Physica C 317–318 Ž1999. 55–72

Superfluid 3 He and unconventional superconductors Peter Wolfle ¨

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Institut fur ¨ Theorie der Kondensierten Materie, UniÕersitut ¨ Karlsruhe, 76128 Karlsruhe, Germany

Abstract The properties of superfluid 3 He as a well-studied model system of unconventional superfluidity are reviewed. This will include a discussion of BCS theory extended to spin triplet p-wave states, the nodal structure of the energy spectrum and its implications for the thermodynamic and transport properties. Similarities with the model states proposed for the heavy fermion and high Tc cuprate superconductors will be pointed out. Then, the existence of textures, vortices and other topological defects in the order parameter field of s-3 He is discussed and contrasted with unconventional superconductivity in metals. Order parameter collective modes in s-3 He and their possible counterparts in superconductors are considered. The role of the electric charge in superconductors and its coupling to the electromagnetic field will be analyzed. The last part will cover some recent results on nonlocal and nonlinear corrections to the magnetic penetration depth. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Superfluid; Superconductors; 3 He

1. Introduction The phenomena of superfluidity and superconductivity of Fermi systems continue to fascinate both experimentalists and theorists even more than 80 years after the discovery of the first superconductor by Kamerlingh Onnes in 1911. While the fundamental reason for both of these phenomena, the macroscopic condensation of quasiparticles obeying Bose statistics, had been known since the early times of quantum mechanics, it was not clear for a long time, how sufficiently stable bosons might arise in the conduction electron system of a metal. It took another 30 years until Bardeen, Cooper and Schrieffer

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Fax: q49-721-698150; E-mail: [email protected]

ŽBCS. w1x formulated the famous pairing theory in 1957, which pointed out that even a weak attraction between the fermions will lead to the simultaneous formation and condensation of quasibosons, the Cooper pairs. Shortly after BCS it was proposed that the Fermi system 3 He should undergo a BCS-transition as well. However, because of the hard core nature of the interaction potential between 3 He atoms, the Cooper pairs were predicted to form in a state of nonzero relative angular momentum w2x. The basic properties of a p-wave superfluid were calculated from generalized BCS theory by Anderson and Morel ŽAM. w3x and by Balian and Werthamer ŽBW. w4x, among others, even before the superfluid phases of 3 He were finally discovered in 1971 w5x Žthe discovery by Lee, Osheroff and Richardson was rewarded by the 1996 Nobel prize in physics.. It soon turned out that the properties of the two superfluid phases could be very well accounted for by the two model

0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 0 4 4 - 1

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P. Wolfler Physica C 317–318 (1999) 55–72 ¨

states considered earlier by AM and BW. A key to the first identification of the phases were their magnetic resonance properties, and the subsequent theoretical interpretation by Leggett w6,7x. In the years following the discovery it turned out that superfluid 3 He is one of the most fascinating condensed matter systems yet discovered w2x. It is a test system for many of the ideas of modern theoretical physics: Ži. the simultaneous occurrence of several broken symmetries, including non-Abelian gauge symmetries; Žii. the existence of a multitude of topological defects of the order-parameter field; and Žiii. the possibility of massless and massive collective modes of the order parameter. In the context of the physics of condensed matter superfluid, 3 He is a model system for anisotropic superfluidity and superconductivity. Being an extremely clean system Žany impurities are deposited at the container walls by virtue of the quantum fluctuations of the 3 He atoms. and having the highest possible symmetry, the phenomenological theory of Fermi superfluids in this case could be refined and developed to give a detailed and accurate description of the vast majority of experimental observations. The concepts of the generalized BCS theory and the quasiclassical transport theory of anisotropic superfluids were thus well tested and ready for use with other systems such as neutron stars and anisotropic, so-called unconventional superconductors. Since the late seventies, a number of superconductors with strongly anisotropic order parameter have been found. The best candidates for unconventional superconductivity Ži.e., a superconducting state in which in addition to gauge symmetry some other symmetry is broken., are the heavy fermion compounds, e.g., UPt 3 , and the cuprate superconductors. In these systems the pairing of electrons Žwhich has been established experimentally beyond doubt. appears to be mainly caused by correlations within the system of electrons itself, not unlike the situation with liquid 3 He. Since a strong local Coulomb repulsing among conduction electrons is an important feature of these systems, it is not implausible that the Cooper pairs are not in the fully symmetric s-wave state, but in a nonzero angular momentum state, which again allows the partners of a Cooper pair to avoid each other Žand thereby the Coulomb repulsion. to a large extent. This argument may be ex-

pected to be true irrespective of the nature of the normal state, be it a Fermi liquid, marginal Fermi liquid, Luttinger liquid, etc. A straightforward application of the theory developed for 3 He to unconventional superconductors is possible, but meets with serious limitations. These are mainly due to Ži. the reduced symmetry of electrons in a crystalline solid, as compared to an isotropic liquid Žsuch as 3 He. and to Žii. the existence of imperfections even in the cleanest samples. Of course, there is also a difference between a charged Fermi system, coupled to the electromagnetic field, and a neutral Fermi system Žwhere rotation can substitute for the action of a magnetic field..

2. The Fermi liquid 3 He Fermi liquid FL theory was introduced by Landau to account for the properties of normal liquid 3 He w8–11x. Indeed, the relations of the Fermi liquid theory are very well obeyed by 3 He in the temperature range from about 50 mK down to the superfluid transition temperature at 1 to 2.6 mK, i.e., in a temperature interval spanning one and one-half decades. The reason for the limited range of validity of FL theory is the emergence of bosonic excitations, in particular spin fluctuations, above 50 mK, well below the Fermi temperature of TF , 1 K. Within Fermi liquid theory it is assumed that the low-lying excitations of a strongly interacting Fermi system are fermionic quasiparticles. The quasiparticles are assumed to be in one-to-one correspondence to the excitations of the noninteracting Fermi gas, ™ and therefore carry spin 1r2, momentum k, and charge e Žfor the charged system.. Their number in momentum space is denoted by n™k s . At zero temperature the quasiparticles occupy a Fermi sphere of radius k F in momentum space, where k F is determined by the particle density n,n s k F3 r3p 2 . The quasiparticle energy may be parameterized as e™k s s Õ F Ž k y k F . y m , where the Fermi velocity Õ F s k FrmU , mU is the effective mass and m is the chemical potential. In d s 3 dimensions, the density of states is given by NF s mU k Frp 2 Žcounting both spin projections.. The average number of quasiparticles in momentum and spin eigenstates, n™k s , thus

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

describes the low energy states of the system completely. At the heart of Fermi liquid theory is the effect of interactions with other excited quasiparticles on the energy of a given quasiparticle d e™k s s Ý f™k s ™k X s X dn™k X s X . ™X

ks

Ž 1.

X

For isotropic systems with short-range interaction the Fermi liquid interaction function may be parametrized by f™k s ,™k X s X s

1 NF

`

Ý Pl Ž kˆ ,kˆX .

Fls q Flass X .

Ž 2.

ls0

™ ™

Here kˆ s kr< k <, s s "1, Pl Ž x . are the Legendre polynomials, and Fls and Fla are the dimensionless spin-symmetric and spin-antisymmetric Landau parameters, which characterize the effect of the interaction on the quasiparticle energy spectrum. It is well known that the thermodynamic properties of the quasiparticle system at low temperature T < T F are those of the Fermi gas, renormalized by the interaction. Thus the specific heat C V , the spin susceptibility x and the density response d nrd m are given by C V s Žp 2r3. NF T, x s m20 NFrŽ1 q F0a ., d nrd m s NFrŽ1 q F0s .. From the experimental data for these quantities one finds for 3 He that while the Landau parameter F0a , y0.7 weakly dependents on pressure, F0s increases from ; 10 to ; 100 when the pressure increases from 0 to the melting pressure. Using the relation F1s s 3Ž mU rm y 1., from Galilean invariance, one finds F1s ; 6–15. The fact that the dimensionless interaction parameters Fls,a are much larger than unity implies that 3 He is a strongly correlated Fermi liquid. The Landau quasiparticles introduced above are not exact eigenstates of the many-body system, but rather have a finite lifetime t k . As long as the Fermi surface exists and the interactions are not too singular Ža counter example is the long-range interaction mediated by the transverse electromagnetic field between electrons wHolstein et al.x., the quasiparticle decay rate may be shown to be proportional to Ž e k y m . 2 or T 2 , whichever is larger, and hence the energy width of the quasiparticle state is much smaller than the energy itself, i.e., the state is well defined.

