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The difficulty in identifying exotic superconductors
Physica B 197 (1994) 472-480
ELSEVIER
The difficulty in identifying exotic superconductors C.M.
Gould
Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA
Abstract
Conventional s-wave superconductors have order parameters with no internal degrees of freedom - they are characterized by an amplitude and a phase. All unconventional superconductors share the property of a nontrivial internal symmetry which makes their behavior more interesting and their identification more difficult. The determination of the order parameter of the longest-studied exotic superconductor, superfluid 3He, remains inconclusive despite the existence of a Standard Model. Using superfluid 3He as a cautionary tale, the experimental evidence for the identification of order parameters of other exotic superconductors is also examined.
1. Introduction
Thirty years ago, superconductivity was simple. T h e BCS theory of superconductivity [1] was still a young theory enjoying success in explaining n u m e r o u s experimental results. The underlying idea was that quasiparticles near the Fermi surface with opposite m o m e n t a find it energetically favorable to pair with each other, with all systems then known pairing in an isotropic (swave) singlet state. This means that in the superconducting state, the anomalous average of two annihilation operators (ak, a ~ + ) develops an isotropically nonzero value. Even strong-coupling systems like lead and mercury were accounted for with this simple pair wavefunction. Just over twenty years ago, however, the discovery of superfluid 3He [2] d e m a n d e d a generalization of this scheme. The discovery of superconductivity in the heavy-fermion system CeCu2Si 2 in 1979 [3], and subsequent discovery of the high-T c cuprate superconductors in 1986 [4] have stimulated a reexamination of the allowed superconducting states.
T h e question I want to focus on is "what are the possibilities?" and then examine the experimental evidence. In the case of 3He, I will try to show that even after all these years, the question is still a legitimate one. As we will see, for all exotic superconductors there is a wide variety of pair wave functions from which Nature can choose. Only when experiment has demonstrated a reason for picking one particular state over all others can we say we have identified it. N o single experiment can do the complete job because in general, several possible pair states will have identical, or nearly identical, responses to any particular probe. Nor is any experiment free from theoretical bias in its interpretation. These problems are the heart of the difficulty in identifying exotic superconductors.
2. What are the possibilities?
The generalization of the BCS isotropic singlet gap function to a direction- and spin-dependent one was given by Balian and W e r t h a m e r in 1963
0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI: 0921-4526(93)E0173-E
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C.M. Gould / Physica B 197 (1994) 472-480 [5]. The gap now becomes a matrix (in spin space) of functions of direction in momentum space:
evidence of this in 3He [6]. The objective in this paper is to identify experimentally the correct irreducible representation at T c, and for multidimensional representations to discover how Nature has lifted the remaining degeneracy.
=
3. Superfluid 3He
_ 0 d0(k )
+
d0(0/¢))
- dx(I¢) + idy(/¢)
dz(/¢)
dz(f¢)
dx(fc)+idy(f¢) J
where we have explicitly broken the matrix into antisymmetric and symmetric parts, which correspond to spin-singlet and -triplet wave functions. Singlet pair wave functions have a single component, while triplets naturally have three. Writing the triplet part as we have done has the advantage that the construct d transforms as a vector under rotations in spin space: A a ( k ) = (d(/¢)-o- ~io-(2) , re" Because the total pair wavefunction must be antisymmetric under interchange of particles, symmetric spin states (triplets) must be coupled with antisymmetric orbital states (L = odd), and vice versa. In 3He spin-orbit coupling (in the form of nuclear dipole-dipole interactions) is small, as it is in the cuprate superconductors. It is not small in the heavy fermion superconductors, or in nuclear matter. We can now classify all possible solutions to the BCS gap equation according to the irreducible representation of the symmetry group of the system into which they fall. In 3He, an isotropic liquid, we get an infinite set labelled by the orbital angular momentum, neatly separating into singlets (l = even) and triplets (l = odd). In crystals we get a finite set of irreducible representations which is different for every crystal structure. The point of this classification scheme is that asymptotically as T - ~ T c the exact pair wavefunction must fall into one, and only one, of these irreducible representations, the one with the highest T c. As the temperature is lowered there may be mixing of different representations, but in general this is small. There is some
