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Symmetry of the mixed state of superconductors with anisotropic pairing
PHY!llCA E
Physica C 206 ( 1993) 373-386 North-Holland
Symmetry of the mixed state of superconductors with anisotropic pairing I. Luk’yanchuk and M. Zhitomirsky L.D. Landau Institutefor Theoretical Physics, Kosygina St. 2, Moscow, 117940, Russian Federation Received 23 November 1992
The theory of mixed state symmetry of superconductors with anisotropic pairing is developed. Such a theory employs the symmetric Landau approach to the second order phase transitions for the case of the normal metal-superconductor transition in a magnetic field. It is shown that in most cases the Abrikosov vortex lattice near H., has the hexagonal structure. The transformation properties of such a lattice under translations and rotations are derived. From these the mixed state phase classification for unconventional superconductors is determined. The implication of such ideas for the phase transitions in UPt3 is discussed.
1. Introduction One of the most intriguing mion
physics
is the peculiar
features properties
of heavy-ferof the super-
state. Thermodynamic characteristics of it led one to hope that the superconducting gap can have zeros on the Fermi surface and therefore it should have the anisotropic form. Another confirmation of this anisotropy was done by the investigation of the superconducting phase transition in UPt, [ 1,2]. It was observed that the transition consists of two superconducting phase transitions with critical temperatures T,=O.S K, T,,=O.45 K. One of the most convincing explanations of such splitting is that the superconducting phases corresponding to T, and T,, have a different symmetry of the order parameter. To express this idea in more explicit terms we will present the superconducting order parameter A,,( r, r’ ) = ( ‘k(,( r) !Pp( r’ ) ) in the Wigner variables: Cooper pair coordinates with center R= (r+ r’ ) 12, and the position on the Fermi surface k, which can be obtained by the Fourier transformation r-r’ +k. If the superconducting state is uniform (i.e. the order parameter AaB(k, R ) does not depend on R ), the conducting
’ Present address: Institut fur Theoretische Physik, RWTHAachen, Templergraben 55, 5100 Aachen, Germany.
anisotropic behavior of the superconducting gap means that some nontrivial k-dependence of A,, occurs. The most natural way to formulate the phenomenological theory of the normal metal-unconventional superconductor phase transition is to discuss it in the frame of Landau theory of second order phase transitions. According to this approach [ 3,4] the appearance of superconductivity means breaking of the normal metal symmetry group which includes the following operations: U( 1 ), the gauge group of multiplication of electron wave functions Y with an arbitrary phase factor: Y+e’*YX, K, the point crystal group symmetry: in UPt, K=D6,,, for the case of an isotropic metal K= SO, x f (I^is space inversion); and &, time inversion. The residual symmetry group L defines the symmetry of the order parameter A&k). If such a breaking includes not only the U ( 1) elements, as in the case of the usual s-wave pairing, giving L a more complicated structure than KX R, the superconducting order parameter AaB(k) can be defined as an unconventional one. The important feature is that A&k) just below the phase transition can be expanded by the basis functions of irreducible representations (IR) of the group K. This gives the possibility to classify all possible types of pairing in accordance with the IR of the group K. In the case
0921-4534/93/%06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
374
I. Luk’yanchuk. h4. Zhitomirsky /Symmetry
of an isotropic crystal this fact leads to s-, p-, d-, ... wave superconductivity. A many-dimensional IR corresponds to multicomponent superconducting order parameters. On the other hand, the IR-classification strongly restricts the allowed residual groups L. Finally, L can be obtained by the minimization of the GinzburgLandau functional for the corresponding pairing type. This procedure was followed in refs. [ 31 and [ 41 for the crystal point groups K relevant to the heavy-fermion compounds, all the possible residual superconducting groups L having been enumerated. Another classification feature of d,,(k) is given by the space symmetry I^and can be obtained from the identity: &r(k) = -dm( -k) [ 31. It leads to the singlet: d,@(k) =d(k)iaY and triplet: d&k) = (d( k)a) i&’ types of superconductivity. One more interesting feature of UPt, is the T,-T, splitting behavior in a magnetic field. It is well established now [ 1 ] that the H-T phase diagram contains at least three different superconducting phases with phase transitions between them. These phase transitions meet at some thetacritical point which is also the kink point in the Hc2( T) dependence. Since the transitions seem to be phase transitions between different vortex state phases, it is interesting to construct the theory of the Abrikosov mixed state for the case of unconventional superconductors. There are two regions on the H-T phase diagram of the superconductor where significant simplifications are possible. In the first one, close to H,,, the starting point of the theory is the consideration of separate vortices. Surprisingly, it was found [ 5 ] that cores of the vortices in unconventional superconductors can have nonaxisymmetric form. However, until now it is unclear what type of lattice they do form. The second region lies near the upper critical field Hc2, where one could construct the theory beginning from solutions of the linearized GinzburgLandau equations. This consideration will be in the center of our attention. The most straightforward way to investigate the mixed state of the unconventional superconductors is to solve the GL equations for the nontrivial order parameter A&k, R) in a magnetic field and in such a way to generalize the Abrikosov vortex lattice theory. However, for the multicomponent order parameter such a procedure becomes much more
of the mixed state
complicated. Even solving linearized GL equations (an example of it for the p-wave pairing will be done in section 2 ) is not so simple [ 6,7 1. At the same time the procedure is strongly dependent on the coefficients in the GL equations, and so it is difficult to find all the possible solutions. Instead we will consider another approach based on the symmetry arguments. By such a treatment it was already shown [ 81 that the phase transitions in the mixed state of UPt, has a natural explanation as a structural phase transitions in the vortex lattice: distortions or/and period multiplication. Here we will fix our attention mainly on the symmetry properties of the lattice below Hc2. We will use the fact that the appearance of the Abrikosov vortex lattice near H,, ( T) is a second order phase transition which breaks the normal metal symmetry group. So, we will apply the above mentioned Landau phase transition theory and we will find the possible residual symmetry groups of the Abrikosov vortex lattices in unconventional superconductors. The approach will be outlined in detail in section 3. For now, we will stress only that the main difference between the phase classifications in an uniform state at H= 0 [ 3,4] and in a mixed state near Hc2 is the difference between the initial symmetry groups of the normal state since: ( 1) the magnetic field breaks the crystal point symmetry group; (2 ) in the mixed state the order parameter A,,( k, R) is no longer uniform and the initial group should involve also the translations in the magnetic field. Following the symmetry approach we will find the basis functions from which the order parameter A,,(k, R) near Hc2 should be constructed (section 4). In section 5 we will show that for the arbitrary pairing type the Abrikosov vortices form a hexagonal lattice at least in a wide region of the coefficients in the GL equations. This fact means that under translations and rotations by 60” the vortex structure A&k, R) does not change with the accuracy up to multiplication with some phase factors. These factors will be determined in sections 6 and 7 and therefore the symmetry of the lattice will be defined exactly. In the Conclusion (section 8 ) we summarize our results and discuss the connections between unconventional superconductivity at H=O and near K2.
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I. Luk’yanchuk, h4. Zhitomirsky /Symmetry of the mixed state
2. Ginzburg-Landau
,
theory for p-wave pairing near
2 HD
( 1+C)V2-
HCZ 2C(V,-iv,)’
X For clarity of presentation, we will consider the problem which has to be solved for the example of p-wave pairing in the isotropic metal (K= SO,xi) with strong spin-orbit coupling when the order parameter is defined as: d&k, R) = ioY[ n(R) xk]a. This is the simplest example of a superconducting state with a multicomponent order parameter. The GL functional for the vector order parameter q(R) can be written as F=A(
T- T,)rlrt*+a(rttl*)2+b’12rl*2
(1) The gradient Vi includes the vector-potential of the magnetic field A in the usual way: Vi= ai - i ( 2e/fic)Ai. In a H-T phase diagram there exist two regions interesting for us: ( 1) At fields H< H,, the order parameter A(k, R) is R-independent. The k-dependence of A(k) can be obtained from the uniform terms of eq. ( 1). For b < 0 it will be the arbitrarily directed real vector q= (A(T-T,)/2(a+b))“‘n where n2= 1; as discussed in the Introduction the residual symmetry group of A(k, R) breaks the initial SOS space symmetry: L= {C,, e%, 6,,, R}. Here C, are rotations by the angle @ around the n-direction; 6 and 6” are reflections in the planes perpendicular and parallel to n, ri being combined with the element ei” from U ( 1). If b> 0 one has the complex vector q= (A ( T- T,) / nT=n:=t, where nin2=0; 2a)L’2(n,+inz), L={e-‘“C,, 6, &,I?} (here n= [nlxn2]). (2) The region near the upper critical field which will be discussed in the article. The procedure of the calculation of Hc2 for ( 1) is standard: we neglect the fourth order terms a( q~*)~, bv2f2 and find the eigenvalues T-T, of the corresponding linear GL equation [ 7 ] :
0
2C(V,+iV,)’ (1+C)V2+
0
2e %HDO
0
V1
L
Here C= (K2+K3)/2K,, D= (K2-K3)/2K,; in eq. (2 ) we also assume that Va= 0. The “eigenvalues” A( T- T,) depend on the magnetic (eigen) field. We can choose from them the eigenlevel which has a largest value of H at a given T-T,. Such dependence defines the upper critical field Hc2( T). Below it there appears a nonzero solution of eq. (2). To find them, the nonlinear terms u(~q*)~, bq2q*2 should be taken into account. The solution can be constructed from the eigenfunctions of eq. (2 ) which in the case of the s-wave superconductor are the wave functions of a zero Landau level. In the mixed state the order parameter depends both on R and k. The candidates for the Hc2 solutions of eq. (2 ) are [71: A&k, R ) = iaYJZfO(R ) , polar phase , A,,(k, R)=iaY(JX-iJ,,)jdR),
(3a)
A phase,
(3b)
A,,(k,R)=iaY((J,+il,lfo(R) + (Jx - i.J,% CR) ), SK phase ,
(3c)
where J= [kxa]
.
corresponding eigenfields The 2mcA( T,- T)/ 1el hK,tliy where
are
Hi=
Ap=l, AA=l+C-D,
The solution with the largest eigenfield Hi does correspond to Hc2 = max (Hi). By changing K,, K2, KS all variants are possible.
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I. Luk’yanchuk, M. Zhitomirsky /Symmetry
The characteristic of the anisotropic pairing feature which can be seen from eq. (3~) is the nonfactorized dependence of A on k and R. Scharnberg and Klemm (SK) were the first to find this [ 91 by the solution of the Gorkov equations for p-wave pairing in the weak coupling limit. In section 4 we will show the symmetry reasons for such a form of A&k, R). Equations (2) are quite analogous to the corresponding equations in the model of the two-component Ginzburg-Landau functional with cylindric symmetry appropriate to UPT3 [ 71. Therefore, the analysis given below could be straightforwardly transferred also in this case. To solve the problem of the vortex lattice structure near Hc2 it is needed to take into account the nonlinear terms in the GL equations. This is not easily done straightforwardly. Even the numerical calculations encounter difficulties. Instead, we will use the symmetry approach which will allow us to speak about the properties of the mixed state of superconductors with anisotropic pairing in a qualitative fashion. We will outline this method in the next section.