57

In order to explore the instability of the Fermi liquid state with respect to pair formation, one needs to calculate the effective interaction between quasi™ ™ particles in momentum states < k : and < y k : on the Fermi surface. In principle the full information on the interaction is contained in the scattering ™ ampliŽ1,2;3,4., of quasiparticles in states < k 1 s 1 :, tude A ™ ™ ™ < k 2 s 2 : scattering into states < k 3 s 3 :, < k 4 s4 :. Isotropy in spin space leaves only two independent amplitudes A J, J s 0,1 for the initial and final state being a spin singlet or™triplet. Isotropy in orbital space and the conditions < k i < s k F , i s 1,2,3,4 leave only two independent orbital parameters, which may be chosen as the dimensionless momentum transfer Ž1 ™ 3. Q s qrk F and the ŽLandau. angle u between the ™ ™

™ ™

planesŽ k 1 ,k 3 . and k 2 ,k 4 , i.e., . ™ ™ The scattering amplitude of q.p. in states < k ) , < y k : is then given by taking the limit u ™ p , as A J Ž Q, u ™ p .. When the pair instability sets in, e.g., upon lowering the temperature T ™ Tc , A J Ž Q, u ™ p . will diverge, signalling the phase transition into the superconducting state. Outside a narrow range of width D u < 1 around u s p , the scattering amplitude is a measure of the pair interaction, and may be interpolated to give a regular function A reg Ž Q, u .. One may construct the relevant part of A by extrapolating from areas of Ž Q, u .-space where information is available to the desired regime Ž Q, u s p .. In doing this one has to take care that the Pauli principle is respected, requiring A0,1 Ž1,2;3,4. s "A0,1 Ž2,1;3,4. and A1 Ž Q s 0, u s 0. s 0. The scattering amplitude may be expanded in a complete basis of eigenfunctions X l mŽ Q, u . of the exchange operator satisfying these constraints w1,12,13x. Further, making use of the fact that the Landau parameters are related to the Q ™ 0 limit of the scattering amplitude as A sl s Flsrw1 q FlsrŽ2 l q 1.x s Ž3 A1l q A0l .r4 and A al s Flarw1 q FlarŽ2 l q 1.x s Ž A1l y A0l .r4, one may determine the first three terms in the expansion of A in terms of X l m from the measured values of F0s, F0a and F1s. Defining the dimensionless pair interaction parameters l l by

V™ky™k X s

2 NF

`

ˆˆX . Ý Ž 2 l q 1. ll Pl Ž kk ls0

Ž 3.

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

58

one finds the general expression

ll s

1 8

2

2

H0 dQQP Ž Q r2 y 1. A l

J reg

Diagonalizing the mean field Hamiltonian one finds Fk in terms of Dk :

Ž Q,p .

Ž 4.

and in the above approximation

l0 s y 12 Ž A s0 q 3 A a0 . y A1s , 32 , 1 6

l1 s Ž

A s0 q A a0

1 3

. ,y .

Dk ss X ™

F™k ss X s

2 E™k s

Ž 1 y 2 f Ž Ek s . .

Ž 8.

™

where f is the Fermi function and

Ž 5. q E™k s s j k2s q Ž D™k D™ k . ss

A more complete analysis involving 6 eigenfunctions and additional information on the scattering amplitude from transport measurements shows that in the whole pressure range l0 and l2 are repulsive, while l1 is definitely attractive. From this one has to conclude that liquid 3 He is undergoing a transition into a pair correlated state below a temperature Tc , T0 expŽ1rl1 ., where T0 may be identified with the temperature beyond which fluctuation effects begin to modify the pure Fermi liquid behavior, or T0 , 50 mK. Obviously, the determination of the transition temperature requires a full microscopic theory, which reaches beyond the phenomenological quasiparticle description considered above. Such a theory is not available yet. Nonetheless, the above demonstrates that at least the qualitative nature of the effective pair interaction may be inferred from a judicious application of Fermi liquid theory.

1r2

Ž 9.

is the energy of fermionic excitations ŽBogoliubov quasiparticle. in the superconducting state Žhere we have assumed that Dk ss X is either proportional to a unitary matrix or else is diagonal.. The solution of the gap equation is most conveniently discussed in terms of the eigenfunctions of the pair potential on the Fermi surface, assuming V™ky™k X to vanish outside a shell of width 2 e 0 about the Fermi energy. These eigenfunctions may be classified according to the representations of the symmetry group of the Hamiltonian. In the case of 3 He the Ž .™ Ž . symmetry group is G s SO Ž3.™ L = SO 3 S = U 1 u corresponding to the full rotation groups in orbital space and spin space and the gauge group Žwe neglect the small spin–orbit coupling for the present.. The representations are labelled by the angular momentum L and by the spin S of the Cooper pairs, and hence are Ž2 L q 1 = .Ž2 S q 1. fold degenerate. We may write the eigenvalue equation

3. Superfluid 3 He

VˆcmL,,Sm s VL ,S cmL,,Sm

3.1. Equilibrium properties in weak coupling theory According to BCS, any attractive interaction component between the quasiparticles in 3 He should lead to the formation of a condensate of pairs, expressed by a nonzero expectation value

where m s yL,y L q 1, . . . , L and m s yS,y S q 1, . . . ,S. The Pauli principle requires the gap parameter to be antisymmetric, or D™k ss X s yDyk™s X s , and therefore only even Žodd. values of L are allowed for spin S s 0Ž S s 1.. The eigenfunctions are given by

² ayk™s X a™k s s F™k ss X :

cmL,0,0 Ž k ; ss X . s itssy X ,YL m Ž kˆ . ,

™

Ž 6.

where a™k s annihilates a quasiparticle of momentum k and spin s . This allows a mean field approximation of the interaction term in the Hamiltonian, leading to the off-diagonal Žin particle-hole space. energy or gap function

™

Dk ss X s Ý Vkyk X Fk X ss X . ™

™

™X

k

™

™

Ž 7.

Ž 10 .

L even,

™

cmL,,1m Ž k ; ss X . s i Ž t yt m . ss X YL m Ž kˆ . , L odd,

m s x , y, z

Ž 11 .

where the t m are the Pauli matrices and the YL m are spherical harmonics. For liquid 3 He, the L s 1 pair interaction component is the strongest attractive one, and hence there are 3 = 3 s 9 degenerate eigenfunc-

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

59

™

X 1,1 Ž . available, in which the gap functions cm, m k: ss tion may be expanded:

3

Dk ss X s ™

3

dm m Y1 m Ž kˆ . Ž it yt m . ss X

Ý Ý ms1 m s1 3

s

3

Ý Ý

dm j kˆ j Ž it yt m . ss X

Ž 12 .

js1 m s1 ™ ™

where kˆ j are the components of kr< k <. The matrix dm j is the order parameter proper of the system. The considerable freedom associated with the large number of components gives rise to a rich variety of behavior, in particular the existence of several different equilibrium phases as a function of pressure, temperature and magnetic field, the appearance of new broken symmetries and collective modes. The critical temperature Tc follows from the gap equation in the limit D ™ 0, when the different components are independent. For each attractive pair interaction component one finds a nonzero transition temperature TcL s 1.13 e c exp y

1 N Ž 0 . VL

.

Ž 13 .