Having found all of the possible states, we now eliminate the wrong ones. In 3He, finding the correct irreducible representation is easy; removing the residual degeneracy is not. The first experiment to reveal that the phase transition in 3He was in the liquid [7], looked at the change in the NMR resonance frequency relative to the normal liquid. The fact that the superfluid is in this sense magnetically active means that it cannot be a spin singlet state. Half of all possible representations just got eliminated. Among the triplets, the simplest, p-wave, was considered first. Balian and Werthamer in their 1963 paper also solved the weak-coupling limit for p-wave superconductors and found the lowest energy state. (Earlier, Vdovin [8] showed this state was the solution of a different formulation of the BCS problem which, unfortunately, is not ill general correct.) This BW state has surprising and nontrivial properties that match all known properties of the B phase of superfluid 3He. There are no competing alternatives. This is powerful evidence that we have identified the correct irreducible representation. Since T c for the A and B phases is the same, their representations must also be the same. Identifying the order parameter of the A ~hase is more difficult. Fortunately, superfluid He is a good mean field system, so near Tc we can apply a simple Landau expansion, suitably generalized for our multicomponent order parameter. Our gap parameter d(/~) can be rewritten in terms of a 3 by 3 matrix of complex constants: d(k) = A k =
k, kz
C.M. Gould I Physica B 197 (1994) 472-480
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which explicitly exhibits the 18 degrees of freedom. In forming the second- and fourth-order terms in the Landau expansion, the fact that the rows and columns of the A matrix transform separately as spin and orbital vectors constrain the ways we can combine four of these matrices. There are precisely five ways they can be combined [9], and in the Landau expansion these five terms each have a separate material dependent coefficient, labelled/31,... ,/35. Here is the means by which microscopic theory makes contact with experiment. Rainer and Serene showed [10] that within Fermi liquid theory, knowledge of the quasiparticle scattering amplitudes in the normal fluid is sufficient to uniquely determine these /3-parameters. If only we understood the normal fluid well enough, we could calculate the/3-parameters and predict the superfluid state at each pressure. What gap parameters are consistent with the A phase? Mermin and Stare [9] showed that in comparing the free energies of A phase states, for purposes of finding the lowest, one need not search a five-dimensional parameter space, but instead only a two-dimensional space of ratios and combinations of the/3-parameters shown in Fig. 1. They also showed that given the static
[33 I ~,[
3
axial 2
axi-planar /~weak- coupl;ng
1~ planar 0 polar "~ -1 0 1 IL+I3,
Fig. 1. Phase diagram of candidate states for the A phase, given the physically relevant condition /31 < 0. Solid lines denote a first-order phase transition, dashed lines a secondorder one. Note the weak-coupling (no interactions) point at the critical intersection of three states with different symmetries.
magnetic properties of the A phase, only six states could ever be extrema of the free energy. NMR frequencies (at all pressures) exclude all but three states: the axial, the axi-planar and the planar. Thermodynamic studies exclude the planar state by showing that the phase transition between the A and B phases in a magnetic field is always first order. Thus, we are down to only two candidates. A word about this phase diagram is now in order. At any specific pressure the/3-parameters take on a particular set of values, and will be projected from the five-dimensional/3-parameter space onto this two-dimensional Mermin-Stare diagram. As we change the pressure, the /3parameters will change. Since interactions are expected to increase monotonically with pressure (varying as the ratio of Tc/TF), we expect to find the physical point approaching the weak-coupling point as pressure is reduced to zero, although how close it can come is not known a priori. At this point, theory took a decisive hand. Spin fluctuations are known to be important in explaining the properties of the normal liquid. Models incorporating spin fluctuations were constructed, modelling all normal state properties, and used to predict /3-parameters [11]. Despite their differences in detail, all of these calculations agree that the/3-parameters lie exclusively in the region favoring the axial state. Experiments of the time supported this conclusion, and the Standard Model was established. Unfortunately, thermodynamic data checking the values of/3-parameters directly was not then available. Now that we have such data it is clear that there is a serious problem with the theoretical understanding of the normal liquid. As we turn on the interactions in 3He, the directions that theory and experiment take through/3-space are nearly orthogonal [12]. Theory cannot, therefore, be a reliable guide to selecting between the axial and axi-planar states. There are several probes that have been used as evidence in favor of the axial state. Curiously, much of this evidence also supports the axiplanar state, for the simple reason that they can be continuously deformed into each other, with an energy cost fixed by the/3-parameters. Never-
C.M. Gould / Physica B 197 (1994) 472-480
theless, the two phases can be distinguished with appropriate probes. The best tool presently is probably NMR. The A phase of the superfluid exhibits both a longitudinal resonance at a frequency that depends only upon temperature, OL(T), and a transverse resonance at a frequency to given (in large fields) by to
2
2 2 = ( T H o ) 2 + R T ~ Q L ( T 1.