3. Landau phase transition theory for the normal metal-superconductor transition in a magnetic field
The appearance of a superconducting vortex lattice in a magnetic field is a second order phase transition, and we will consider it in terms of the Landau phase transition theory. Such an approach gives the possibility to formulate the mixed state theory for an arbitrary pairing without considering a specific form of the nonlinear GL equations. In the Appendix we will consider the standard procedure of obtaining the Abrikosov vortex lattice solution for an s-wave superconductor as a particular case of this approach. Note, that the symmetry approach should also be valid in the low temperature region where the GL equations for A(k, R) are not applicable. So, in accordance with the Landau symmetry approach we will hold to the following program.
of the
3.1. Definition group G
mixed state of the hamiltonian
symmetry
The appearance of the vortex lattice means spontaneous breaking of the group symmetry G of the normal metal state in a magnetic field, G={U(
l), ‘ib, _&,, 6, r&l?, I),
(4)
where U( 1) is the gauge group, fa are Cooper pair magnetic translations on an arbitrary vector a; f_.A,,(k,
R)=A,,(k,
R+a)
exp(is&(R))
(5)
(for the magnetic field vector-potential we will use the Coulomb gauge: A(R)= [HXR]/Z). &, is a rotation around the magnetic field direction H. In the case of singlet pairing or triplet pairing with weak spin-orbital coupling, & includes rotations in the k and R spaces. If the pairing is a triplet one and the spin-orbital coupling is strong, & includes also rotation of the spin space. For simplicity we will only consider the first possibility and further we will omit the spin indices. The generalization for the triplet pairing with strong spin-orbital coupling can be done straightforwardly by the formal substitution: k-J= [kxa] as was done in the previous section. 6 and eH are the reflections in the planes perpendicular and parallel to magnetic field: &k,+ -k,, &Jk,+ik,,)-k,-ik,,. Since H is an axial vector, BH changes sign and b does not. To compensate the sign change the time inversion l? is combined with c?~. f is space inversion. 3.2. Order parameter
definition
The wave function of the superconducting vortex lattice state just below H,, should be constructed from the basis functions of a particular irreducible representation (IR) of the hamiltonian symmetry group G. The IRS of group G were found in ref. [ 10 1. Such IR basis functions are the eigenfunctions of the linearized GL equations.like (2 ). In general form the linearized equation for A(k, R) can be symbolically written as 2(H,
T)A(k,
In section
R)=O.
4 we will briefly
(6) outline
the IR classifl-
31-l
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
cation procedure and the properties of their basis functions. The IR basis functions are isomorphic to the Landau level wave functions, the zero Landau level corresponding to s-wave superconductivity. For other types of superconductivity other Landau levels are involved. 3.3. Vortex state symmetry
coordinate
space R= (x, y).
a=a,-ia,, a=a,+ia,.
(8)
Sometimes we will also use the complex variables the vector k: k= (k,, k,, kz) = (K, IC,k,) where
for
K=
To find the symmetry properties of the vortex lattice near Hc2 we need to know the coefficients in the expansion of A( k, R) over the basis functions of the corresponding IR. The coefficients can be defined by the minimization of the fourth order terms in the Landau functional which describes this transition. (The third order terms are forbidden by U( 1) symmetry; this is the reason why the normal metal-superconducting vortex lattice transition is a second order one.) However, due to infinite degeneracy of every IR of G (which is the analog of the infinite degeneracy of the Landau levels), the order parameter has an infinite number of components and the direct minimization is rather complicated. Some geometrical ideas presented in section 5 simplify this procedure and allow one to find the conditions at which A( k, R) has the structure of a hexagonal vortex lattice. We will use such a geometrical approach to obtain the transformation rules of A(k, R) under rotations of 60” and translations by the lattice periods. In sections 6 and 7 it will be shown that such rules depend on the IR to occur at H=H,,. The transformation properties define the symmetry group H of A(k, R). Obtaining H c G solves the problem of the normal metal-superconductor phase transition in a magnetic field and serves as further background of the phase transitions theory between different phases within the mixed state of the superconductor with unconventional pairing. To avoid overcomplicated notation we will use in this paper the dimensionless variables using f,,= (cYi/ 2eH) ‘I2 as a unit of length. It will be more convenient to use the complex variables
(k,+ik,)/2,
R= (k,-ik,)/2.
(9)
In some parts of this paper we will omit the nonessential variables in the order parameter A( K, R, k,,
z, 2).
4. Classification of the linearized GL equations solutions near Hc2: IR of the hamiltonian symmetry group The Abrikosov vortex lattice near H,, is formed from the eigenfunctions which belong to some eigenlevel of the linearized GL equations (6). This level is classified by the IR of the hamiltonian symmetry group G (4) which can be found if the commutations rules of operators in G are known. Since G contains the continuous operations T; and i,, the formal specification of such rules should be done by using the infinitesimal generators f?, [of i=Aand ~$1 F* =
eiJ5(Ti+
+A.?- ) ,
f_
= ei@f
9
where i+ = -i(a+z)/fi,
t^_ = -i(&-t)/JZ,
i=f,+f,=f,+za-3.
(7)
(10) (11)
The generators G, fR correspond to rotations of the k and R spaces around H. We treat t^+, f in the Coulomb gauge. Generalization to an arbitrary gauge is given in ref. [ 81. The symmetry properties of the eigenfunctions of eq. (6 ) are defined by the commutation rules of the generators i, , f: [t;,tl-]=l,
[i&,r]=+i,)
In particular, we can introduce the commutator t^+, f with the Casimir operator
z= (x+iy)/2, Z= (x-iy)/2,
for the two-dimensional In these variables
a=r+i_t;
)
(12) with
(13)
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I. Luk’yanchuk, M. Zhitomirsky /Symmetry
which enumerates the eigenlevels of the operator 2. More exactly: the eigenfunctions of2 should also be the eigenfunctions of the operator 3. Using the creation and annihilation operators of the Landau levels: 8+= -i(a-z)/Jz,
ii= -i(a+Z)/Jz,
(14)
[B,8+]=1;
(15)
we can write the operator2
in another
3=G+ci+(i.
form: (16)
This form is useful to find the general form of the eigenfunctions of the operator 2: dS(k,R)=
2 fn(R) N=rn-+ll
‘Y;(k),
(17)
where
s Y,,(k) are the spherical harmonics in k-space: (f,Y,,,(k)=mY,,,(k)) andf,(R) the nth Landau level wave functions (d +&= (n + 4 )fn). The wave functions dg (k, R) are characterized by the two quantum numbers: N= 0, + 1, IL 2, .... and (T= + . All the coefficients (Y,in eq. ( 17 ) should be real (or have the same phase); this property follows from the &J? symmetry from eq. (4). The number N generalizes the Landau level number n: when the wave function does not depend on the second variable k the N-level classification reduces to the usual Landau levels. This result is quite natural since Landau levels are also an IR of the group {T,, L,]. The quantum number rr defines the wave function parity under the operation 6 from G, eq. (4). This classification explains the nontrivial SK solution for the p-wave pairing (3~). The phases (3) are described by the (a, N) quantum numbers: Pphase (-, 0), A-phase (+, -l), SKphase (+, 1). Another property of the eigenstates of eq. (6) is the infinite degeneracy of the N-levels which is a consequence of the noncommutative group relations ( 12). This degeneracy is analogous to the degeneracy of the Landau levels. If some solution of eq. (6) is known, all other eigenfunctions of the same N-level can be obtained by applying the magnetic translation
of the mixed state
operators T;. Some possible ways to parametrization of the independent N-level eigenfunctions will be given in sections 5 and 6. One important thing which can be seen from eq. ( 17) is the overall pairing type in the Abrikosov lattice near Hc2. If the spin-orbital coupling is strong, the mixing of singlet and triplet superconductivity is possible. E.g. the phases (a,+ ia,)k,fi (R) ia,, (p-wave pairing) andfo (R )ia, have the same quantum numbers ( +, 0) and hence they are mixed in the vortex state (a,+ icr, changes sign under reflection in the xy plane since cr is an axial vector). Of course the pairing which occurs at H=O in the mixed state has the dominant amplitude. So, if in the uniform state two superconducting transitions do exist, the corresponding phases can be mixed in the Abrikosov state. From the group G generator properties ( 12) we can obtain relations between the elements i,, ?=Aof this group: ^,. ^^ L, TA= TxL4 , (18) where
This equation can be verified by the expansion of L,, Tl in a series and applying the commutation rules ( 12) for ?+ , f_ , f. Such relations are the same as the moving operation for the usual Euclidean plane (group E,). The difference between these two groups is in the noncommutative magnetic translations: ~J.-,2C~--&0~JJ~ The operators lations since [t^,,a+]=o,
d+, 6 commute
[?&]=O.
The commutations [&+,[I=-Li+,
(19) with magnetic
trans-
(20)
rules of Li+, ci with fare as follows: [fi+,f]=ri+.
(21)
Using eq. (2 1) we can also deduce the useful relations which will be used in the next sections: ~,B=e-i~B~,,~,ri+=ei~~i+%.
(22)
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
5. The vortex lattice in unconventional superconductors
magnetic translations by the lattice some phase factors: eiS, eiY:
The solutions of the CL equations near H,, for the s-wave superconductors were studied by Abrikosov who found that the superconducting order parameter A(R) forms a periodic lattice with one magnetic quantum flux per cell. The hexagonal structure of such a lattice is most stable. The purpose of this section is to show how this result can be generalized for an arbitrary pairing type. In accordance with the Landau phase transition theory, the appearance of superconductivity below H,, means a break of the initial hamiltonian symmetry group (4). It is quite natural to assume that at H< Hc2 A(k, R) also has a lattice structure. (Note that the mixed state (A(k, R) ) cannot have the laminar structure since, as was shown in ref. [ 111, such a symmetry is incompatible with the commutation rules for the magnetic translations. ) We will also assume one-flux quantization of the elementary lattice cell. The residual symmetry group of A(k, R) : H c G is defined by the elementary cell basis vectors a and b which satisfy the condition 1allb 1 sin CC=297 ( CY=U-b). So, the lattice can be determined by two independent parameters. We will introduce the complex parameter 7= ( )b I / Ia I )eh which completely determines the lattice since it also has two components. The parameters 7,~ i and 7,,= e2ri/6 correspond to square and hexagonal lattices. In the complex variables the basis vectors can be written as: a, 7u where ~=a(7)=(ix/(7-5))‘/~,
379
(23)
to satisfy the condition Iallb I sin (Y= 2x. To find what type of the lattice will occur we should find the solutions A(k, R, 7) of the nonlinear CL equations near HC2 for each 7 and then, after minimization of the free energy F(r), choose the most stable one. To avoid complicated calculations we will use symmetry arguments which generalize the ideas of ref. [ 121. Let us consider the order parameter as a function of 7: A=A(k, z, 7). The form A(k, z) also implies dependence on 2, unless explicit stated otherwise. The lattice symmetry of the order parameter means that A( k, z, 7) is an invariant function under
faA(k,
z, 7)=en(r-z)A(k,
z+a,
periods
7)=ei8A(k,
up to
z, 7) ,
f=,A(k, z, 7) =en(~Z-rZ) A(k, z+7u, 7)=eiYA(k,
z, 7). (24)
We will discuss the meaning of the factors eiS, eiy in the next section. Let us now introduce two groups of scaling transformations of the parameters K, z, 7: (1)
K-+K, z+z, 7+7+
(2)
lc+lCm7,
z-z-,
1; 7-+ - l/7.
(25)
It is possible to see that the functions A(k,, IC,z, 7+ 1 ), A(k,, K-, zfi, - l/7) are also the invariants under the same group of magnetic translations {pa,, pTa} with the same phase factors eia, eiy as A( k,, K, z, 7). It serves as the background of the following theorem: Adkz,
JGz, 7) =M7)&(kr,
&(k
K, z, 7) =&(7)&(kr,
K-3
K, z, 7+ 1) 3
zfi,
-l/7)
.