Tc does not depend on the magnetic quantum numbers m and m ; all components cm,L m for given L emerge at the same Tc . In the case of several attractive VlX s, the gap parameter may have very small admixtures from these other channels, appearing only at T < Tc , except if these VlX s are accidentally very close. In the case that the largest Tc occurs for nonzero L Žfor 3 He, L s 1., the gap parameter is given by that superposition of the eigenfunction cm,L m leading to the lowest possible free energy. In the weak coupling limit one may show that the minimum is obtained for the least anisotropic state, i.e., the one q ™ 2 .x :rw²Trs Ž Dq .:x2 is closfor which ²w Trs Ž D™ k Dk k Dk est to unity. For L s 1 there is actually one exactly isotropic state, the BW state, of the form W

DkBss X s De i f Ý Rm j Ž it yt m . ss X kˆ j ™

Ž 14 .

m, j

where Rm j is a rotation matrix of relative spin–orbit rotations. Thus the BW-state has an isotropic energy gap just like an ordinary s-wave state. It can be shown to describe the B phase of 3 He, which in the

absence of a magnetic field occurs in the larger part of the phase diagram. A magnetic field is found to suppress the B phase in favor of the Ždistorted. A phase, such that for fields in excess of ; 5 T only A phase is left. While the BW state conforms to the predictions of weak coupling theory, the second observed phase, the A phase, which exists only at high pressure and high temperature, does not. As first shown by Anderson and Brinkman w14x its existence can be derived from a strong coupling effect, namely the feedback of the order parameter into the effective pair interaction in the superfluid. At high pressurertemperature, when ferromagnetic spin fluctuations become more pronounced, an anisotropic spin state is favored, which unlike the BW state with even distribution of magnetic substates S z s 0," 1, only has S z s "1 components Ž‘equal spin pairing’.. This is the Anderson–Brinkman–Morel ŽABM. state M y™ ˆ X ˆ DkAB ˆ q inˆ . ss X s D0 d P Ž it t . ss k P Ž m ™

Ž 15 .

where the unit vectors d,ˆ m ˆ and nˆ specify preferred directions in spin space Ž dˆ. and orbital space Ž m ˆ and nˆ .. The orientation of m ˆ and nˆ is perpendicular, nm ˆ ˆ s 0, and m, ˆ nˆ and lˆs mˆ = nˆ span a triad, with lˆ the orientation of the orbital angular momentum of the Cooper pairs. The energy gap of the ABM state is anistropic 2

2 ˆˆ Ž Dq k D k . ss s D0 1 y Ž kl . dss X

X

Ž 16 .

with two point nodes at kˆ s "l.ˆ The low temperature behavior of the thermodynamic properties and those transport properties involving thermal excitations is governed by the node structure of the gap parameter. Whereas for a state with isotropic gap D such as the BW state Žor an s-wave state. the familiar thermally activated behavior A expŽyDrT . is observed, e.g., in the specific heat, the change in spin susceptibility x ŽT . y x ŽT s 0., the normal fluid density rn , for a state with nodes in the gap function the quasiparticles will predominantly populate a region around the nodes u ; u 0 of linear extension dkrk F s ŽTrD0 .1r n, if the gap vanishes as a function of angle u as D A Ž u y u 0 . n. In the case of linear point nodes, as far as the ABM state, the fraction of the Fermi surface populated at

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

60

temperature T is Ž dkrk F . 2 A ŽTrD0 . 2 . It follows that the specific heat of the ABM state at low T varies as C VAB M ; C N Ž dkrk F . 2 ; NF T 3rD02 , and similarly for the other quantities. The anisotropy of the order parameter also implies that tensor quantities are in general anisotropic in these states. For example, the normal fluid density rn , characterizing the flow of thermal Žquasiparticle. excitations, is defined by ™ g n s l™ n Õn .

r

Ž 17 . ™ gn ,

The normal fluid mass current density which is equal to the momentum density, may be calculated for a steady state, uniform quasiparticle flow of velocity ™ Õn from ™ gn s

™

™

™

™

k

Ef

™

Ý k f ž Ek y kÕ™n / s Ý k ™

k

ž

y

E Ek

™

/

™ ™ kÕ nq0

Ž Õn3 . . Ž 18 .

Here f Ž E . is the Fermi function and the quasiparticle ™ ™ energies Ek acquire a ‘Doppler shift’—kÕ n in the ™ frame moving with velocity—Õn . By comparing Eqs. Ž17. and Ž18., one finds that the components of l rn are given by

rn ,i j s Ý k i k j Ž yE frEEk . .

Ž 19 .

™

ks

Thus, the local preferred directions Že.g., m, ˆ n, ˆ lˆ in the ABM state. are the principal axes of the tensor l rn . In the BW state l rn is isotropic. The eigenvalues of l rn in the ABM state are very anisotropic, in particular at low temperatures, where rn 5 s lˆl rn lˆ; 2 4 ŽTrD0 . , whereas rn H ; ŽTrD0 . . The Fermi liquid interaction Ž F1s . renormalizes rn in an important way. A magnetic field is observed to have two effects. The strongest effect is the suppression of Cooper pair spin components with magnetic quantum number m s 0. This effect disfavors the BW state Žwhich is an equal mixture of all three spin components. as compared to the Žproperly rotated. ABM state. In fields of ; 0.6 T the B phase is completely suppressed in favor of the A phase. The second, much smaller effect is due to the small particle–hole asymmetry of the density of states and the pair interaction, which leads to a small splitting of the A transition into two transitions. First, the A 1 phase forms, con-

sisting of Cooper pairs of the favorable spin orientation, before the A 2 phase follows with both spin species condensed albeit with different weight. 3.2. Ginzburg–Landau expansion of the free energy Near a second order phase transition the order parameter OP is small and the change in free energy induced by the formation of order may be expanded in powers of the OP. The general form of this expansion is determined by the symmetry of the system. In the case of liquid 3 He, the free energy F must be invariant under rotations in orbital space and spin space neglecting again the small spin–orbit coupling, and F must be a real quantity, of course. In terms of the normalized order parameter matrix Am j s 3y1 r2 dm jrD Ž T . ,

Ž 20 .

the G–L free energy takes the form w14,15x F s FN q aD 2 q 12 D 4 b 1
½

qb 3 tr Ž AAT . Ž AAT .

U

qb5 tr Ž AAq . Ž AAq .

U

q b4 tr Ž AAq .

5.

2

Ž 21 .

There are five possible fourth order invariants in this case, obtained by contracting the orbital and spin indices of AAUAAU in all possible ways, multiplied by phenomenological parameters bi . By comparison with the weak coupling result, one finds the values b 2 s b 3 s b4 ss b5 s y2 b 1 s Ž6r5. b 0 in this limit. The problem of finding the stable phase by minimizing F for a given set of b-parameters has not been solved in general so far. However, a combination of analytical and numerical studies has provided a rather complete picture: there are 16 possible order parameter structures characterized by their residual symmetry groups, which fall into two classes. The first class comprises 8 so-called ‘inert’ states. These states retain a fixed structure within their respective domain of stability in b-parameter space. The BW and ABM states are of this type. The remaining ‘noninert’ states depend continuously on the bparameters.

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

The free energies of the BW state and the ABM state are given by F BW s FN y

a2 1 3

2 Ž b 12 q b 345 .

F AB M s FN y

a

2

2 b 245

For the BW state where the OP is given by a real quantity times a phase factor, dm j Ž ™ r . s < DŽ ™ r . < Rm j Ž ™ r . exp i f Ž ™ r.

,

Ž 22 .

Ž 25 .

the phase f Ž™ r . is seen to transform as ™ ™ f Ž™ r . ™f Ž™ r . y 2 mu Pr.

,

61

Ž 26 .

From this relation, one concludes that the quantity

where b 12 s b 1 q b 2 , etc. In the weak-coupling limit, b BW s b 12 q Ž1r3. b 345 s b 0 , whereas b ABM s b 245 s Ž6r5. b 0 . Thus, a 20% relative change of b BW and b AB M is needed to stabilize the ABM phase relative to the BW phase. There are several microscopic models involving the effect of spin fluctuations or transverse current fluctuations, and other excitations, which can qualitatively account for the stabilization of the ABM state. The intuitively most appealing one is due to Anderson and Brinkman w14x. It emphasizes the importance of spin fluctuation exchange in producing the attractive interaction in the p-wave channel. In the superfluid state, the spinfluctuation spectrum, and hence the pair interaction is modified. The equal spin-paring configuration of the ABM state enhances the spin-fluctuations at higher pressure Žlarge mU rm. and not too low temperature, relative to the BW state.

