconducting paramagnetic phase, NC- FM normal conducting ferromagnetic, and phase II in all possible varieties like helical, spiral or si- nusoidal magnetic order. All these states may be found in a material like ErHoRh,B 4. Our main interest is for the possible merging points of these phases, i.e. the endpoints of the corresponding transition lines, and we have specifically raised the question under what circumstances these points can be Lifshitz points. Quite generally, a conventional Lifshitz point se- perates three phases: a disordered phase (SC-PM or NC-PM in our case) from two ordered phases: a phase with uniform magnetic order (NC-FM) and a modulated phase (phase II) characterized by a finite wave vector qo. An essential characteristic of a LP is the fact that q0 continuously goes to zero when the LP is approached from the phase II-segment of the order disorder transition line (ODOT). From mean field considerations in Sect. II the following picture emerges: in absence of any superconductivity ( 0 - 0 ) the ODOT T~,o (contained in c( = ~ ( T - T _ ' ] /T ' ] is shifted to a higher transition m o i l m o J

temperature Tmo by the coupling to the magnetic induction B, i.e. the interaction of the spin system with is own magnetic field. The influence of superconductivity is twofold: it reduces the aforementioned shift due to the Meissner effect; and it introduces a new length scale 2 L such that a new phase (II) becomes possible. The wavelength of the modulations of the magnetic order parameter in this phase is given by q2 = ( 1 - l/2L)/(12L) (see (17)), where l is a magnetic interaction length related to the stiffness 7~ in (1) by 4rclZ=7~. The transition into phase II occurs at Tmo d which lies between T,~ ~ and T,, 0 �9 Returning to the question of LP's we see from the expression for qo that basically two possibilities exist to achieve qo~0. The first is / / 2 r~ l . Since 2 L must be finite in this case this situation might be encountered in real materials at the reentrant transition line, if one assumes that the superconducting order parameter does not vanish continuously there. However, the mean field transition temperature Tmo d equals T'o at 1/2 L= 1; this means that the real transition temperature T m would have to be as low as T'o for a LP to be present in reality. For illustration see Fig. 5c, d. More probable appears the possibility


that phase II extends below the real transition line and becomes metastable (if it was stable at all) before the phases join the LP on T~'o, see Fig. 3. On the other hand, a drastic variation of 1/2 L with e.g. concentration is conceivable in particular in ErHoRh4B r where an additional competition of two different magnetic order parameters is present [9c]. One may also think of influencing both magnetic (1) as well as electronic (2L) parameters by alloying in a suitable way. The second possibility for diminishing the wavevector, which characterizes the modulated phase, continuously to zero is 1/2f~O. Obviously this occurs near a second order superconducting transition line (2c~ oo). The corresponding transition temperature Tmo a comes out to be T,, o. Therefore a possible candidate is the merging point of the superconducting (upper) transition line with the reentrant transition line and the magnetic O D O T separating the phases NC-PM and NC-FM, see Fig. 5. From the a-correlation function one can see that two different behaviours are displayed at this point. This is due to the fact that the length )~f set by superconductivity for spatial modulations in the magnetic order parameter is itself a correlation length which diverges at this point. An essentially LP behaviour is displayed by the susceptibility )~• (see (23)) if one looks for correlations on a length scale much larger than 2L, whereas usual ferromagnetic correlations dominate if q - 2 ~ 2 2 . We have termed this point 'modified Lifshitz point ' (MLP). In both cases (LP and MLP) the determination of the location of this point in the (T,x)-plane is a selfconsistent problem since 2 c depends on temperature via the superconducting density 0g. We give a simple, (28), and a more refined (32) ansatz for this dependence from which the location of the LP (MLP) can be calculated if all system parameters (represented by the GL-coefficients ~/0, V~, c% etc.) are known. We find that phase II merging in the MLP can be located as indicated in Fig. 5a. To study fluctuation effects we have generalized the mean field investigation by a selfconsistent decou- pling of all fourth order terms in Sect. III. There are two important fluctuation effects. The vector potential A can be decomposed into a mean field part and a fluctuating part Arl. Since the functional is Gaussian in Afl this field can be treated exactly by simply performing the functional inte- gration in the partition function. This gives rise to a contribution to the new effective free energy which - for constant superconducting order parameter 0 - is proportional to - 0 3 . It indicates that the superconducting transition is a fluctuation induced first order transition, also in the coupled system.