(26)
This theorem follows from the statement that AN( k,, K, z, 7) is the unique solution of eq. (6) which satisfies the condition $AN = NAN, or, in other words, from the wave functions of the given infinitely degenerate IR ( ?, N) of group G it is possible to construct only one function AN( k,, K, z, 7) which transforms like eq. (24) under pro, To,. The proof of this fact for the s-wave superconductor which correspond to IR ( + , 0) is given in the Appendix. Since the bases of the wave functions for another IR which are defined by the same operators t^+, fare isomorphic to the basis of ( +, 0) the statement is also valid for an arbitrary IR with quantum numbers ( ?, N). Properties (26) have a simple physical meaning [ 8,12 ] : we used the parameters (a, 7~) as the basis of the elementary cell of the vortex lattice. Transformations (25) imply other possibilities for the choice of basis: (a, 7u + a) and (7~2, -a). According to eq. (26) the wave function A,(k,, K, z, 7) should be invariant under such basis changes. The elementarycells (a, zu), (a, zu+u), (zu, -a) have the same areas.
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I. Luk’yanchuk, M. Zhitomirsky /Symmetry
More explicit expressions for the factors cN( r), &(r) will be given in section 7. From properties (26) it follows that not all the points of the complex plane T are independent. To define the nonequivalent points T it is sufficient to use the fundamental region r which is shown in fig. 1. All other points t could be obtained by the operations (25 ). Properties (26) leads to the following symmetry of F( r) at the r-plane:
(27) The last equation was obtained by the lattice basis inversion + complex conjugation of the real function F(r). Due to eq. (27) the symmetry points ri, and rs of the boundary of the fundamental region should be the stationary points (maximal, minimal or saddle) of F( r). Since near the point ri, the energy F(r) locally has triangular symmetry defined by eq. (27), the point r,, can only be either a local maximum or local minimum of the function F(r). In fig. 1 the lines with equal free energy F(T) for the s-wave superconductor near H,, are shown (see Appendix). The energy minimum corresponds to the points of the hexagonal vortex lattice rh. The saddle point of F(r) corresponds to the square lattice ‘Imr
of the mixed state
( z=T~). The point Im r-+ +03 obviously corresponds to the energy maximum. Turning to the case of unconventional superconductivity, we note that for every multicomponent GL functional there exists some special choice of coefficients under which the multicomponent nonlinear GL equations decouple to a number of more simple equations, each of them looking like a GL equation for an s-wave superconductor. E.g. at K2= - K3, a= - 2b, the GL equations for p-wave pairing originating from eq. ( 1) decouple to three such equations for the nX+iq,,, ~X-i~Y, Q components. For the above particular choice of coefficients the r-dependence of the energy is given by eq. ( 45 ) . Since the minimum in the hexagonal lattice point rh iS conditioned by the symmetry, this minimum should be stable under small variations of the coefficients in the multicomponent GL functional. This means that in the coefficients space there exist at least some region around the coefficients of special choice at which the lattice is hexagonal. The explicit form of F(r) for the SK-phase was given in ref. [ 13 1. The numerical analysis for such a case [ 141 shows that the region of coefficients were the hexagonal lattice is most stable is rather large. Only this case will be under our consideration. We can conclude also that the statement given in the previous papers [ 71 that the vortex lattice should always be nonhexagonal for any nontrivial pairing with multicomponent order parameter is incorrect.
6. Translation properties of A@, R)
Zn, =
~1
~112
U
112
e6
1
Fig. 1. Fundamental region r which parametrizes the vortex lattices with different lattice parameters. T, corresponds to the square lattice and r, to the hexagonal one. The distribution of equal energy lines F(r) is shown.
The transformation rules of A(k, R) near Hc2 under magnetic translations by the lattice periods a and Ta are defined by eq. (24). To understand the meaning of the phase factors eis and eiy, we will show that for each pair (8, y) from the different eigenfunctions corresponding to the degenerate level Nit is possible to choose wave functions that transform like eq. (24). Consider some eigenfunction AN(z, 7) which transforms like eq. (24) for definite (fi, y), for simplicity say ( --n, --n). Let us consider the function Al,,,(z, T) = f,d,(z, 7). Using eq. (19) we find the transformation rules:
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I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
=e2a(1-A)f+iFaA,(z, RA,N(Z?
7)=LmN(z,
7)~ -e2a(L--“)A5N(z,
a
7) ,
7)
M
Kl
wn
c-e
2a(d--d)AlN(z,
7) .
Solving the equations _e*a(r--1)=ei/3 , -e
2n(rT-fA)=eiy
9
ai2
M2
r
(28)
(29)
we can find the vector I for each pair (j?, y). Since magnetic translations commute with the hamiltonian, all the functions A&z, 7) also belong to the eigenlevel N. Applying any periodic potential with periods (a, 7~) removes this degeneration and splits the Nth eigenlevel into a band. The A&z,r) are the correct wave functions of the approximation of zero splitting. The transformation rules eq. (24) are the analog of the Bloch representation for a one electron wave function in a periodic potential, the pair (j?, y) usually defining the wave vector K in the Brillouin zone. Using the analogy with Bloch electrons we can introduce the Brillouin zone as a parametrization region of such functions: -A < j?< x, -A < y< n. In the L-plane it corresponds to some region defined by eq. (29 ). To find this region we use the equivalence of points A, 1+ a, A+ ra, which follows from eq. (29). The procedure is the same as for choosing the Brillouin zone in the K-space of the usual periodic crystal. For our purposes it will be convenient ‘to use as the Brillouin zone the WignerSeitz cell which has the same symmetry as the point group of the Abrikosov vortex lattice. The Brillouin zone for the hexagonal lattice coincides with the elementary cell of this lattice and is a hexagon of side a/$ (see fig. 2). The functions A,&z, 7) from the Brillouin zone for given T form the basis of the eigenfunctions of the degenerate N-level (see also ref. [ 15 ] ) since: ( 1) due to the different transformation rules these functions are orthogonal and hence they are linearly independent, and (2) all functions from the N-level can be obtained by the magnetic translations of some arbitrary function from this level. As we showed, every magnetic translation transforms the function Ar,N(z, T) to some other function A,&z, 7) in the Brillouin zone.
Fig. 2. The elementary cell of a hexagonal Abrikosov vortex lattice. It coincides with the Brillouin zone, defined in the text.
The operations from the hexagonal point group transform the wave function Ali,N(z, 7,,)to some other function which corresponds to another 1j of the Brillouin zone. All points lj defined in such a way form the star of vector Li-{;ij>. In a usual crystal the electron wave functions corresponding to the symmetrical points of the Brillouin zone have a so-called small symmetry group. The operations from the small group do not change the wave vector K of this function (or transform it to an equivalent one ) and multiply the wave function by some phase factor. Analogous symmetry properties of the wave functions A,,( z, T,,) for the symmetry points of the hexagonal Brillouin zone will be obtained in the next section. Now we will give the magnetic translation transformation rules for these points. The wave function AN(z, TV)zA~,~(z, zh) corresponding to the hexagon center r (A = 0) transforms as: ~~JN(z, ~&,4~,
71,)=-&(z, 7,,)= -&(z,
7,,),
(30)
7,,).
The wave functions corresponding alent hexagonal vertices transform point K,: A=a(z,,+ 1)/3:
faAK1(z,
-e-2irr/3AK,(z,
7h),
pTnAK1(z,q,)= -e2in’3AK1(z,
z,,) ;
7,,)=
point K2: L=a(2r,-
FoA,(z, fraA,(z,
7h)= TV) =
to the nonequivas
(3la)
1)/3:
-ezix/3AKZ(z, T,,), -e-2in/3AKZ(z,
TV) ;
(31b)
382
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
the wave functions of the midpoints lent hexagon sides transform as: point M,: I=az,,/2:
of nonequiva-
we will work with A,(z, T) which can be obtained fromA,(z,r)as:A,(~,r)=(B+)~A,(z,t).Usingeq. (22 ) we have A,(z,
~CIAW,(z, G) =A.w (z, G), ~wzAA4,(z, G) =AlcI,(z, %) >
(32a)
point M2: L = a/2:
r) = (c?+)~A~(z,
=~o(z)(d+)NAo(~,
z) 7+1)=&,(s)-AN(z,
AN(z, T)= (B+)NAo(~,
T)
=&,(~)(ii+)~A,,(z,,/%, (32b)
point MJ: I=a(
-l/r)
=&(r)(ri+)NL,&(z,
1 -q,)/2:
-l/r)
=&,(7)(~/i)""~&i+)~A,(z,
=&,(r)(~/s)~‘*A,(zJr/r, %7Ah4&
7h)
=&~(z,
%)
.
(32~)
In accordance with eq. (28 ), any magnetic translation of coordinates will change the rules (30). We chose the origin of coordinates in such a way as to have consistency with the s-wave superconductor Abrikosov lattice where the origin of coordinates is situated at the position of some vortex core and the magnetic translation rules of A(k, R) have the form (30a).
7. Point symmetry of A(k, R) In section 5 we proved that the Abrikosov nuclei lattice A(k, z, z) near Hc2 has the hexagonal structure. The translation properties of A(k, z, th) were obtained in section 6. However, to know the full symmetry group of the lattice one also needs to know the rotation properties of A(k, z, q,) around the symmetry axis of the hexagonal lattice. Here we will show that these properties depend on the starting quantum numbers N (defined in section 4) of the solutions of eq. (6) at H= Hc2. To find the A,(k, z, rh) transformation rules we should find first the factors iN( r), and MT) in eq. (26 ). The following formulas are valid: G(r)=
(rlVv’*&l(r)
.
(33)
To prove eq. (33) we note that the Abrikosov lattice wave function AN( z, z) of an s-wave superconductor corresponding to the Nth Landau level has the same symmetry properties as an arbitrary eigenfunction A,(k, z, 7) of eq. (6) with quantum number N. So,
-l/r)
-l/z),
where the angle @ is defined by: e pi@= ,/$. Specification of eq. (33) for the hexagonal lattice parameter ?h = e2in’3 gives the transformation rule for the hexagonal lattice A,(k,, IC, z, q,) when rotating by 2x16 around the center of the hexagonal elementary cell; ,$x,6AN(k,
Z,
T,,)=e2irr(N+1)‘6AN(k,
Z, rh)
.
(34)
Property (34) was obtained by two successive transformations T+ - l/7+ - 1/T+ 1 which leave the point TVimmovable and rotates the Z-K spaces: z, rc+e-2ia/6z, e-2ix% In eq. (34) we used the coefficients Co(r) =emin14, and L(r) = id/Jr which can be calculated from the explicit Abrikosov solution A,(z, 7) (see Appendix). The wave functions AKm,N(k, z, th) and A,,, (k, z, T,,), discussed in section 6, should also have some symmetry properties under transformations by small groups of the points K,,, and Mj in the Brillouin zone. Using eqs. (18), (32) and (34) we have points K,: m= 1, 2 (e.g. K,) L 2n/3 A KI,N- -i
2n/3
f
a(r+1)/3
AN
= Fa’a(r-I ,,3L2rr/3A~ =e2ia(N+1)'3~~f~_,),~AN =e2ix(N+1)/3F
Mr)=Co(r),
r+l)
E-e
a(r-1),3~-
47+1)/3fa%d~,,~
2ix’3e2i”‘N+“‘3~~;1(,_1)/3~_a(r+1)/3~~’adK,
.N’ )
the closed loop faa(+, j,3~-aCr+1 j,3F0 sweeps the magnetic flux - @,/6 which brings in the additional phase factor eezi*16. So, the transformation rule for AK,,N( bush)as well as for A,,,(?,) has the form
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
&&,,Ntk
z, ?I,)= e2i”N’3&,,Ntkz, %h);
(35)
points M; j= 1, 2, 3 (e.g. M2) L&Z,~
= L, &AN
= F-_n&&
=ein(N+I)~_-n,2AN=eix(N+l)~
--II
A
M&N-
iaNA
M2.N.