™ Õs s

1 2m

™

=f Ž ™ r.

Ž 27 .

transforms as a velocity It ™is referred to as the ‘superfluid velocity’. Note = =™ Õs s 0. The corresponding super current is given by ™ g s s l™ rs Õs , with l rs the superfluid density tensor. By invoking the twofluid model, we can determine l rs in the following way. The total mass current is obtained by adding the superfluid and normal fluid currents ™

g sl rs P™ Õs q l rn P™ Õn .

Ž 28 .

By Galilean invariance, the mass current in a refer™ ence frame moving with velocity yu is ™ g X s™ g q r™ u l ™ ™ l ™ ™ l s rsŽ Õs q u. q rnŽ Õn q u., and hence rs q rnl s r 1l . For the ABM state things are more subtle, as the OP is intrinsically complex,

3.3. Superflow and textures

dm j Ž ™ r . s < D0 Ž ™ r . < dˆm Ž ™ r. m r . q inˆ j Ž ™ r. . ˆ j Ž™

The most spectacular property of a pair-correlated Fermi system is of course the superfluidity. This property is related to the complex-valuedness of the order parameter, which in turn is a consequence of the broken UŽ1. gauge symmetry. Let us consider the local pair amplitude

Multiplication of dm j by a phase factor is equivalent to a rotation of m ˆ and nˆ in their plane by the angle yf , or m ˆ X q inˆX s e i f Ž mˆ q inˆ .. Taking the gradient and letting f ™ 0 yields ™

™

=f s y Ý m ˆ j= nˆ j .

Ž 29 .

Ž 30 .

j

™™

F™k ss X Ž ™ r . s d 3 r X eyk r² cs Ž ™ r q 12 ™ r . cs X Ž ™ r y 12 ™ r . :.

H

Ž 23 . A Galilean transformation into a frame of reference moving with velocity ™ u, under which the single-par™ . ticle momentum eigenstate w™k Ž™ r . s expŽ ik™P ™ r™ ™ ™ transforms into w™k X s w™k expŽyimuP r . Žusing kX s k ™ . causes F to change as follows y my ™ ™ F™k ss X Ž ™ r . ™ F™k ss X Ž ™ r . exp Ž y2 imu Pr . .

Ž 24 .

The local gap parameter Dk ss X transforms in the same way. ™

Inserting Eq. Ž30. into Eq. Ž27., one can see that the superfluid velocity does no longer describe potential flow. Rather, the flow depends on the local orientation of the OP. In other words, changing the local orientation of the OP can have a major effect on the superflow w16x. Two important consequences are Ži. superflow is less stable in the ABM state: a continuous motion of the orientation of the preferred direction lˆ can ‘unwind’ the phase and dissipate the superflow. The cure for this disastrous effect is the pinning Žor ‘locking’. of the lˆ vector field at boundaries or by external fields Že.g., magnetic fields.. Žii.

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

62

ˆ there may exist defects or ‘ vortices’ in the l-field, which carry a finite Žquantized. circulation, but do not have a normal core. These are actually energetically favored. The defects in the order parameter field configuration have been studied both theoretically and experimentally in rotating liquid 3 He w17,18x. Of particular interest are the so-called topological defects, which are stable in the sense that there removal requires the superfluid to go normal. These defects may be classified in an elegant way by using the methods of topological algebra. In general, the configuration of the preferred directions, e.g., d,ˆ l,ˆ will not be uniform, but will vary smoothly to form a so-called ‘texture’. The textures are determined by the interaction of the OP with boundaries and with external fields. For example, the lˆ-vector of the ABM state is oriented normal to a boundary, because the quasiclassical orbit of the partners of the Cooper pair obviously prefers to be in ˆ a plane parallel to the surface. The d-vector of the ABM state, being perpendicular to the spin vector of the Cooper pair will orient itself perpendicular to an applied magnetic field. The so-called ‘bending’ of the OP costs energy, which causes a certain stiffness of the preferred directions. This is described by the so-called ‘gradient free energy’, which has to be added to the GL free energy in nonuniform states. It takes the form FG s

1 2

3

Hd r  K Ž= d 1

j ml

.Ž =j dmUl .

qK 2 Ž =j dm l .Ž =l dmU j . q K 3 Ž =j dm j .Ž =l dmUl . 4

Ž 31 . with coefficients K i , which in the weak coupling limit take the value K 1 s K 2 s K 3 s Ž1r5. NF j 02 , where j 0 s w7z Ž3.r48p 2 x1r2 Õ FrTc is the zero temperature coherence length characterizing the extension of a Cooper pair. 3.4. CollectiÕe modes The dynamical properties of a macroscopic system are dominated by the existence of collective modes. These modes involve the weakly damped coherent motion of a macroscopic number of parti-

cles in the system. Generically, collective modes may appear only as a consequence of some continuous symmetry or the dynamical breaking of a symmetry. In the absence of long-range order collective modes exist as a consequence of the macroscopic conservation laws of the system Že.g., conservation of particle number and momentum in a liquid leads to the existence of weakly damped longitudinal sound waves.. In a classical liquid, sound waves are only well defined in the hydrodynamic regime, defined by the condition vt < 1, where v is the frequency of sound and t is a typical microscopic relaxation time. In a quantum liquid, where the thermal energy T is much less than the energy per particle Žapproximately equal to the Fermi energy e F in a Fermi system., the relaxation rate ty1 is much less than e F and collective modes, called zero sound, may even exist in the collisionless regime vt 4 1. These modes may be viewed as wave-like excitations of the meanfield experienced by the quasiparticles. A zero sound mode may exist for every angular momentum channel l ŽŽ2 l q 1.-fold degeneracy., provided the corresponding Landau parameter is sufficiently large Žroughly, Fls,a R 2 l q 1 is required.. Since the Fermi liquid interaction is usually a smooth function of the angle u , only the first few Fl ’s may be expected to satisfy the requirement. In the case of liquid 3 He only F0s and Žat high pressure. F1s qualify and correspondingly one observes longitudinal and transverse zero sound w2x. In the presence of long-range order additional possibilities for collective modes arise. For every continuous broken symmetry in the ordered state a gapless collective mode ŽGoldstone mode. emerges. In this case the ground state is invariant with respect to rotations of the preferred direction of the corresponding symmetry variable. A long-wavelength excitation of the symmetry variable only costs the bending energy of the order parameter field, proportional to the wave-vector squared. In addition, the internal structure of Cooper pairs may be capable of oscillations about the equilibrium configuration, the so-called pair vibrations. Their frequency is typical of the order of the gap frequency. In the following, collective modes in the hydrodynamic regime and in the collisionless regime will be reviewed in more detail.

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

3.4.1. Hydrodynamic modes The most common type of collective modes are the modes in the hydrodynamic regime. This is the regime where interaction processes, or in a gas kinetic picture collisions between the quasiparticles are so frequent that the system is close to local equilibrium. In other words, the frequency of the external field driving the system is much lower than typical microscopic frequencies. Most degrees of freedom are relaxed to their equilibrium value, except for certain collective degrees of freedom, which are not allowed to relax because there is a conservation law or because there exists long range order associated with a spontaneously broken symmetry. The number of conserved quantities follow from the symmetry of the Hamiltonian. In the case of liquid 3 He, one has conservation of particle number Žor mass., of momentum, of energy, and of spin. The corresponding hydrodynamic variables are the mass density r , the momentum density g, the energy density e and the spin density s. Instead of the energy density, it is useful to consider the entropy density s, related to e by the first law of thermodynamics. For any spontaneously broken continuous symmetry of the thermodynamic equilibrium state there exists a symmetry variable f , which labels the manifold of degenerate ground states. For example, the phase of the order parameter f is the symmetry variable corresponding to the broken gauge symmetry. In local equilibrium, the equation of motion of the symmetry variable is given by df

EE s

dt

EG

Ž 32 .

where EErEG is the derivative of the internal energy with respect to the thermodynamic variable corresponding to the generator G of the symmetry transformation in question. In the example of the gauge symmetry, the generator of gauge transformations is the number operator of Cooper pairs Nr2 and the derivative EErEN s m , where m is the chemical potential. Hence, the phase of the order parameter obeys the equation of motion df dt

s y2 m .