The second important fluctuation effect concerns the transition into phase II. If the correlation function )~• diverges on a whole spherical shell lql =q0 at Tmo d the renormalization of the transition temperature c~l~(T-Tmo)/Tmo by a-fluctuations - which is proportional to J'ddq)~• - renormalizes the transition temperature to - o o . Thus a-fluctuations would to- tally suppress the transition into phase II. Both fluctuation effects are strongly affected by the presence of anisotropies. If anisotropy is taken into account such that spatial variations of the magnetic order parameter are allowed in certain fixed direc- tions only both effects are considerably reduced. In the system ErHoRh4B 4 anisotropies exist which lead to an ordering of moments in the basal plane of the tetragonal unit cell on the Er-rich side and along the c-axis on the Ho-rich side. One may therefore expect that the nonuniform state is characterized by wave vectors with limited directional degeneracy. Having assured that the mean field picture is not qualitatively changed by fluctuations if anisotropies are taken into account in the way mentioned above we turn to a more complete treatment of fluctuations with an e-expansion around mean field theory. The existence of nonuniform states is taken into account - as is standard in LP literature - by a decomposition of d-dimensional space into two subspaces of dimension rn and d - m respectively and an anisotropic scaling of the corresponding lengths. This ansatz also takes into account the anisotropy necessary to prevent the aforementioned fluctuation effects, as long as d ~ m. The renormalization group analysis (RG) of the LP (Sect. IV) which is located in the neighbourhood of the reentrant transition line is straightforward. Since the superconducting order parameter is assumed not to be critical the functional is basically mapped onto a purely magnetic functional corresponding to a LP in dc=4+m/2 dimensions. The critical properties of such a point have been discussed in the literature. The RG analysis of the MLP (Sect. V) is more complicated. Since already the mean field picture of this point displays two completely different kinds of critical behaviour we must not expect the MLP to be describable as a unique fixed point. The presence of two critical fields interacting via the gauge coupling leads to the existence of two critical masses or likewise two differently diverging correlation lengths; thus depending on the scale on which fluctuations are considered one arrives at two different asymptotic characterizations of the MLP: if the magnetic correlation length is assumed to be much larger than the superconducting correlation length the fixed point is essentially a LP for the magnetic subsystem in 4+m/2 dimensions; the superconduct-


ing order parameter behaves mean field like to order e and is only weakly coupled to the magnetic system. In the other limiting case the functional is mapped onto an ordinary critical point for the magnetic subsystem in d = 4 dimensions, but with a runaway behaviour in the gauge coupling charge e indicating the first order nature of the superconducting transition. Thus we have shown that the electromagnetically coupled superconductor-magnet system can exhibit superconductivity induced Lifshitz point behaviour. It would be very interesting to test for this sort of behaviour experimentally by paying specific attention to the magnetic properties of reentrant ferromagnetic superconductors in the neighbourhood of the reentrant transition line. In particular the merging point of the reentrant transition line with the (upper) superconducting transition line and the normalcon- ducting paramagnetic-ferromagnetic phase boundary should always display the behaviour which we call modified LP behaviour. The existence of a modulated magnetic phase at this point - even if metastable - may be expected to show up in e.g. neutron scattering experiments. We would like to suggest that such studies on e.g. DyRuRh4B 4, GdErRh4B 4 or ErHoRh4B 4 are initiated in the future.

References

1. A recent experimental review: Maple, M.B.: In: Proceedings of the Indo-U.S. Conference on the Science and Technology of the Rare Earths, held March 3-8, 1980, Cochin, Kerala, India

2. A recent theoretical review: Fulde, P., Keller, J.: In: Springer Series: Topics in current physics, superconductivity in ternary compounds. Fischer, O., Maple, M.B. (eds.), Vol. B, Chap. 8. Berlin, Heidelberg, New York: Springer Verlag

3. Krey, U.: Int. J. Magnetism 3, 65 (1972); 4, 153 (1973) 4. Blount, E.I., Varma, C.M.: Phys. Rev. Lett. 42, 1079 (1979) 5. Kuper, C.G., Revzen, M., Ron, A.: Phys. ,Rev. Lett. 44, 154

(1980) 6. Matsumoto, H., Umezawa, H., Tachiki, M.: Solid State Com-

mun. 31, 157 (1979) 7, Greenside, H.S., Blount, E.I., Varma, C.M.: Phys. Rev. Lett.