The same rule also applies to the points Mi, M3: &AM,N(k, z, rh) =eiXNAMj,,v(k, z, rh) . In conclusion, we also point properties of A(k, z, T):
(36)
out the trivial
parity
~A’d+(k,z,~~)=A’(-k,,K,~,r~) =?A’(k,z,q,).
(37)
383
(5 ) The vortex lattice should also have some symmetry properties under rotations around symmetry points (the hexagon vertices, K,, K2 and midpoints of hexagon sides, M,, A4,, M3) of an elementary cell (see fig. 2): since the hamiltonian is invariant under magnetic rather then the usual translations, the formal definition of such operations is: P_AlZiFA where fA is the magnetic translation of A(k, z) on the vector l=l& or TMj, and Li is an element of the point symmetry group of points Kj or A+ In this connection it is more convenient to speak about the symmetry of the functions FAA(k, z): A, (k, z), A, (k, z). E.g. the symmetry groups of AK,,N (k, z) (A=a(2r,+1)/3) and AM,,N(k, z) (~=a~,$?) for the quantum numbers (0, N) are
8. Conclusion The results obtained in this paper are as follows. ( 1) In most cases near HC2 the mixed state of superconductors with unconventional pairing is the hexagonal vortex lattice with periods a, ~,,a. (2) The symmetry properties of the lattice are determined by two quantum numbers (a, N) of the solutions of linearized GL equations. cr= ? 1 is the parity under reflection B in the plane perpendicular to the magnetic field; N the generalized Landau level number. (3) There is no one-to-one correspondence between pairing type at the uniform state at H=O and the symmetry of mixed state near HC2: for the same pairing type the different quantum numbers (a, N) of A( k, z) can be realized. However, the pairing type restricts the possible choice of (cr, N). E.g. for the swave pairing there exists only one possibility: ( +, O),forp-wavepairingthree: (+, -l), (-,O), (+, +l), etc. (4) The mixed state properties can be described by the symmetry group H of the order parameter A( k, z). If the position of the coordinate center r is chosen at the center of some vortex, this group H = Hr is: H={ei=FO, eixF;hn, e--2ix(N+i)/6L2rr,6, a& &sH, eixCN+‘)ao . The last element
comes from the identity:
(38)
f=&8.
(40) The symmetry groups for the points K,, M2, M3 are defined by the formulas ( 32)) ( 35 ) and ( 36). (6) Symmetry properties (38)-(40) were obtained by constructing the hexagonal lattice wave function A(k, z) from the eigenfunctions of the infinite degenerate Nth eigenlevel of the linearized eq. (6) and by using the properties of these functions. Strictly speaking, such functions are the solution of the nonlinear problem only when H is the intinitesimally close to HC2. However, we can guarantee that sufficiently close to HC2the symmetry of A(k, z) will be the same. This follows from the fact that the eigenfunctions of the Nth level are the basis functions of the IR of the initial symmetry group (4). To break the lattice symmetry one needs to admix to A(k, z) some other IR. Since for the two different IRS the probability to have the same critical temperature (field) is accidentally small, the break of the symmetry (38)-( 40) is only by virtue of another phase transition at some critical field H* < H,,. Of course, below HC2 it will be the mixing of another IR, but without symmetry violation. To specify it we will point out that from eqs. (38)-(40) it follows that (as obtained in section 4) the quantum number N classification of the solutions of linearized GL equations in the mixed state reduces to the classification
384
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
over mod 6 since the phases AN( k, z) and AN+6 (k, z) have the same symmetry groups. So, the basis functions from the N+ 6, N+ 12, ... eigenlevels are mixed with AN( k, z) . For the usual s-wave superconductor this means that the hexagonal Abrikosov lattice for arbitrary fields can be constructed from the 0, 6, 12, ... Landau levels wave functions, and other Landau levels do not participate in it. Numerically this fact was obtained in ref. [ 16 1. In accordance with phase transition theory the mixing is as small as ( 1- T/T,) 3 or ( 1 - H/Hc,)3 if we go from H,, to the lower fields. Now we are able to give the formal definition of the unconventional superconductivity in the mixed state since the definition given in ref. [ 31, as was pointed in section 1 is not valid in a nonuniform state. The state with symmetry group ( 38 ) - ( 40 ) with rr= +, N=O(mod 6) can be defined as the usual Abrikosov lattice since an s-wave superconductor has the same symmetry properties. The state with another o and N( mod 6) can be formed only by superconductors with anisotropic pairing and hence they should be defined as unconventional. In this paper we considered the nontrivial superconducting mixed state in an isotropic metal. However, the results can be straightforwardly generalized to the case of an anisotropic crystal where the magnetic field is directed along some n-fold symmetry axis. The quantum number N classification of the eigenfunctions of eq. (6 ) will reduce to the classification over mod it [ lo], which in turn will reduce the classification scheme of the vortex lattice state in an appropriate way. The hexagonal symmetry of the vortex lattice near H,, and different classifications of superconducting phases at H=O and near Hc2 can be the reason of phase transitions in the mixed state. As it was shown in ref. [ 8 1, the eigenlevels of linearized eq. (6) with eigenfields lower then Hc2 are responsible for the phase transitions. Depending on (rr, N) these transitions, which are structural phase transitions in the vortex lattice, are distortions and/or period multiplications and/or a-parity violations. Here we will only illustrate this possibility by two examples for the case of p-wave pairing, using he solutions of the GL equations which were studied in section 2. (a) Let us assume that in eq. ( 1) b < 0 and that at H=O the small crystal field anisotropy fixes the real vector ?,Jalong the z axis. If at Hc2 the phases (3b)
or (3~) occur, there should be another phase transition with parity r~violation at a field H*, lower than Hc2. This phase transition is conditioned by the eigentield of solution (3a). At H-H,, the odd part of A( k, R) will vanish and at H-c H,, only the even part will exist. There will be the same scenario if b> 0 and if at H,, the phase (3a) occurs. The more nontrivial conclusion following from the different quantum numbers N of the phases (3) is that the phase transition at H=H* should always be accompanied by some structural phase transition in the vortex lattice. (b) Let b < 0; the small anisotropy fixes real 11in the x-y plane, and the solution (3~) near H,, occurs. For such a case (as was shown in ref. [ 17 ] ), the vortex lattice near H,, has a form distorted along q. However, the vortex lattice which occurs near Hcz, delinitely has the hexagonal structure (at least at I& = K3, a = - 2b). Therefore; in the mixed state at some field H=H* there should be the phase transition with vortex lattice deformation. Solution (3b) is responsible for this transition. It is possible that the (a) or (b) scenarios take place in UBe,3 where a line of specific heat peculiarities was observed [ 18 1. We can assume also that the instability to distortions in the vortex cores in unconventional superconductor [ 5 ] originates from the described effect. There must be a structural phase transition in the vortex lattice if the linear equation (6) has two close eigenlevels. This is the case in UPt, where the kink in Hc2 seems to be due to the intersection of the eigenfields with different quantum numbers (a, N)
[lOI. Acknowledgements We would like to thank V. Mineev and G. Volovik for the helpful discussions, and H. Capellmann for his hospitality and interest in the work. This work was supported by BMFT, Bonn, Germany (Grant No. FKZ.: 13N5750) and by the Academy of Science of Russia.
Appendix In this appendix
we will formulate
in the terms of
I. Luk’yanchuk, M. Zhitomirsky /Symmetry
our article the Abrikosov theory of the s-wave superconductor nuclei lattice near Hc2 which is described by the usual GL equation a(T-T,)B(R)+bld(R)I*d(R) (41)
+ Maxwell equations. The mixed state order parameter d(R) near Hc2 should be constructed from the eigenfunctions of the lowest energy level of he linearized GL equation (4 1) which in the complex notation can be written as
( Id(R)
14>
(Id(R)
12)2
= [(~-7)/2i]“~(
]e2(0, 2~)]‘+&(0,
(
(a-z,
(0-Z)
+ ; d(z, Z) > =(l-T/T,)d(z,Z).
2r)]12)
(46)
over 7. The level lines for p(7) are shown in fig. 2. The minimum corresponds to the point 7h= ezxi16 in the hexagonal nuclei lattice. The transformation rule under the rotation of @,,(z, Z, TV) by 60” also can be obtained from the theory of the &function: c2x/3@o(z, 5, Th) =ezrri’%&(z, 2, th) .
-
385
the Maxwell equations. This procedure can be reduced to the minimization of the Abrikosov parameter /3(r) [ 121
‘(‘)=
-K(V-igA)2d(R)=0
of the mixed state
(47)
(42) References
This equation formally coincides with the Schrodinger equation for an electron in a magnetic field. The eigenfunctions of the lowest Landau level (which corresponds to the highest H,, = (U/~&K) ( T, - T) ) have the form Y=f(z)e-ZZ,
(43)
wheref( Z) is an arbitrary analytic function of Z (but not z! ). This arbitrariness means the infinity degeneration of the Landau levels occur. The Abrikosov lattice d(R) near Hc2 is the solution of eq. (4 1) which is invariant under magnetic translations {T,, T,,}. So, in the parallelogram (a, 7a) (a=a(r) is defined by eq. (23)),f(Z) should satisfy the quasiperiodic boundary conditions: f(Z+u) f(
z+
.fu)
= -e2faea2f(Z) =
,
_e-ix~e+a*e2ffcxf(~~
.
(44)
From the theory of functions it is known that there exists only one analytic function which transforms like eq. (44). It can be written by the &function. The solution d(R) = c&,(z, F, 7) has the form ~~(z,Z,~)=~,(z/u,7)e’~e-“. To find the lattice parameter the free energy functional F(
(45) 7 7)
one must minimize taking into account
[ 1 ] Review of experiments: K. Hasselbach, A. Lacerda, K. Behnia, L. Taillifer, J. Flouquet and A. de Visser, J. Low Temp. Phys. 81 (1990) 299; A. de Visser and J.J.M. Franse, J. Magn. Magn. Mater. 100 (1991) 204; L. Taillifer, J. Flouquet and G.G. Loncarich, Physica B 169 (1991) 257. [ 21 Review of theory: M. Sigrist and K. Ueda, Rev. Mod. Phys. 63 (1991) 239. [3] G.E. Volovik and L.P. Gor’kov, Zh. Eksp. Teor. Fiz. 88 (1985) 1412 (Sov. Phys. JETP 61 (1985) 842). [4] K. Ueda and T.M. Rice, Phys. Rev. B 31 (1985) 7114. [ 51 T.A. Tokuyasu, D.W. Hess and J.A. Sauls, Phys. Rev. B 41 (1990) 8891; Yu.S. Barash and A.S. Mel’nikov, Zh. Eksp. Teor. Fiz. 100 (1991)307 (Sov.Phys.JETP73(1991) 170). [ 61 L.I. Burlachkov, Zh. Eksp. Teor. Fiz. 89 ( 1985 ) 1382 (Sov. Phys. JETP 62 (1985) 800). [ 71 M.E. Zhitomirsky, Pis’ma Zh. Eksp. Teor. Fiz. 49 ( 1989) 333 (Sov. Phys. JETP Lett. 49 (1989) 378); S.K. Sundaram and R. Joynt, Phys. Rev. B 40 ( 1989) 8780. [8] M.E. Zhitomirsky and LA. Luk’yanchuk, Zh. Eksp. Teor. Fiz. 101 (1992) 1954 (Sov. Phys. JETP 74 (1992) 1046). [9] K. Scharnberg and R.A. Klemm, Phys. Rev. B 22 (1980) 5233. [lo] I.A. Luk’yanchuk, J. Phys. (Paris) I 1 ( 1991) 1195; I.A. Luk’yanchuk and M.E. Zhitomirsky, Physica C 185 (1991) 2629. [ 1 I ] M.E. Zhitomirsky, Pis’ma Zh. Eksp. Teor. Fiz. 55 ( 1992) 460 (Sov. Phys. JETP Lett. 55 (1992) 472). [ 121 D. Saint-James, G. Sarma and E.J. Thomas, Type II Superconductivity (Pergamon, Oxford, 1969 )
386
I. Lukyanchuk,
M. Zhitomirsky /Symmetry
[ 131 M.E. Zhitomirsky, Zh. Eksp. Teor. Fiz. 97 (1990) 1346 (Sov. Phys. JETP 70 (1990) 760). [ 141 AS. Mel’nikov, Zh. Eksp. Teor. Fiz. 101 (1992) 1978 (Sov. Phys. JETP 74 (1992) 1059). [ 151 G. Eilenberger, Phys. Rev. 164 (1967) 628.
of the mixed state
[ 161 G. Lasher, Phys. Rev. 140 (1965)
523A.