Ž 33 .

In the B phase the symmetry with respect to relative rotations of spin and orbital space is broken.

63

The corresponding symmetry variables are the three angles umso , m s 1,2,3 describing Žinfinitesimal. rotations about three mutually orthogonal axes. The generators of these rotations are the three components of spin Sm . Correspondingly, one finds d umso

EE s ESm

dt

s vmSP .

Ž 34 .

The three ‘chemical potentials’ vmSP are obtained from the expression for the energy in terms of the spin

g2 EM s

2x

S2 yg SPH

Ž 35 .

where g is the gyromagnetic ratio x is the spin susceptibility and H is a magnetic field, as

vnSP s g

g Sn

ž

x

/

y Hm .

Ž 36 .

For a singly connected superfluid, only the gradients of the symmetry variables are observable quantities. It is useful to introduce the quantities 1 Õs s

™

2m

=f

Ž 37 .

and ÕmSP s

1 2m

™

=umso ,

m s 1,2,3.

Ž 38 .

It can be shown that these quantities transform as velocities under Galilean transformations. They characterize the motion of the superfluid condensate, more precisely the mass and the spin of the condensate and are therefore called superfluid velocity Õs and spin superfluid velocity ÕmSP . The hydrodynamic equations of the B phase neglecting dissipation and the spin–orbit interaction are Ži. the continuity equation, Žii. the momentum conservation law, Žiii. the equation of motion for the superfluid velocity, Živ. the spin conservation law, Žv. the equation of motion for the spin superfluid velocity components, and Žvi. the equation of motion for the entropy density s. The nonmagnetic hydrodynamic modes obtained as a solution of Ži., Žii., Žiii. and Žvi. are first sound and second sound, i.e., a density wave and a temperature wave. Whereas first sound is a very strong and easily excited mode in superfluid 3 He, second sound

64

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

is difficult to excite due to its small velocity and large damping. The collective modes associated with the variables spin and ÕmSP are three spin–orbit waves with linear spectrum v A q w19x. They are made possible by the stiffness of the order parameter with respect to local spin–orbit rotations. The spin superfluid density tensor rmSPÕ is a measure of this stiffness. These spin waves have been detected in magnetic resonance experiments in a restricted geometry. 3.4.2. Goldstone modes and pair Õibration modes in the collisonless regime In a classic isotropic superconductor there is only one well-defined order parameter collective mode, associated with the phase of the condensate. The modulus of the order parameter may in principle oscillate around its equilibrium value, but these oscillations are usually overdamped by pair-breaking processes, impurity scattering and other effects. By contrast the p-wave order parameter of superfluid 3 He is capable of several other types of collective motion. These may be grouped into two classes, depending on whether the internal structure of a Cooper pair is changed or not. If the internal structure is changed, we can view the collective motion as a ‘molecular’ vibration of the Cooper pairs, called pair vibration, which is actually phase locked throughout the system, much like the optical phonon excitation in a crystal lattice. A change in structure of the Cooper pair, e.g., a quadrupolar or ‘squashing’ distortion of the spherical Cooper pair in the BW state, or an oscillation of the preferred vectors m ˆ and nˆ against each other in the ABM state, will cost an energy of the order of the condensation energy and the typical mode frequencies will therefore be of the order D. If the internal structure of the Cooper pairs is not changed, the situation is quite different. Examples are a change of the phase of the order parameter, or a change induced by a rotation in orbital or spin space. Quite generally this can only happen if a continuous symmetry is broken in the superfluid state, or in other words, if the order parameter is not invariant against some such symmetry operation. By definition of a superfluid, gauge invariance is always broken. This causes a ‘massless’ collective mode to appear, called Anderson–Bogolyubov mode. According to a theorem by Goldstone w20x, a massless mode is

generated by each broken continuous symmetry. In the case of superfluid 3 He there are additional Goldstone modes, caused by the breakdown of rotational symmetries. Take the BW state, for example. According to Eq. Ž14. the order parameter involves a matrix Rm j describing relative rotations of spin and orbital space. Since the free energy is independent of relative rotations of spin and orbit space Žneglecting the tiny spin–orbit interaction., all states described by Eq. Ž14., with different matrices Rm j , are energetically degenerate. Thus the spin–orbit symmetry is spontaneously broken. A collective motion of the particles involving a global spin–orbit rotation does not cost any energy. In other words, there is no restoring force and the corresponding mode frequency is zero. On the other hand, a spatial variation of the spin–orbit orientation requires energy, the so-called bending energy, since the order parameter prefers a spatially completely uniform configuration and reacts like an elastic medium to any spatial distortions. Thus, the mode frequency will be a function of the wave vector q, tending to zero as q ™ 0. The power law in q cannot be inferred from this simple argument. Also the question whether the mode is well-defined Žreal valued prefactor. or damped Žcomplex valued prefactor. requires a more in-depth consideration. One finds for the BW state three spin–orbits modes with linear dispersion v s cso q. The weak spin–orbit interaction fixes the relative orientation of spin and orbit space to the extent that the angle of rotation is given by the so-called ‘Leggett angle’ u L s cosy1 Žy1r4., while the rotation axis nˆ remains undetermined. Consequently, one of the modes acquires a gap of the order of the dipole energy. In the ABM state, the relevant broken symmetries are the rotation symmetry in spin space, as indicated by the preferred direction d,ˆ and the rotation symmetry in orbital space, which are both broken down from the full rotation group SO Ž3. to the group UŽ1. of rotations about dˆ or l,ˆ respectively. One finds two spin wave modes with linear spectrum v s cs q and two orbital modes with nonlinear, complex spectrum, describing the damped motion of the lˆ vector. Gauge transformations are equivalent to orbital rotations about l,ˆ so that a combination of gauge symmetry and rotation symmetry is broken, rather than pure gauge symmetry.

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

All of the above applies in the so-called collisionless regime, i.e., in the case when collisions among the quasiparticles are so scarce as to lead only to small corrections to the behavior described above. This is typically the case when the mode frequency is much larger than the scattering rate. In the opposite limit, in the so-called hydrodynamic regime, the dynamics of the system is completely governed by symmetry and the collective modes are derivable from phenomenological theory as has been discussed in Section 3.4.1. The collective modes in the collisionless regime are obtained within time-dependent mean-field theory. The equation determining the change of the order parameter d D™k induced by an external field in weak coupling approximation takes the form w21x d Dp q Ý Vp k u k d Dk k

½

ž

s Ý Vp k Dk Ž v q u . 1 q k

u q2

v

1 y 2

2jk

v

2

ž

Ž v yu . 1q

2jk

v

/

lk

d Dk q

u2

v

5

u kX d Dk .

Ž 39 . Here we have defined u s q P krm ) , E u kX s ™ u k , Ej k

uk s

1 2 Ek

tanh

Y1Um Ž pˆ . Y1 m Ž kˆ .

Ý

Ž 43 .

msy1

where the Y1 mŽ kˆ . are the spherical harmonics. Neglecting spin–orbit interaction, the spin-dependence of the spin-triplet order parameter may be separated out by expanding 3

d Dk ss X Ž q, v . s

Ý

d dm Ž kˆ ; q, v . Ž tm it 2 . ss X

Ž 44 .

ms1

where Žtm .ss X , m s 1,2,3 are the Pauli spin matrices. Expanding d dmŽ kˆ . in the basis of eigenfunctions of the pair interaction:

Ý

d dm m Y1 m Ž kˆ .

Ž 45 .

Ž 40 .

one derives from Eq. Ž46. two sets of linear equay. tions for d dmŽq, s Ž1r2.Ž dm mŽ q, v . " dmUmŽyq, m yv ... For definiteness, we assume the preferred direction dˆ to point along the z-axis. Then, the variable d Žy1. and d Žq. z1 x, y1 , associated with gauge transformations and spin rotations about the x and y axes have eigenfrequencies tending to zero in the limit q ™ 0. They are the Goldstone modes discussed above. Žy1. The remaining m s 1 variables d Žq. z1 , d x , y1 do not show any resonant behavior. Žy. However, the six variables dmŽq. ,y 1 , dm ,y 1 show a collective resonance at the frequency

Ek

vcl , 1.23 D0 Ž T . .