46, 49 (1981) 8, Moncton, D.E., McWhan, D.B., Schmidt, P.H., Shirane, G.,

Thomlinson, W., Maple, M.B., MacKay, H.B., Woolf, L.D., Fisk, Z., Johnston, D.C.: Phys. Rev. Lett. 45, 2060 (1980)

9. Grewe, N., Schuh, B.: (a) Z. Phys. B - Condensed Matter 36, 89 (1979); (b) Phys. Rev. B22, 3183 (1980); (c) Solid State Commun. 37, 145 (1981)

10. Hornreich, R.M., Luban, M., Shtrikman, S.: Phys. Rev. Lett. 35, 1678 (1975)

11. Selke, W.: Z. Phys. B - Condensed Matter 29, 133 (1978) 12. Hornreich, R.M.: J. Mag. Magn. Mater. 15-18, 387 (1980) 13. In order to distinguish between the field B which may be a

fluctuating variable in the functional (1) and the equilibrium field satisfying the Ginzburg-Landau equations, the latter and its vector potential carry a subscript 0. B 0 is used here to express M~ through a in a mean field context

14. This line of reasoning apparently excludes formation of vortices or domains and can only be applied to describe transitions to states with slight modulations in 00, such as the spiral or the linearly polarized coexistent states

15. Kleinert, H.: Phys. Lett. 83A, 294 (1981) 16. Michelson, A.: Phys. Rev. B16, 577; 585 (1977) 17. Lynn, J.W., Shirane, G., Thomlinson, W., Shelton, R.N.: Phys.

Rev. Lett. 46, 368 (1981) 18. Michelson, A.: Phys. Rev. B16, 5121 (1977) 19. In the following we drop the fluctuations in the longitudinal

part AlE of the vector potential. They are connected with local fluctuations of the charge density p since ~i~ - 17.js~ V.A in the gauge used (and for constant 00.). These fluctuations are controlled by an electric field term in the free energy which we have not included explicitely

20. Halperin, B.I., Lubensky, T.C., Shang-keng Ma: Phys. Rev. Lett. 32, 292 (1974). See also Refs. 9a and 22

21. Wilson, K.G., Kogut, J.: Phys. Rev. 12C, 75 (1974) 22. Jing-Huei Chen, Lubensky, T.C., Nelson, D.R.: Phys. Rev.

B17, 4274 (1978) 23. In the course of the first renormalization steps couplings of

the type craB, aZB2, ... are generated since the propagator for the field ~b is q-dependent. A possible route to these couplings is shown in Fig. 8a. At criticality they are of order e and contribute also to the mass coefficients r and g. This is shown in Figs. 8b and 8c

24. Hu, C.-R., Ham, T.E.: Physica B + C 108, 1041 (1981) 25. Odoni, W., Keller, G., Ott, H.R., Hamaker, H.C., Johnston,

P.C., Maple, M.B.: Physica B + C 108, 1227 (1981) 26. Brazovskii, S.A.: Soy. Phys. JETP 41, 85 (1975) 27. Amit, D.J.: J. Phys. C7, 3369 (1974) 28. We drop the longitudinal fluctuations at this point since these

are largely suppressed by electrostatic forces which are not contained in our description explicitely; see also [-19]

29. Unless the upper critical dimensionality were given by dc=4 +m, which is not the case as shown later

30. Due to gauge invariance the relation for the three vertex V~A leads to the same result

31. We refer to (76) as "standard" choice because for m ~ 0 it 2 d+ 2--~4~ yields the standard rescaling G0-2 b

32. Comparing the two terms a02 and 7oe2OZA 2 in the free energy functional the Ginzburg criterion states that

(A2)~j'deqL4(q) must be of order a for mean field theory to

be valid. Since , u~O~a and (A2)~Id m/4-1 if y~=0 or

(AZ)~#d/4 1 if 7~=1 we get in both cases de=m=8

B. Schuh N. Grewe Institut ffir Theoretische Physik Universit~it zu K/51n Ziilpicher Strasse 77 D-5000 K61n 41 Federal Republic of Germany

Lifshitz-point behaviour of ferromagnetic superconductors

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