[ 171 G.E. Volovik, J. Phys. C 21 (1988) L221. [ 181 B. Ellman, T.F. Rosenbaum, J.S. Kim and G.R. Stewart, Phys. Rev. B 44 (1991) 12074.
Physica C 206 ( 1993) 373-386 North-Holland
Symmetry of the mixed state of superconductors with anisotropic pairing I. Luk’yanchuk and M. Zhitomirsky L.D. Landau Institutefor Theoretical Physics, Kosygina St. 2, Moscow, 117940, Russian Federation Received 23 November 1992
The theory of mixed state symmetry of superconductors with anisotropic pairing is developed. Such a theory employs the symmetric Landau approach to the second order phase transitions for the case of the normal metal-superconductor transition in a magnetic field. It is shown that in most cases the Abrikosov vortex lattice near H., has the hexagonal structure. The transformation properties of such a lattice under translations and rotations are derived. From these the mixed state phase classification for unconventional superconductors is determined. The implication of such ideas for the phase transitions in UPt3 is discussed.
1. Introduction One of the most intriguing mion
physics
is the peculiar
features properties
of heavy-ferof the super-
state. Thermodynamic characteristics of it led one to hope that the superconducting gap can have zeros on the Fermi surface and therefore it should have the anisotropic form. Another confirmation of this anisotropy was done by the investigation of the superconducting phase transition in UPt, [ 1,2]. It was observed that the transition consists of two superconducting phase transitions with critical temperatures T,=O.S K, T,,=O.45 K. One of the most convincing explanations of such splitting is that the superconducting phases corresponding to T, and T,, have a different symmetry of the order parameter. To express this idea in more explicit terms we will present the superconducting order parameter A,,( r, r’ ) = ( ‘k(,( r) !Pp( r’ ) ) in the Wigner variables: Cooper pair coordinates with center R= (r+ r’ ) 12, and the position on the Fermi surface k, which can be obtained by the Fourier transformation r-r’ +k. If the superconducting state is uniform (i.e. the order parameter AaB(k, R ) does not depend on R ), the conducting
’ Present address: Institut fur Theoretische Physik, RWTHAachen, Templergraben 55, 5100 Aachen, Germany.
anisotropic behavior of the superconducting gap means that some nontrivial k-dependence of A,, occurs. The most natural way to formulate the phenomenological theory of the normal metal-unconventional superconductor phase transition is to discuss it in the frame of Landau theory of second order phase transitions. According to this approach [ 3,4] the appearance of superconductivity means breaking of the normal metal symmetry group which includes the following operations: U( 1 ), the gauge group of multiplication of electron wave functions Y with an arbitrary phase factor: Y+e’*YX, K, the point crystal group symmetry: in UPt, K=D6,,, for the case of an isotropic metal K= SO, x f (I^is space inversion); and &, time inversion. The residual symmetry group L defines the symmetry of the order parameter A&k). If such a breaking includes not only the U ( 1) elements, as in the case of the usual s-wave pairing, giving L a more complicated structure than KX R, the superconducting order parameter AaB(k) can be defined as an unconventional one. The important feature is that A&k) just below the phase transition can be expanded by the basis functions of irreducible representations (IR) of the group K. This gives the possibility to classify all possible types of pairing in accordance with the IR of the group K. In the case
0921-4534/93/%06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
374
I. Luk’yanchuk. h4. Zhitomirsky /Symmetry
of an isotropic crystal this fact leads to s-, p-, d-, ... wave superconductivity. A many-dimensional IR corresponds to multicomponent superconducting order parameters. On the other hand, the IR-classification strongly restricts the allowed residual groups L. Finally, L can be obtained by the minimization of the GinzburgLandau functional for the corresponding pairing type. This procedure was followed in refs. [ 31 and [ 41 for the crystal point groups K relevant to the heavy-fermion compounds, all the possible residual superconducting groups L having been enumerated. Another classification feature of d,,(k) is given by the space symmetry I^and can be obtained from the identity: &r(k) = -dm( -k) [ 31. It leads to the singlet: d,@(k) =d(k)iaY and triplet: d&k) = (d( k)a) i&’ types of superconductivity. One more interesting feature of UPt, is the T,-T, splitting behavior in a magnetic field. It is well established now [ 1 ] that the H-T phase diagram contains at least three different superconducting phases with phase transitions between them. These phase transitions meet at some thetacritical point which is also the kink point in the Hc2( T) dependence. Since the transitions seem to be phase transitions between different vortex state phases, it is interesting to construct the theory of the Abrikosov mixed state for the case of unconventional superconductors. There are two regions on the H-T phase diagram of the superconductor where significant simplifications are possible. In the first one, close to H,,, the starting point of the theory is the consideration of separate vortices. Surprisingly, it was found [ 5 ] that cores of the vortices in unconventional superconductors can have nonaxisymmetric form. However, until now it is unclear what type of lattice they do form. The second region lies near the upper critical field Hc2, where one could construct the theory beginning from solutions of the linearized GinzburgLandau equations. This consideration will be in the center of our attention. The most straightforward way to investigate the mixed state of the unconventional superconductors is to solve the GL equations for the nontrivial order parameter A&k, R) in a magnetic field and in such a way to generalize the Abrikosov vortex lattice theory. However, for the multicomponent order parameter such a procedure becomes much more
of the mixed state
complicated. Even solving linearized GL equations (an example of it for the p-wave pairing will be done in section 2 ) is not so simple [ 6,7 1. At the same time the procedure is strongly dependent on the coefficients in the GL equations, and so it is difficult to find all the possible solutions. Instead we will consider another approach based on the symmetry arguments. By such a treatment it was already shown [ 81 that the phase transitions in the mixed state of UPt, has a natural explanation as a structural phase transitions in the vortex lattice: distortions or/and period multiplication. Here we will fix our attention mainly on the symmetry properties of the lattice below Hc2. We will use the fact that the appearance of the Abrikosov vortex lattice near H,, ( T) is a second order phase transition which breaks the normal metal symmetry group. So, we will apply the above mentioned Landau phase transition theory and we will find the possible residual symmetry groups of the Abrikosov vortex lattices in unconventional superconductors. The approach will be outlined in detail in section 3. For now, we will stress only that the main difference between the phase classifications in an uniform state at H= 0 [ 3,4] and in a mixed state near Hc2 is the difference between the initial symmetry groups of the normal state since: ( 1) the magnetic field breaks the crystal point symmetry group; (2 ) in the mixed state the order parameter A,,( k, R) is no longer uniform and the initial group should involve also the translations in the magnetic field. Following the symmetry approach we will find the basis functions from which the order parameter A,,(k, R) near Hc2 should be constructed (section 4). In section 5 we will show that for the arbitrary pairing type the Abrikosov vortices form a hexagonal lattice at least in a wide region of the coefficients in the GL equations. This fact means that under translations and rotations by 60” the vortex structure A&k, R) does not change with the accuracy up to multiplication with some phase factors. These factors will be determined in sections 6 and 7 and therefore the symmetry of the lattice will be defined exactly. In the Conclusion (section 8 ) we summarize our results and discuss the connections between unconventional superconductivity at H=O and near K2.
375
I. Luk’yanchuk, h4. Zhitomirsky /Symmetry of the mixed state
2. Ginzburg-Landau
,
theory for p-wave pairing near
2 HD
( 1+C)V2-
HCZ 2C(V,-iv,)’
X For clarity of presentation, we will consider the problem which has to be solved for the example of p-wave pairing in the isotropic metal (K= SO,xi) with strong spin-orbit coupling when the order parameter is defined as: d&k, R) = ioY[ n(R) xk]a. This is the simplest example of a superconducting state with a multicomponent order parameter. The GL functional for the vector order parameter q(R) can be written as F=A(
T- T,)rlrt*+a(rttl*)2+b’12rl*2
(1) The gradient Vi includes the vector-potential of the magnetic field A in the usual way: Vi= ai - i ( 2e/fic)Ai. In a H-T phase diagram there exist two regions interesting for us: ( 1) At fields H< H,, the order parameter A(k, R) is R-independent. The k-dependence of A(k) can be obtained from the uniform terms of eq. ( 1). For b < 0 it will be the arbitrarily directed real vector q= (A(T-T,)/2(a+b))“‘n where n2= 1; as discussed in the Introduction the residual symmetry group of A(k, R) breaks the initial SOS space symmetry: L= {C,, e%, 6,,, R}. Here C, are rotations by the angle @ around the n-direction; 6 and 6” are reflections in the planes perpendicular and parallel to n, ri being combined with the element ei” from U ( 1). If b> 0 one has the complex vector q= (A ( T- T,) / nT=n:=t, where nin2=0; 2a)L’2(n,+inz), L={e-‘“C,, 6, &,I?} (here n= [nlxn2]). (2) The region near the upper critical field which will be discussed in the article. The procedure of the calculation of Hc2 for ( 1) is standard: we neglect the fourth order terms a( q~*)~, bv2f2 and find the eigenvalues T-T, of the corresponding linear GL equation [ 7 ] :
0
2C(V,+iV,)’ (1+C)V2+
0
2e %HDO
0
V1
L
Here C= (K2+K3)/2K,, D= (K2-K3)/2K,; in eq. (2 ) we also assume that Va= 0. The “eigenvalues” A( T- T,) depend on the magnetic (eigen) field. We can choose from them the eigenlevel which has a largest value of H at a given T-T,. Such dependence defines the upper critical field Hc2( T). Below it there appears a nonzero solution of eq. (2). To find them, the nonlinear terms u(~q*)~, bq2q*2 should be taken into account. The solution can be constructed from the eigenfunctions of eq. (2 ) which in the case of the s-wave superconductor are the wave functions of a zero Landau level. In the mixed state the order parameter depends both on R and k. The candidates for the Hc2 solutions of eq. (2 ) are [71: A&k, R ) = iaYJZfO(R ) , polar phase , A,,(k, R)=iaY(JX-iJ,,)jdR),
(3a)
A phase,
(3b)
A,,(k,R)=iaY((J,+il,lfo(R) + (Jx - i.J,% CR) ), SK phase ,
(3c)
where J= [kxa]
.
corresponding eigenfields The 2mcA( T,- T)/ 1el hK,tliy where
are
Hi=
Ap=l, AA=l+C-D,
The solution with the largest eigenfield Hi does correspond to Hc2 = max (Hi). By changing K,, K2, KS all variants are possible.