2T

The corresponding collective motion can be visualized as a ‘clapping’ of the two preferred directions m ˆ and nˆ of the order parameter Žthe two ‘hands’., hence the name ‘clapping mode’. The resonance has a finite width even at T s 0 due to pair-breaking processes in the vicinity of the nodes of the gap, where the condition v ) 2 < Dk < for their occurrence can be met. The m s 0 variables show resonance behavior at three different frequencies. The corresponding eigenvalue equation has a solution at v s 0, corresponding to the Goldstone mode associated with the breaking of rotation symmetry in orbital space.

and the bare pair propagator, sometimes called the Tsuneto function, < Dk < 2 l k ,

1

Vp k s V1

ms0,"1

q u kX d e k q l k Dk Dq k d Dk q Dk d Dk Dk

2

regime in the A phase. In the case of 3 He the l s 1 component is dominant. We shall neglect l / 1 components for the present, such that

d dm Ž kˆ . s

/

65

y4v 2u k q u 2j k u kX Dk

Ž 41 .

with D k , v 2 Ž v 2 y 4 Ek2 . y u 2 Ž v 2 y 4j k2 . .

Ž 42 .

For concreteness, we here sketch the calculation of the collective modes obtained in the collisionless

Ž 46 .

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

66

There are two further zeros of the real part, one with a strongly temperature dependent eigenfrequency,

vnfl ,

T Tc

D0

Ž 47 .

called normal-flapping mode, and one with eigenfrequency ;

vsfl - a Ž T . D0 Ž T . ,

Ž 48 .

the super-flapping mode. Here aŽT . is a nearly constant function which varies between 1.6 and 2. The names are thought to characterise the motion of the two ‘wings’ m ˆ and n, ˆ oscillating out of the plane perpendicular to the equilibrium orientation of l.ˆ All of these modes have been observed as peaks in the ultrasound absorption. The calculated values of the absorption are in good agreement with experiment w2x.

approach is greatly reduced w24x. In special cases, when a simple model of the Fermi surface, e.g., a sphere or cylinder shape is appropriate, it may, however, still be useful to establish Fermi liquid relations between different quantities. An example are the models of almost antiferromagnetic Fermi liquids proposed to describe heavy fermion w25x and cuprate superconductors w26x. In the case of cuprate superconductors this obviously requires assuming a Fermi liquid state at temperatures above the superconductive phase, which is difficult to justify in view of the extraordinary properties of the cuprates in the normal state. In the following, we will take the point of view that the superconductive state in both the heavy fermion and cuprate superconductors is a BCS-like state, characterized by Cooper pairs of extension substantially larger than the lattice spacing.

4. Unconventional superconductivity

4.1. Low temperature properties and node structure of the gap

The existence of superconductors with unconventional order parameter is by now fairly well established. Clear experimental indications exist for two classes of strongly correlated electron systems, the heavy fermion compounds w22x and the cuprate superconductors w23x. The term ‘unconventional’ here means states with order parameter structure violating a rotation or reflection symmetry of the system in addition to gauge symmetry. As in the case of superfluid 3 He, the reason for pairing in states with reduced symmetry is a strongly repulsive short range interaction, which can be largely avoided if the partners of the pair are in a state with effectively finite angular momentum. While in the case of the isotropic fermion system 3 He Landau Fermi liquid theory allowed to infer the effective pair interaction to a considerable extent from experimental data on the thermodynamic and transport properties, in the case of unconventional superconductors this is hardly possible. The reason is that the reduced symmetry of Bloch electrons in the crystal lattice compared to free fermions requires introducing many more Fermi liquid parameters and components of the quasiparticle scattering amplitude, such that the predictive power of the Fermi liquid

As we have seen in the section on 3 He, the additional symmetry breaking usually leads to very anisotropic gap structures, characterized by nodes of the gap on the Fermi surface. The node structure governs the low temperature behavior, leading to temperature power laws in the themodynamic and transport properties. A qualitative determination of the temperature power can be given following the discussion preceding Eq. Ž17.. Essentially, the normal state result has to be multiplied by a T-dependent reduction factor to account for the fraction of Bogolyubov quasiparticles with energy gap < Dk < - T. In principle, experimental observation of the temperature power laws would allow the complete determination of the node structure of the gap. Unfortunately, the power laws tend to be masked by additional effects, such as caused by impurity scattering, which can destroy the node structure at low energies. A classification of the possible order parameter structures can be given in analogy to the earlier studies of 3 He. Starting point is the symmetry group of the system, which consists of the point symmetry group of the crystal, the group of rotations in spin space, the gauge group and time reversal symmetry.

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

In the case of heavy fermion compounds, the spin– orbit scattering is strong, and the symmetry operations of Ždiscrete. rotationrreflection in position space and rotations in spin space are combined into a single finite group. In this case only one continuous symmetry is left, the gauge symmetry. Again a natural framework for classifying different gap structures is in terms of the eigenfunctions of the pair interaction V on the Fermi surface VˆcnŽ,Gn . s lŽnG .cnŽ,Gn . .

Ž 49 .

The eigenstates cn,Ž Gn . may be grouped in multiplets labelled by the irreducible representations G of the point group. Different sub-states of an dG -dimensional representation, labelled by n s 1,2, . . . dG have the same eigenvalue lŽnG .. For each representation G there is an infinite number of eigenstate multiplets labelled by n. With increasing n, the eigenstates cn,Ž Gn . acquire an increasing number of nodes. Usually, for given G , we may expect the eigenvalue lŽ0G . Ž n s 0. with the smoothest eigenfunction to be largest Žin modulus.. However, in the case of small point groups this need not be the case. This is clearly seen by observing that under a reduction in symmetry Žsymmetry group G ™ GX . eigenfunctions out of several representations Gi of the larger symmetry group G and with n s 0 are grouped into the same representation G X of the smaller symmetry group GX , however, representing now excited states:  c 0Ž G i.4 X Ž G .4 ™ cn . The obvious example is the reduction of symmetry from free space to the point group of a crystal, under which infinitely many higher angular momentum states c 0Ž lm. s Yl mŽ u , f . Žwith Yl m the spherical harmonics. are grouped into excited states  cn,Ž Gn ., n s 0,1,2 . . . 4 . The largest negative eigenvalue, lŽ0G 0 ., determines the transition temperature. Then, the equilibrium state Žnear Tc at least. will have the gap parameter structure dG

Dk s ™

™

Ý cn c 0,Ž Gn . Ž k . 0

Ž 50 .

ns1

where the cn are determined by minimizing the ŽG–L. free energy. Let us now consider two examples. The first will be the heavy fermion superconductor UPt 3 , probably the best-studied example of its class. The crystal

67

structure is hexagonal, the point group D6 h . Due to the inversion symmetry of the lattice, the single particle states are two-fold degenerate, so that in spite of the strong spin–orbit coupling there is a pseudo-spin dependence. One observes a splitting of the transition of about 10% of Tc ŽTc , 0.5 K. w27x. The splitting has been shown to disappear under applied pressure, at about 4 kbar w28x. There is a weak antiferromagnetic transition at T , 5 K, which seems to be absent at pressures R 3 kbar. This is very suggestive of a two-dimensional representation with a weak symmetry-breaking field due to the anti-ferromagnetic order. The low-temperature behavior of the specific heat, the thermal conductivity, the ultrasound absorption indicates a line of gap nodes in the basal plane. There are two possible representations left, E1 g and E2 u , an even parity, pseudospin-singlet and an odd-parity, pseudospin-triplet state, with order parameters E1 g : D™k "s D0 kˆ z kˆ x " ikˆ y ,

ž

/

E2 u : D™k "s D0 ck ˆˆ z kˆ x " ikˆ y

ž

2

Ž 51 .