The factor R 2 can distinguish the axial and axiplanar states. In the axial state it must be identically 1, whereas in the axi-planar state it may range from 1 to 1/4 depending upon /3parameters. At the melting pressure this relation has been carefully checked and found to be equal to 1. At lower pressures it has not been done as well. We can try to look at the angular anisotropy required for both states. Ultrasonic attenuation has been studied as a function of the angle between the sound direction and the magnetic field. Unfortunately, the A phase orbital anisotropy axis does not lie along the magnetic field, but rather is perpendicular to it in most experiments. Measurements at a few discrete angles have been used to compare to theory [13], but detailed angular studies do not exhibit the expected anisotropy [14]. This is not evidence against the axial state, but rather evidence that the probe is inadequate. The angular anisotropy of the A phase also shows up in its superfluid density tensor. The ratio of the magnitude of the anisotropy to the maximum magnitude of the superfluid density must be identically 1/2 in the axial state, while in the axi-planar state it is always less than or equal to 1/2. Again because we do not have a handle acting directly on the anisotropy, but only on something perpendicular to it, these studies are difficult. The ratio has long been known [15] to be approximately 1/2, but the best determination was made only a few years ago by the Manchester group [16] who discovered that the ratio is less than 1/2 by a statistically significant margin. Before leaping to any conclusions, however, I note that this measurement is exceedingly
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difficult, was done at only one pressure, and has not yet been repeated. The major difference between the axial and axi-planar states is the symmetry of the orbital state. This ought to show up in flow properties. Bhattacharyya, Ho and Mermin [17] discovered that the axial state is, in fact, superfluid in the sense of being able to support dissipationless flow, but only barely so. Were it not for the support of the weak nuclear dipole-dipole interactions, the axial state would not be superfluid in the classical sense. Experiments have probed the limits of this stability and have found them much greater than expected from the axial state [14]. Recently, Narasimhan and Ho [18] have performed the first calculation of the stability of the axi-planar state and find that it is significantly greater than that of the axial state, though still less so than experiment indicates. Given that direct measurements have not answered our question, an indirect approach is indicated. If we measure the specific heat jump at the normal-to-superfluid transition, that number will not by itself tell us anything about the symmetry of the order parameter. What it does give us is one combination of the five/3-parameters. If we are sufficiently clever in finding five thermodynamically independent quantities that can be measured in the limit as T---* T~-, we will determine our location in the Mermin-Stare diagram and thereby identify the superfluid. Presently there are four independent quantities that have been measured: (1) the specific heat jump at the normal-to-superfluid transition [19], (2) the ratio of temperature dependences of the A 1 and A 2 phase transitions in very large magnetic fields [20], (3) the magnitude of the quadratic suppression of the B phase relative to the A phase in small magnetic fields [21], and (4) the magnitude of the magnetization discontinuity at the A - B transition in small magnetic fields [22]. Sadly, I do not now have a fifth experiment which would satisfy our needs. Nonetheless, with four experiments we can constrain the/3-parameters at any one pressure to a single line through five-dimensional /3-space. When this line at any one specific pressure is projected onto the Mer-