376
I. Luk’yanchuk, M. Zhitomirsky /Symmetry
The characteristic of the anisotropic pairing feature which can be seen from eq. (3~) is the nonfactorized dependence of A on k and R. Scharnberg and Klemm (SK) were the first to find this [ 91 by the solution of the Gorkov equations for p-wave pairing in the weak coupling limit. In section 4 we will show the symmetry reasons for such a form of A&k, R). Equations (2) are quite analogous to the corresponding equations in the model of the two-component Ginzburg-Landau functional with cylindric symmetry appropriate to UPT3 [ 71. Therefore, the analysis given below could be straightforwardly transferred also in this case. To solve the problem of the vortex lattice structure near Hc2 it is needed to take into account the nonlinear terms in the GL equations. This is not easily done straightforwardly. Even the numerical calculations encounter difficulties. Instead, we will use the symmetry approach which will allow us to speak about the properties of the mixed state of superconductors with anisotropic pairing in a qualitative fashion. We will outline this method in the next section.
3. Landau phase transition theory for the normal metal-superconductor transition in a magnetic field
The appearance of a superconducting vortex lattice in a magnetic field is a second order phase transition, and we will consider it in terms of the Landau phase transition theory. Such an approach gives the possibility to formulate the mixed state theory for an arbitrary pairing without considering a specific form of the nonlinear GL equations. In the Appendix we will consider the standard procedure of obtaining the Abrikosov vortex lattice solution for an s-wave superconductor as a particular case of this approach. Note, that the symmetry approach should also be valid in the low temperature region where the GL equations for A(k, R) are not applicable. So, in accordance with the Landau symmetry approach we will hold to the following program.
of the
3.1. Definition group G
mixed state of the hamiltonian
symmetry
The appearance of the vortex lattice means spontaneous breaking of the group symmetry G of the normal metal state in a magnetic field, G={U(
l), ‘ib, _&,, 6, r&l?, I),
(4)
where U( 1) is the gauge group, fa are Cooper pair magnetic translations on an arbitrary vector a; f_.A,,(k,
R)=A,,(k,
R+a)
exp(is&(R))
(5)
(for the magnetic field vector-potential we will use the Coulomb gauge: A(R)= [HXR]/Z). &, is a rotation around the magnetic field direction H. In the case of singlet pairing or triplet pairing with weak spin-orbital coupling, & includes rotations in the k and R spaces. If the pairing is a triplet one and the spin-orbital coupling is strong, & includes also rotation of the spin space. For simplicity we will only consider the first possibility and further we will omit the spin indices. The generalization for the triplet pairing with strong spin-orbital coupling can be done straightforwardly by the formal substitution: k-J= [kxa] as was done in the previous section. 6 and eH are the reflections in the planes perpendicular and parallel to magnetic field: &k,+ -k,, &Jk,+ik,,)-k,-ik,,. Since H is an axial vector, BH changes sign and b does not. To compensate the sign change the time inversion l? is combined with c?~. f is space inversion. 3.2. Order parameter
definition
The wave function of the superconducting vortex lattice state just below H,, should be constructed from the basis functions of a particular irreducible representation (IR) of the hamiltonian symmetry group G. The IRS of group G were found in ref. [ 10 1. Such IR basis functions are the eigenfunctions of the linearized GL equations.like (2 ). In general form the linearized equation for A(k, R) can be symbolically written as 2(H,
T)A(k,
In section
R)=O.
4 we will briefly
(6) outline
the IR classifl-
31-l
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
cation procedure and the properties of their basis functions. The IR basis functions are isomorphic to the Landau level wave functions, the zero Landau level corresponding to s-wave superconductivity. For other types of superconductivity other Landau levels are involved. 3.3. Vortex state symmetry
coordinate
space R= (x, y).
a=a,-ia,, a=a,+ia,.
(8)
Sometimes we will also use the complex variables the vector k: k= (k,, k,, kz) = (K, IC,k,) where
for
K=
To find the symmetry properties of the vortex lattice near Hc2 we need to know the coefficients in the expansion of A( k, R) over the basis functions of the corresponding IR. The coefficients can be defined by the minimization of the fourth order terms in the Landau functional which describes this transition. (The third order terms are forbidden by U( 1) symmetry; this is the reason why the normal metal-superconducting vortex lattice transition is a second order one.) However, due to infinite degeneracy of every IR of G (which is the analog of the infinite degeneracy of the Landau levels), the order parameter has an infinite number of components and the direct minimization is rather complicated. Some geometrical ideas presented in section 5 simplify this procedure and allow one to find the conditions at which A( k, R) has the structure of a hexagonal vortex lattice. We will use such a geometrical approach to obtain the transformation rules of A(k, R) under rotations of 60” and translations by the lattice periods. In sections 6 and 7 it will be shown that such rules depend on the IR to occur at H=H,,. The transformation properties define the symmetry group H of A(k, R). Obtaining H c G solves the problem of the normal metal-superconductor phase transition in a magnetic field and serves as further background of the phase transitions theory between different phases within the mixed state of the superconductor with unconventional pairing. To avoid overcomplicated notation we will use in this paper the dimensionless variables using f,,= (cYi/ 2eH) ‘I2 as a unit of length. It will be more convenient to use the complex variables
(k,+ik,)/2,
R= (k,-ik,)/2.
(9)
In some parts of this paper we will omit the nonessential variables in the order parameter A( K, R, k,,
z, 2).
4. Classification of the linearized GL equations solutions near Hc2: IR of the hamiltonian symmetry group The Abrikosov vortex lattice near H,, is formed from the eigenfunctions which belong to some eigenlevel of the linearized GL equations (6). This level is classified by the IR of the hamiltonian symmetry group G (4) which can be found if the commutations rules of operators in G are known. Since G contains the continuous operations T; and i,, the formal specification of such rules should be done by using the infinitesimal generators f?, [of i=Aand ~$1 F* =
eiJ5(Ti+
+A.?- ) ,
f_
= ei@f
9
where i+ = -i(a+z)/fi,
t^_ = -i(&-t)/JZ,
i=f,+f,=f,+za-3.
(7)
(10) (11)
The generators G, fR correspond to rotations of the k and R spaces around H. We treat t^+, f in the Coulomb gauge. Generalization to an arbitrary gauge is given in ref. [ 81. The symmetry properties of the eigenfunctions of eq. (6 ) are defined by the commutation rules of the generators i, , f: [t;,tl-]=l,
[i&,r]=+i,)
In particular, we can introduce the commutator t^+, f with the Casimir operator
z= (x+iy)/2, Z= (x-iy)/2,
for the two-dimensional In these variables
a=r+i_t;
)
(12) with
(13)
378
I. Luk’yanchuk, M. Zhitomirsky /Symmetry
which enumerates the eigenlevels of the operator 2. More exactly: the eigenfunctions of2 should also be the eigenfunctions of the operator 3. Using the creation and annihilation operators of the Landau levels: 8+= -i(a-z)/Jz,
ii= -i(a+Z)/Jz,
(14)
[B,8+]=1;
(15)
we can write the operator2
in another
3=G+ci+(i.
form: (16)
This form is useful to find the general form of the eigenfunctions of the operator 2: dS(k,R)=
2 fn(R) N=rn-+ll
‘Y;(k),
(17)
where
s Y,,(k) are the spherical harmonics in k-space: (f,Y,,,(k)=mY,,,(k)) andf,(R) the nth Landau level wave functions (d +&= (n + 4 )fn). The wave functions dg (k, R) are characterized by the two quantum numbers: N= 0, + 1, IL 2, .... and (T= + . All the coefficients (Y,in eq. ( 17 ) should be real (or have the same phase); this property follows from the &J? symmetry from eq. (4). The number N generalizes the Landau level number n: when the wave function does not depend on the second variable k the N-level classification reduces to the usual Landau levels. This result is quite natural since Landau levels are also an IR of the group {T,, L,]. The quantum number rr defines the wave function parity under the operation 6 from G, eq. (4). This classification explains the nontrivial SK solution for the p-wave pairing (3~). The phases (3) are described by the (a, N) quantum numbers: Pphase (-, 0), A-phase (+, -l), SKphase (+, 1). Another property of the eigenstates of eq. (6) is the infinite degeneracy of the N-levels which is a consequence of the noncommutative group relations ( 12). This degeneracy is analogous to the degeneracy of the Landau levels. If some solution of eq. (6) is known, all other eigenfunctions of the same N-level can be obtained by applying the magnetic translation
of the mixed state
operators T;. Some possible ways to parametrization of the independent N-level eigenfunctions will be given in sections 5 and 6. One important thing which can be seen from eq. ( 17) is the overall pairing type in the Abrikosov lattice near Hc2. If the spin-orbital coupling is strong, the mixing of singlet and triplet superconductivity is possible. E.g. the phases (a,+ ia,)k,fi (R) ia,, (p-wave pairing) andfo (R )ia, have the same quantum numbers ( +, 0) and hence they are mixed in the vortex state (a,+ icr, changes sign under reflection in the xy plane since cr is an axial vector). Of course the pairing which occurs at H=O in the mixed state has the dominant amplitude. So, if in the uniform state two superconducting transitions do exist, the corresponding phases can be mixed in the Abrikosov state. From the group G generator properties ( 12) we can obtain relations between the elements i,, ?=Aof this group: ^,. ^^ L, TA= TxL4 , (18) where
This equation can be verified by the expansion of L,, Tl in a series and applying the commutation rules ( 12) for ?+ , f_ , f. Such relations are the same as the moving operation for the usual Euclidean plane (group E,). The difference between these two groups is in the noncommutative magnetic translations: ~J.-,2C~--&0~JJ~ The operators lations since [t^,,a+]=o,
d+, 6 commute
[?&]=O.
The commutations [&+,[I=-Li+,
(19) with magnetic
trans-
(20)
rules of Li+, ci with fare as follows: [fi+,f]=ri+.
(21)
Using eq. (2 1) we can also deduce the useful relations which will be used in the next sections: ~,B=e-i~B~,,~,ri+=ei~~i+%.
(22)
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
5. The vortex lattice in unconventional superconductors
magnetic translations by the lattice some phase factors: eiS, eiY:
The solutions of the CL equations near H,, for the s-wave superconductors were studied by Abrikosov who found that the superconducting order parameter A(R) forms a periodic lattice with one magnetic quantum flux per cell. The hexagonal structure of such a lattice is most stable. The purpose of this section is to show how this result can be generalized for an arbitrary pairing type. In accordance with the Landau phase transition theory, the appearance of superconductivity below H,, means a break of the initial hamiltonian symmetry group (4). It is quite natural to assume that at H< Hc2 A(k, R) also has a lattice structure. (Note that the mixed state (A(k, R) ) cannot have the laminar structure since, as was shown in ref. [ 111, such a symmetry is incompatible with the commutation rules for the magnetic translations. ) We will also assume one-flux quantization of the elementary lattice cell. The residual symmetry group of A(k, R) : H c G is defined by the elementary cell basis vectors a and b which satisfy the condition 1allb 1 sin CC=297 ( CY=U-b). So, the lattice can be determined by two independent parameters. We will introduce the complex parameter 7= ( )b I / Ia I )eh which completely determines the lattice since it also has two components. The parameters 7,~ i and 7,,= e2ri/6 correspond to square and hexagonal lattices. In the complex variables the basis vectors can be written as: a, 7u where ~=a(7)=(ix/(7-5))‘/~,
379
(23)
to satisfy the condition Iallb I sin (Y= 2x. To find what type of the lattice will occur we should find the solutions A(k, R, 7) of the nonlinear CL equations near HC2 for each 7 and then, after minimization of the free energy F(r), choose the most stable one. To avoid complicated calculations we will use symmetry arguments which generalize the ideas of ref. [ 121. Let us consider the order parameter as a function of 7: A=A(k, z, 7). The form A(k, z) also implies dependence on 2, unless explicit stated otherwise. The lattice symmetry of the order parameter means that A( k, z, 7) is an invariant function under
faA(k,
z, 7)=en(r-z)A(k,
z+a,
periods
7)=ei8A(k,
up to
z, 7) ,
f=,A(k, z, 7) =en(~Z-rZ) A(k, z+7u, 7)=eiYA(k,
z, 7). (24)
We will discuss the meaning of the factors eiS, eiy in the next section. Let us now introduce two groups of scaling transformations of the parameters K, z, 7: (1)
K-+K, z+z, 7+7+
(2)
lc+lCm7,
z-z-,
1; 7-+ - l/7.