/

where cˆ is a unit vector in pseudospin space pointing along the cˆ axis. A general state in the two-dimensional space may be denoted by a 2-component vector ™ h s Žh1 , h 2 ., which specifies the state

Dk ,h s 12 Ž h1 q ih 2 . Dkqq Ž h1 y ih 2 . Dky . ™™

™

™

Ž 52 .

The phenomenological G–L free energy is a functional of ™ h Ž™ r .,

½

F s d 3 r a Ž T . ™™ h hU q b 1 Ž™™ h hU .

H

2

U

qb 2 <™™ h h < 2 q k 1 Ž Dihj .Ž Dihj . U

U

qk 2 Ž Dihi . Ž Djhj . q k 3 Ž Dihj .Ž Djhi . U

qk 4 Ž Dz hj .Ž Dz hj . q

1 8p

™

5

B2 .

Ž 53 .

Here a , b 1 , b 2 , k 1 , . . . k 4 are material parameters, which can be calculated from microscopic theory. There are two fourth order terms and four gradient terms, due to the hexagonal symmetry of UPT3 . The Di ’s are the components of the gauge invariant gradi™ ent operator Di s ErE x i q iŽ2erc . A i , where AŽ™ r . is the vector potential. The last term is the energy of the magnetic field in the sample.

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

68

One finds two possible homogeneous equilibrium states. For -b 1 - b 2 - 0, ™ h s h 0 Ž1,0., and for b 2 ) 0, ™ U Žnote that hqs h 0 Ž1,i ., or equivalently ™ hys ™ hq b 1 ) 0 for stability reasons.. As already mentioned, one observes a splitting of the transition, which would be caused by the symmetry breaking field due to the weak antiferromagnetic order. ™ Assuming that the corresponding staggered field, M s , transforms as a vector in the basal plane, one obtains the additional symmetry breaking contribution to the free energy FSBF s e Ms2 d 3 r Ž

H

Ž 54 .

This term favors an order parameter ™ h s h 0 Ž0,1.Ž™ h s h 0 Ž1,0.. for e ) 0 Ž e - 0.. If b 2 ) 0, this state becomes unstable w.r.t. condensation of the other component at a second critical temperature Tc2 , and develops into ™ hq for T < Tc2 . This would be somewhat similar to the splitting of the transition into the A phase of 3 He by a magnetic field. Whether one of the two-dimensional representations, E1 g or E2 u , is realized in UPt 3 or else two nearly degenerate one-dimensional representations cannot be decided with certainty at this time. Let us now turn to the other example, the high-Tc superconductors w23x. Theoretical model calculations identify a spin-singlet state with d-wave symmetry

Dk s D0 cos 2 w ™

Ž 55 .

as the most stable superconducting state Žhere w is ™ the azimuthal angle of k .. If the crystal point group is assumed to be tetragonal, this state has a reduced symmetry as it is not invariant with respect to rotations by pr2 Ž w ™ w q pr2.. However, if a small orthorhombic distortion of the lattice along the xaxis, say, is taken into account, the state Ž55. does not violate any of the symmetry operations of the lattice and belongs therefore to the identity representation G 0 . In the classification scheme introduced above, Dk of Eq. Ž55. is then proportional to an eigenfunction cnŽ G 0 ., n / 0. The qualitatively distinguishing feature then is not an additional broken symmetry, but the sign change of D™k around the Fermi surface. The state Ž55. has four lines of nodes parallel to the z-axis on the cylindrical Fermi surface. The low

temperature properties may therefore be derived from the fact that the fraction of thermal excitations ŽBQP. near the line nodes is proportional to T. The specific heat is varying as C V ; T 2 , while the largest eigenvalue of the normal fluid density tensor rn ; T. Since the magnetic penetration depth lambda is related to the superfluid density by 1rl2 s Ž8p e 2rc 2 m2 . rs and rs s r y rn , one finds lŽT . s const q a T. A linear dependence of l with temperature has been observed in very clean samples. This is one of the important indications of unconventional superconductivity in high-Tc superconductors. It has recently been questioned that the d-wave state can be thermodynamically stable at low temperatures w29x. The linear T dependence of lŽT . at low temperatures has been argued to give rise to a linear T term in the free energy, implying a nonzero entropy in the limit T ™ 0. However, it may be shown w30x that both, nonlocal effects Ža q-dependence of l. and nonlinear effects remove the precarious linear T dependence. There are two energy scales, the nonlocal scale Enonloc ; Õ F q and the nonlinear scale Enonlin ; Õ S k F , on which the singularity in the current response function leading to the linear T term is cut off. Since the characteristic wave vector is the inverse penetration depth, q ; 1rl0 and using the definition of the coherence length j 0 s Õ FrD0 , one finds Enonloc ; Ž j 0rl0 . D0 . The energy scale for nonlinear effects to be important in the screening is obtained from the estimate of the superfluid velocity generated by the penetrating magnetic field, ÕsŽ y . s Ž erc . H Ž0. = Ž l 0 rm . e y y r l 0 , as E nonlin ; Ž erc . Õ F l 0 H ; Ž HrH0 . D0 , with H0 the thermodynamic critical field. At temperatures T less than either Enonloc or Enonlin , whichever is larger, the free energy Ž56. varies as T 2 , and hence the conflict with the third law of thermodynamics is avoided. A related problem is the so-called nonlinear Meissner effect w31x, i.e., the influence of the screening currents at the surface of the superconductor on the energy spectrum of the excitations and hence on the magnetic penetration length. In the local approximation, i.e., in the limit q ™ 0 one finds a magnetic field dependent correction to l, d lloc

3 ,

l0

2'2

zu

H H0

Ž 56 .

P. Wolfler Physica C 317–318 (1999) 55–72 ¨

69 ™

where H and H0 are the applied and the thermodynamic critical field, respectively. It has been suggested that the dependence on the angle u contained in the function zu s Ž1r2.Ž
velocity ™ Õs and the gradient of the phase =u . In a charged superfluid the coupling to the electromagnetic field and the requirement of local gauge invariance dictate the form of ™ Õs as the gauge invariant combination of the gradient of the phase and the ™ ™ vector potential, ™ Õs s Ž1r2 m. =u y Ž 2 erc . A . The action of a magnetic field on a superconductor is similar to that of the rotation of ™ a neutral superfluid, the angular velocity of rotation V taking the role of the magnetic field. Vorticity penetrates a sufficiently rapidly rotating neutral superfluid in the same way, i.e., in the form of a lattice of vortices, as the magnetic field penetrates a superconductor in the Shubnikov phase.

4.2. Superflow and Õortices

4.3. CollectiÕe modes in unconÕentional superconductors

Much of the fascination of the superfluid 3 He derives from the existence of textures of the order parameter preferred directions, as indicated above. The symmetry space of electrons in a crystal lattice does not allow for continuous rotations of the order parameter in orbital space and, if there is strong spin–orbit interaction, in spin space. The preferred directions of the order parameter are confined to certain discrete orientations with respect to the crystal axes. In the case of degenerate representations, such as the E2 u representation of UPt 3 discussed above, the two components D™kq and D™ky may be linearly combined to the continuous manifold of states ™ h s Žh1 ,h 2 ., where D™k s h1 D™kqq h 2 D™ky, but these states are not degenerate. The equilibrium states are given by only two discrete orientations, e.g., Ž1,0. and Ž0,1. or Ž1r '2 .Ž1, "i . out of this manifold. Thus the long-range, low-energy configurations of the order parameter preferred directions known as textures in superfluid 3 He do not exist in a crystalline superconductor. In the cuprates, where the spin–orbit interaction is weak, the order parameter appears to be a spin singlet, so the possibility of textures in spin space does not arise. The only degree of freedom capable of forming low-energy nonuniform configurations is the phase. In the neutral superfluid 3 He nonuniform order parameter phase configurations u Ž™ r . are associated Ž . with superflow via the relation 27 of the superfluid

ž

/

As discussed in the preceding sections, the possible collective modes of a given equilibrium order parameter depend on Ži. the symmetry of the system under consideration Žsymmetry group G ., Žii. the symmetry of the order parameter Žsymmetry group H ., Žiii. additional information on the dynamical properties of the system, e.g., the form of the pair interaction. In a crystalline superconductor the symmetry group G consists of the point symmetry group of the crystal, the group of rotations in spin space, the gauge group, and time reversal. The spin degrees of freedom are of course only relevant for spin–triplet pairing. In the case of strong spin–orbit coupling, believed to be present in heavy fermion compounds, the symmetry operations in orbital space and spin space are combined into a single finite group. Hence, the only continuous symmetry left is gauge symmetry, which by definition is broken in the superconducting state. Accordingly, in contrast to superfluid 3 He, there is no new Goldstone mode generated by the reduced symmetry of the unconventional superconducting states in either the heavy fermion systems or the cuprates. The Goldstone mode associated with the broken gauge symmetry, however, in a system of charged particles acquires a gap due to the coupling to the electromagnetic gauge field. This mode is identical

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to the well-known plasma resonance. Usually the plasma frequency v pl is several orders of magnitude larger than the gap frequency and is thus a normal state property. However, in special circumstances it may happen that the plasma frequency for charge oscillating along a particular direction is below the superconducting gap frequency. This is apparently the case for polarization along the c-axis in high-Tc ˆ superconductors. As the examples of superfluid 3 He suggests, Ž . well-defined modes with finite gap lim ™ q™ 0 v q s v Ž0. ) 0 may exist in unconventional superconductors. As a rule, the mode frequency should not exceed 2 D, where D is some average over the anisotropic gap, e.g., D s Ž- < Dk < 2 ).1r2 . For higher frequencies the mode would be overdamped by pair-breaking processes. If the anisotropic gap has nodes on the Fermi surface, pair-breaking will be possible for all frequencies v . However, the phasespace for pair-breaking will go to zero for v ™ 0. It is convenient to discuss the order parameter in the representation in terms of eigenfunctions of the pair interactions introduced in Eq. Ž49.. Collective modes of the order parameter are described by a time Žand space. dependent admixture of other eigenfunctions to the equilibrium gap function, Žwhich belongs to the representation G 0 .. Provided that the other eigenvalues lŽnG . of the pair interaction for G / G 0 andror n / 0, are sufficiently far separated in value from the leading eigenvalue lŽ0G 0 ., i.e., there are no accidental degeneracies, the admixture of these states will be costly in energy, and the corresponding collective mode frequencies will be large, such that these modes will be overdamped. It is interesting to note that this includes modes involving a rotation of a given gap structure with respect to the crystal axes, which requires admixture of the appropriate states from representations G / G 0 , if possibly the same n s 0. The only type of collective excitations, which generally have frequencies in the range vcoll Q 2 D are the ones obtained by admixing partner states from the same multiplet G 0 , in the case of a multidimensional representation G 0 . States belonging to a one-dimensional representation like the d-wave state of the cuprate superconductors will in general not have any well-defined order parameter collective modes in addition to the one known from conventional s-wave superconductors. The exception is a

near degeneracy of lŽ0G 0 . with a subleading coupling constant lŽ0G 1 .. In this case admixture of a G 1-component may give rise to a collective mode with frequency vcoll - 2 D if the two coupling constants lŽ0G 1 . and lŽ0G 0 . are sufficiently close, implying that lŽ0G 1 . has to be attractive as well. For the E1 g state of UPt 3 two-dimensional representation introduced above, one finds a collective mode with frequency vcoll , 1.2 D0 , at all temperatures, where D0 is the maximum of the gap w33x. The mode gives rise to a peak in the electromagnetic power absorption at fixed frequency v - 1.2 D0 ŽT s 0. as a function of temperature or at fixed temperature as a function of frequency. A similar mode is expected for the E2 u state. Observation of the mode is difficult because of the inconvenient frequency range of several tens of GHz required to tune into the mode at low T. 5. Conclusion The comparison of the properties of superfluid He and of unconventional superconductors ŽUSC. attempted in the above shows that the similarities are mostly found in the general physics of these two different systems. On the level of the actual behavior observed there are many differences related to both intrinsic properties and external perturbations. Considering the similarities first, both systems are, as far as we know today, pair correlated Fermi systems. The pair interaction, or to put it more cautiously, the correlations inducing the formation of Cooper pairs appear to be generated in both cases from within the system itself. In other words, there is reason to believe that electron correlations effects in the unconventional superconductors are responsible for the pair formation. However, unlike liquid 3 He, where Fermi liquid theory allows to extract information on the pair interaction from thermodynamic and transport measurements, in the crystalline unconventional superconductors Fermi liquid theory is much less powerful due to the reduced symmetry of Bloch electrons in the crystal if the normal state is a Fermi liquid at all. The main reason for non-s-wave pairing in liquid 3 He, the strong repulsive short-range interaction, is expected to work also for USC. There the on-site Coulomb repulsion is the reason for strong electron correlations in the high-Tc superconductors 3

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and in Anderson-lattice models of the heavy fermion compounds. Generically, the tendency of electrons in a Cooper pair or of atoms in liquid 3 He to avoid each other leads to broken symmetries in addition to the gauge symmetry Žalthough extended s-wave states with vanishing amplitude in the limit of small distance of the partners of a Cooper pair are possible, but require a very special pair interaction.. As a rule, these unconventional superfluid states have gap parameters with nodes on the Fermi surface Žan exception being the BW state of 3 He.. The nodes allow for a finite density of states for fermionic excitations even below the gap energy, tending to zero as some power of the excitation energy. Consequently, the temperature dependence of all physical quantities is given by power laws, rather than the activated behavior know from conventional superconductors. One of the consequences of an order parameter belonging to a multidimensional representation of the symmetry group of the system is the existence of collective modes with frequencies v ; D. In the case of 3 He the representation is the L s 1, S s 1 representation and hence is 3 = 3 s 9 dimensional. In the case of heavy fermion superconductors there are candidate states transforming according to a two-dimensional representation for which a collective mode of this type is expected. The differences in the behavior of USC and superfluid 3 He derive mainly from the following three sources. Ži. The full rotation symmetry valid for liquid 3 He is broken down to discrete rotations in the crystalline USC. Žii. The fermions in USC are charged and couple to the electromagnetic field. Žiii. External perturbations of the quasiparticles in USC such as impurity scattering and the interaction with phonons have a major effect. As to Ži., the most important consequence of the discreteness of the allowed rotations is the locking of the preferred directions of the order parameter in ™ k-space to the crystal axes. Therefore, unlike 3 He-A where the preferred direction lˆ may rotate continuously leading to the formation of textures, this is precluded in USC. In USC with negligible spin–orbit interaction, the preferred direction of a spin–triplet order parameter may still rotate continuously, as ˆ does the d-vector in 3 He-A. A further consequence

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of the lack of a continuous symmetry of the space of degeneracy of the order parameter in USC is the absence of ™ Goldstone modes associated with the rotations in k-space. Spin waves may exist in USC with weak spin–orbit interaction and spin-triplet pairing. The coupling of the charge of the electrons in USC to the electromagnetic field gives rise to the Meissner effect. The anisotropy of the order parameter carries over to the Meissner effect as well as the existence of low-lying fermionic excitations, which lead to temperature power laws in the magnetic penetration depth. A second effect of the coupling to the e.m. gauge field is the generation of a gap in the Anderson–Bogolyubov mode or phase-mode which becomes the plasma mode. Finally, unlike liquid 3 He, which is an extremely pure system, the conduction electrons in USC interact with static imperfections and lattice vibrations of the crystal. Impurity scattering has a destructive effect on the order parameter of USC and leads to Tc suppression just like magnetic impurities in a conventional superconductor do. Another effect is the creation of a finite density of states at the Fermi level by impurity scattering, which tends to mask the asymptotic low temperature power laws associated with the node structure of the gap. In reading this, one may get the impression that many of the issues that played and still play a prominent role in understanding the properties of candidate systems for unconventional superconductivity were not discussed in the context of superfluid 3 He, and thus prior knowledge of superfluid 3 He was not essential. Nonetheless, on the conceptional side it was important to have had a model system which could serve to sort out the fundamental issues of broken symmetries, symmetry classifications, the structure of excitations, the collective behavior, in short the phenomenology of unconventional superfluid states.

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