(25)
It is possible to see that the functions A(k,, IC,z, 7+ 1 ), A(k,, K-, zfi, - l/7) are also the invariants under the same group of magnetic translations {pa,, pTa} with the same phase factors eia, eiy as A( k,, K, z, 7). It serves as the background of the following theorem: Adkz,
JGz, 7) =M7)&(kr,
&(k
K, z, 7) =&(7)&(kr,
K-3
K, z, 7+ 1) 3
zfi,
-l/7)
.
(26)
This theorem follows from the statement that AN( k,, K, z, 7) is the unique solution of eq. (6) which satisfies the condition $AN = NAN, or, in other words, from the wave functions of the given infinitely degenerate IR ( ?, N) of group G it is possible to construct only one function AN( k,, K, z, 7) which transforms like eq. (24) under pro, To,. The proof of this fact for the s-wave superconductor which correspond to IR ( + , 0) is given in the Appendix. Since the bases of the wave functions for another IR which are defined by the same operators t^+, fare isomorphic to the basis of ( +, 0) the statement is also valid for an arbitrary IR with quantum numbers ( ?, N). Properties (26) have a simple physical meaning [ 8,12 ] : we used the parameters (a, 7~) as the basis of the elementary cell of the vortex lattice. Transformations (25) imply other possibilities for the choice of basis: (a, 7u + a) and (7~2, -a). According to eq. (26) the wave function A,(k,, K, z, 7) should be invariant under such basis changes. The elementarycells (a, zu), (a, zu+u), (zu, -a) have the same areas.
380
I. Luk’yanchuk, M. Zhitomirsky /Symmetry
More explicit expressions for the factors cN( r), &(r) will be given in section 7. From properties (26) it follows that not all the points of the complex plane T are independent. To define the nonequivalent points T it is sufficient to use the fundamental region r which is shown in fig. 1. All other points t could be obtained by the operations (25 ). Properties (26) leads to the following symmetry of F( r) at the r-plane:
(27) The last equation was obtained by the lattice basis inversion + complex conjugation of the real function F(r). Due to eq. (27) the symmetry points ri, and rs of the boundary of the fundamental region should be the stationary points (maximal, minimal or saddle) of F( r). Since near the point ri, the energy F(r) locally has triangular symmetry defined by eq. (27), the point r,, can only be either a local maximum or local minimum of the function F(r). In fig. 1 the lines with equal free energy F(T) for the s-wave superconductor near H,, are shown (see Appendix). The energy minimum corresponds to the points of the hexagonal vortex lattice rh. The saddle point of F(r) corresponds to the square lattice ‘Imr
of the mixed state
( z=T~). The point Im r-+ +03 obviously corresponds to the energy maximum. Turning to the case of unconventional superconductivity, we note that for every multicomponent GL functional there exists some special choice of coefficients under which the multicomponent nonlinear GL equations decouple to a number of more simple equations, each of them looking like a GL equation for an s-wave superconductor. E.g. at K2= - K3, a= - 2b, the GL equations for p-wave pairing originating from eq. ( 1) decouple to three such equations for the nX+iq,,, ~X-i~Y, Q components. For the above particular choice of coefficients the r-dependence of the energy is given by eq. ( 45 ) . Since the minimum in the hexagonal lattice point rh iS conditioned by the symmetry, this minimum should be stable under small variations of the coefficients in the multicomponent GL functional. This means that in the coefficients space there exist at least some region around the coefficients of special choice at which the lattice is hexagonal. The explicit form of F(r) for the SK-phase was given in ref. [ 13 1. The numerical analysis for such a case [ 141 shows that the region of coefficients were the hexagonal lattice is most stable is rather large. Only this case will be under our consideration. We can conclude also that the statement given in the previous papers [ 71 that the vortex lattice should always be nonhexagonal for any nontrivial pairing with multicomponent order parameter is incorrect.
6. Translation properties of A@, R)
Zn, =
~1
~112
U
112
e6
1
Fig. 1. Fundamental region r which parametrizes the vortex lattices with different lattice parameters. T, corresponds to the square lattice and r, to the hexagonal one. The distribution of equal energy lines F(r) is shown.
The transformation rules of A(k, R) near Hc2 under magnetic translations by the lattice periods a and Ta are defined by eq. (24). To understand the meaning of the phase factors eis and eiy, we will show that for each pair (8, y) from the different eigenfunctions corresponding to the degenerate level Nit is possible to choose wave functions that transform like eq. (24). Consider some eigenfunction AN(z, 7) which transforms like eq. (24) for definite (fi, y), for simplicity say ( --n, --n). Let us consider the function Al,,,(z, T) = f,d,(z, 7). Using eq. (19) we find the transformation rules:
381
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
=e2a(1-A)f+iFaA,(z, RA,N(Z?
7)=LmN(z,
7)~ -e2a(L--“)A5N(z,
a
7) ,
7)
M
Kl
wn
c-e
2a(d--d)AlN(z,
7) .
Solving the equations _e*a(r--1)=ei/3 , -e
2n(rT-fA)=eiy
9
ai2
M2
r
(28)
(29)
we can find the vector I for each pair (j?, y). Since magnetic translations commute with the hamiltonian, all the functions A&z, 7) also belong to the eigenlevel N. Applying any periodic potential with periods (a, 7~) removes this degeneration and splits the Nth eigenlevel into a band. The A&z,r) are the correct wave functions of the approximation of zero splitting. The transformation rules eq. (24) are the analog of the Bloch representation for a one electron wave function in a periodic potential, the pair (j?, y) usually defining the wave vector K in the Brillouin zone. Using the analogy with Bloch electrons we can introduce the Brillouin zone as a parametrization region of such functions: -A < j?< x, -A < y< n. In the L-plane it corresponds to some region defined by eq. (29 ). To find this region we use the equivalence of points A, 1+ a, A+ ra, which follows from eq. (29). The procedure is the same as for choosing the Brillouin zone in the K-space of the usual periodic crystal. For our purposes it will be convenient ‘to use as the Brillouin zone the WignerSeitz cell which has the same symmetry as the point group of the Abrikosov vortex lattice. The Brillouin zone for the hexagonal lattice coincides with the elementary cell of this lattice and is a hexagon of side a/$ (see fig. 2). The functions A,&z, 7) from the Brillouin zone for given T form the basis of the eigenfunctions of the degenerate N-level (see also ref. [ 15 ] ) since: ( 1) due to the different transformation rules these functions are orthogonal and hence they are linearly independent, and (2) all functions from the N-level can be obtained by the magnetic translations of some arbitrary function from this level. As we showed, every magnetic translation transforms the function Ar,N(z, T) to some other function A,&z, 7) in the Brillouin zone.
Fig. 2. The elementary cell of a hexagonal Abrikosov vortex lattice. It coincides with the Brillouin zone, defined in the text.
The operations from the hexagonal point group transform the wave function Ali,N(z, 7,,)to some other function which corresponds to another 1j of the Brillouin zone. All points lj defined in such a way form the star of vector Li-{;ij>. In a usual crystal the electron wave functions corresponding to the symmetrical points of the Brillouin zone have a so-called small symmetry group. The operations from the small group do not change the wave vector K of this function (or transform it to an equivalent one ) and multiply the wave function by some phase factor. Analogous symmetry properties of the wave functions A,,( z, T,,) for the symmetry points of the hexagonal Brillouin zone will be obtained in the next section. Now we will give the magnetic translation transformation rules for these points. The wave function AN(z, TV)zA~,~(z, zh) corresponding to the hexagon center r (A = 0) transforms as: ~~JN(z, ~&,4~,
71,)=-&(z, 7,,)= -&(z,
7,,),
(30)
7,,).
The wave functions corresponding alent hexagonal vertices transform point K,: A=a(z,,+ 1)/3:
faAK1(z,
-e-2irr/3AK,(z,
7h),
pTnAK1(z,q,)= -e2in’3AK1(z,
z,,) ;
7,,)=
point K2: L=a(2r,-
FoA,(z, fraA,(z,
7h)= TV) =
to the nonequivas
(3la)
1)/3:
-ezix/3AKZ(z, T,,), -e-2in/3AKZ(z,
TV) ;
(31b)
382
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
the wave functions of the midpoints lent hexagon sides transform as: point M,: I=az,,/2:
of nonequiva-
we will work with A,(z, T) which can be obtained fromA,(z,r)as:A,(~,r)=(B+)~A,(z,t).Usingeq. (22 ) we have A,(z,
~CIAW,(z, G) =A.w (z, G), ~wzAA4,(z, G) =AlcI,(z, %) >
(32a)
point M2: L = a/2:
r) = (c?+)~A~(z,
=~o(z)(d+)NAo(~,
z) 7+1)=&,(s)-AN(z,
AN(z, T)= (B+)NAo(~,
T)
=&,(~)(ii+)~A,,(z,,/%, (32b)
point MJ: I=a(
-l/r)
=&(r)(ri+)NL,&(z,
1 -q,)/2:
-l/r)
=&,(7)(~/i)""~&i+)~A,(z,
=&,(r)(~/s)~‘*A,(zJr/r, %7Ah4&
7h)
=&~(z,
%)
.
(32~)
In accordance with eq. (28 ), any magnetic translation of coordinates will change the rules (30). We chose the origin of coordinates in such a way as to have consistency with the s-wave superconductor Abrikosov lattice where the origin of coordinates is situated at the position of some vortex core and the magnetic translation rules of A(k, R) have the form (30a).
7. Point symmetry of A(k, R) In section 5 we proved that the Abrikosov nuclei lattice A(k, z, z) near Hc2 has the hexagonal structure. The translation properties of A(k, z, th) were obtained in section 6. However, to know the full symmetry group of the lattice one also needs to know the rotation properties of A(k, z, q,) around the symmetry axis of the hexagonal lattice. Here we will show that these properties depend on the starting quantum numbers N (defined in section 4) of the solutions of eq. (6) at H= Hc2. To find the A,(k, z, rh) transformation rules we should find first the factors iN( r), and MT) in eq. (26 ). The following formulas are valid: G(r)=
(rlVv’*&l(r)
.
(33)
To prove eq. (33) we note that the Abrikosov lattice wave function AN( z, z) of an s-wave superconductor corresponding to the Nth Landau level has the same symmetry properties as an arbitrary eigenfunction A,(k, z, 7) of eq. (6) with quantum number N. So,
-l/r)
-l/z),
where the angle @ is defined by: e pi@= ,/$. Specification of eq. (33) for the hexagonal lattice parameter ?h = e2in’3 gives the transformation rule for the hexagonal lattice A,(k,, IC, z, q,) when rotating by 2x16 around the center of the hexagonal elementary cell; ,$x,6AN(k,
Z,
T,,)=e2irr(N+1)‘6AN(k,
Z, rh)
.
(34)
Property (34) was obtained by two successive transformations T+ - l/7+ - 1/T+ 1 which leave the point TVimmovable and rotates the Z-K spaces: z, rc+e-2ia/6z, e-2ix% In eq. (34) we used the coefficients Co(r) =emin14, and L(r) = id/Jr which can be calculated from the explicit Abrikosov solution A,(z, 7) (see Appendix). The wave functions AKm,N(k, z, th) and A,,, (k, z, T,,), discussed in section 6, should also have some symmetry properties under transformations by small groups of the points K,,, and Mj in the Brillouin zone. Using eqs. (18), (32) and (34) we have points K,: m= 1, 2 (e.g. K,) L 2n/3 A KI,N- -i
2n/3
f
a(r+1)/3
AN
= Fa’a(r-I ,,3L2rr/3A~ =e2ia(N+1)'3~~f~_,),~AN =e2ix(N+1)/3F
Mr)=Co(r),
r+l)
E-e
a(r-1),3~-
47+1)/3fa%d~,,~
2ix’3e2i”‘N+“‘3~~;1(,_1)/3~_a(r+1)/3~~’adK,
.N’ )
the closed loop faa(+, j,3~-aCr+1 j,3F0 sweeps the magnetic flux - @,/6 which brings in the additional phase factor eezi*16. So, the transformation rule for AK,,N( bush)as well as for A,,,(?,) has the form
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
&&,,Ntk
z, ?I,)= e2i”N’3&,,Ntkz, %h);
(35)
points M; j= 1, 2, 3 (e.g. M2) L&Z,~
= L, &AN
= F-_n&&
=ein(N+I)~_-n,2AN=eix(N+l)~
--II
A
M&N-
iaNA
M2.N.
The same rule also applies to the points Mi, M3: &AM,N(k, z, rh) =eiXNAMj,,v(k, z, rh) . In conclusion, we also point properties of A(k, z, T):
(36)
out the trivial
parity
~A’d+(k,z,~~)=A’(-k,,K,~,r~) =?A’(k,z,q,).
(37)
383
(5 ) The vortex lattice should also have some symmetry properties under rotations around symmetry points (the hexagon vertices, K,, K2 and midpoints of hexagon sides, M,, A4,, M3) of an elementary cell (see fig. 2): since the hamiltonian is invariant under magnetic rather then the usual translations, the formal definition of such operations is: P_AlZiFA where fA is the magnetic translation of A(k, z) on the vector l=l& or TMj, and Li is an element of the point symmetry group of points Kj or A+ In this connection it is more convenient to speak about the symmetry of the functions FAA(k, z): A, (k, z), A, (k, z). E.g. the symmetry groups of AK,,N (k, z) (A=a(2r,+1)/3) and AM,,N(k, z) (~=a~,$?) for the quantum numbers (0, N) are
8. Conclusion The results obtained in this paper are as follows. ( 1) In most cases near HC2 the mixed state of superconductors with unconventional pairing is the hexagonal vortex lattice with periods a, ~,,a. (2) The symmetry properties of the lattice are determined by two quantum numbers (a, N) of the solutions of linearized GL equations. cr= ? 1 is the parity under reflection B in the plane perpendicular to the magnetic field; N the generalized Landau level number. (3) There is no one-to-one correspondence between pairing type at the uniform state at H=O and the symmetry of mixed state near HC2: for the same pairing type the different quantum numbers (a, N) of A( k, z) can be realized. However, the pairing type restricts the possible choice of (cr, N). E.g. for the swave pairing there exists only one possibility: ( +, O),forp-wavepairingthree: (+, -l), (-,O), (+, +l), etc. (4) The mixed state properties can be described by the symmetry group H of the order parameter A( k, z). If the position of the coordinate center r is chosen at the center of some vortex, this group H = Hr is: H={ei=FO, eixF;hn, e--2ix(N+i)/6L2rr,6, a& &sH, eixCN+‘)ao . The last element
comes from the identity:
(38)
f=&8.
(40) The symmetry groups for the points K,, M2, M3 are defined by the formulas ( 32)) ( 35 ) and ( 36). (6) Symmetry properties (38)-(40) were obtained by constructing the hexagonal lattice wave function A(k, z) from the eigenfunctions of the infinite degenerate Nth eigenlevel of the linearized eq. (6) and by using the properties of these functions. Strictly speaking, such functions are the solution of the nonlinear problem only when H is the intinitesimally close to HC2. However, we can guarantee that sufficiently close to HC2the symmetry of A(k, z) will be the same. This follows from the fact that the eigenfunctions of the Nth level are the basis functions of the IR of the initial symmetry group (4). To break the lattice symmetry one needs to admix to A(k, z) some other IR. Since for the two different IRS the probability to have the same critical temperature (field) is accidentally small, the break of the symmetry (38)-( 40) is only by virtue of another phase transition at some critical field H* < H,,. Of course, below HC2 it will be the mixing of another IR, but without symmetry violation. To specify it we will point out that from eqs. (38)-(40) it follows that (as obtained in section 4) the quantum number N classification of the solutions of linearized GL equations in the mixed state reduces to the classification
384
I. Luk’yanchuk, M. Zhitomirsky /Symmetry of the mixed state
over mod 6 since the phases AN( k, z) and AN+6 (k, z) have the same symmetry groups. So, the basis functions from the N+ 6, N+ 12, ... eigenlevels are mixed with AN( k, z) . For the usual s-wave superconductor this means that the hexagonal Abrikosov lattice for arbitrary fields can be constructed from the 0, 6, 12, ... Landau levels wave functions, and other Landau levels do not participate in it. Numerically this fact was obtained in ref. [ 16 1. In accordance with phase transition theory the mixing is as small as ( 1- T/T,) 3 or ( 1 - H/Hc,)3 if we go from H,, to the lower fields. Now we are able to give the formal definition of the unconventional superconductivity in the mixed state since the definition given in ref. [ 31, as was pointed in section 1 is not valid in a nonuniform state. The state with symmetry group ( 38 ) - ( 40 ) with rr= +, N=O(mod 6) can be defined as the usual Abrikosov lattice since an s-wave superconductor has the same symmetry properties. The state with another o and N( mod 6) can be formed only by superconductors with anisotropic pairing and hence they should be defined as unconventional. In this paper we considered the nontrivial superconducting mixed state in an isotropic metal. However, the results can be straightforwardly generalized to the case of an anisotropic crystal where the magnetic field is directed along some n-fold symmetry axis. The quantum number N classification of the eigenfunctions of eq. (6 ) will reduce to the classification over mod it [ lo], which in turn will reduce the classification scheme of the vortex lattice state in an appropriate way. The hexagonal symmetry of the vortex lattice near H,, and different classifications of superconducting phases at H=O and near Hc2 can be the reason of phase transitions in the mixed state. As it was shown in ref. [ 8 1, the eigenlevels of linearized eq. (6) with eigenfields lower then Hc2 are responsible for the phase transitions. Depending on (rr, N) these transitions, which are structural phase transitions in the vortex lattice, are distortions and/or period multiplications and/or a-parity violations. Here we will only illustrate this possibility by two examples for the case of p-wave pairing, using he solutions of the GL equations which were studied in section 2. (a) Let us assume that in eq. ( 1) b < 0 and that at H=O the small crystal field anisotropy fixes the real vector ?,Jalong the z axis. If at Hc2 the phases (3b)
or (3~) occur, there should be another phase transition with parity r~violation at a field H*, lower than Hc2. This phase transition is conditioned by the eigentield of solution (3a). At H-H,, the odd part of A( k, R) will vanish and at H-c H,, only the even part will exist. There will be the same scenario if b> 0 and if at H,, the phase (3a) occurs. The more nontrivial conclusion following from the different quantum numbers N of the phases (3) is that the phase transition at H=H* should always be accompanied by some structural phase transition in the vortex lattice. (b) Let b < 0; the small anisotropy fixes real 11in the x-y plane, and the solution (3~) near H,, occurs. For such a case (as was shown in ref. [ 17 ] ), the vortex lattice near H,, has a form distorted along q. However, the vortex lattice which occurs near Hcz, delinitely has the hexagonal structure (at least at I& = K3, a = - 2b). Therefore; in the mixed state at some field H=H* there should be the phase transition with vortex lattice deformation. Solution (3b) is responsible for this transition. It is possible that the (a) or (b) scenarios take place in UBe,3 where a line of specific heat peculiarities was observed [ 18 1. We can assume also that the instability to distortions in the vortex cores in unconventional superconductor [ 5 ] originates from the described effect. There must be a structural phase transition in the vortex lattice if the linear equation (6) has two close eigenlevels. This is the case in UPt, where the kink in Hc2 seems to be due to the intersection of the eigenfields with different quantum numbers (a, N)
[lOI. Acknowledgements We would like to thank V. Mineev and G. Volovik for the helpful discussions, and H. Capellmann for his hospitality and interest in the work. This work was supported by BMFT, Bonn, Germany (Grant No. FKZ.: 13N5750) and by the Academy of Science of Russia.
Appendix In this appendix
we will formulate
in the terms of
I. Luk’yanchuk, M. Zhitomirsky /Symmetry
our article the Abrikosov theory of the s-wave superconductor nuclei lattice near Hc2 which is described by the usual GL equation a(T-T,)B(R)+bld(R)I*d(R) (41)
+ Maxwell equations. The mixed state order parameter d(R) near Hc2 should be constructed from the eigenfunctions of the lowest energy level of he linearized GL equation (4 1) which in the complex notation can be written as
( Id(R)
14>
(Id(R)
12)2
= [(~-7)/2i]“~(
]e2(0, 2~)]‘+&(0,
(
(a-z,
(0-Z)
+ ; d(z, Z) > =(l-T/T,)d(z,Z).
2r)]12)
(46)
over 7. The level lines for p(7) are shown in fig. 2. The minimum corresponds to the point 7h= ezxi16 in the hexagonal nuclei lattice. The transformation rule under the rotation of @,,(z, Z, TV) by 60” also can be obtained from the theory of the &function: c2x/3@o(z, 5, Th) =ezrri’%&(z, 2, th) .
-
385
the Maxwell equations. This procedure can be reduced to the minimization of the Abrikosov parameter /3(r) [ 121
‘(‘)=
-K(V-igA)2d(R)=0
of the mixed state
(47)
(42) References
This equation formally coincides with the Schrodinger equation for an electron in a magnetic field. The eigenfunctions of the lowest Landau level (which corresponds to the highest H,, = (U/~&K) ( T, - T) ) have the form Y=f(z)e-ZZ,
(43)
wheref( Z) is an arbitrary analytic function of Z (but not z! ). This arbitrariness means the infinity degeneration of the Landau levels occur. The Abrikosov lattice d(R) near Hc2 is the solution of eq. (4 1) which is invariant under magnetic translations {T,, T,,}. So, in the parallelogram (a, 7a) (a=a(r) is defined by eq. (23)),f(Z) should satisfy the quasiperiodic boundary conditions: f(Z+u) f(
z+
.fu)
= -e2faea2f(Z) =
,
_e-ix~e+a*e2ffcxf(~~
.
(44)
From the theory of functions it is known that there exists only one analytic function which transforms like eq. (44). It can be written by the &function. The solution d(R) = c&,(z, F, 7) has the form ~~(z,Z,~)=~,(z/u,7)e’~e-“. To find the lattice parameter the free energy functional F(
(45) 7 7)
one must minimize taking into account
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of the mixed state
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