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m i n - S t a r e diagram, it passes through the heart of both candidate states. At an earlier stage in this quest, I offered the Principle of Least Astonishment [12]. This Principle is just a common sense notion recognizing that as interactions are turned off (as pressure decreases) the /3parameters must approach their weak-coupling values. You should therefore be astonished if Nature selected a point in /3-space that was not approximately as close to the weak-coupling point as possible, subject to the four experimental constraints we found above. The application of this appealing, but unjustified, Principle discovers a series of points that hug the boundary between the axial and axi-planar states, and so, curiously, makes no prediction for the solution. Before leaving superfluid 3He I want to mention two experiments which will be announced at this conference. The first is from the Manchester group which, stimulated by their superfluid density anisotropy results, has tried to check the relationship between the longitudinal and transverse N M R frequencies. Their results, at pressures above the polycritical point, appear to rule out any significant distortion away from the axial state [23]. The second is from Bill Halperin's group at Northwestern which has measured the transverse N M R shift with extreme precision in the A phase at low pressures [24]. These values are compared to measurements done elsewhere at the same pressure, temperature and field in the B phase. These two measurements, after suitable normalization, must be equal if the A phase is the axial state. The fact that the data do agree to within a few percent is strong support for the axial state at low pressures. These two measurements from Manchester and Northwestern are the first good evidence uniquely supporting the axial state at pressures below the melting curve.
4. High-temperature superconductors H e r e there is no space to catalogue all of the materials that have been discovered to date. Instead, for present purposes I will make use of another unjustified (but appealing) assumption
that the symmetry of the order parameter of all systems in this class is the same, and use experimental evidence from whichever system is convenient. The cuprate superconductors are orthorhombic, but since the distortion from tetragonal is exceedingly small, these systems are usually treated as tetragonal. Because the important physics occurs on the CuO 2 planes, many theoretical models focus on only a square two-dimensional lattice. The irreducible representations for this symmetry may be tabulated [25] again cleanly separating into spin-singlet and -triplet groups. Unlike 3He, here the difficult task is to identify which representation is manifested, because it now appears that the correct representation is one-dimensional leaving no residual degeneracy. If this discussion was presented several years ago, the conclusion would have been that conventional s-wave pairing is adequate to explain all experiments. Today there is growing evidence that the dx2_y2 s t a t e is the correct answer, though controversy remains. The danger in the linear approach I will now take in reaching an answer is that it diminishes the significance of other conflicting evidence. For every experiment that I will cite below (save one) there are other experiments, albeit usually earlier, with conflicting results. We will just have to wait to see whether the present evidence will hold up over time. First, we distinguish between the triplet and singlet order parameters. The central evidence against triplet pairing is the zero-temperature magnetic susceptibility as inferred from Knight shift measurements in Y B C O [26]. It is important to recognize, however, that the susceptibility in those experiments is not directly measured. Instead, it is inferred from a subtraction of terms which leads to the spin susceptibility only if a series of assumptions is satisfied, including the neglect of spin-orbit coupling, and the isotropy of the spin susceptibility in the superconductor. The latter assumption is known to be incorrect for all p-wave triplet states proposed for the A phase of superfluid 3He. It is also incorrect for all triplet states I am aware of, which could appear in a square lattice. It could be that the spin susceptibility is isotropic in the
C.M. Gould / Physica B 197 (1994) 472-480
high-T~ materials, but that is not tested by experiments. Assuming that the order parameter is a spin singlet, there are still several irreducible representations to consider. Two experiments argue against a conventional s-wave order parameter. First there are measurements of the nuclear spinlattice relaxation rate in YBCO [27]. In conventional superconductors at temperatures immediately below T~, the relaxation rate increases to a peak (the Hebel-Slichter peak [28]) before falling to zero as T---~0. In contrast to this, in YBCO the relaxation rate decreases monotonically below T~. A second argument against an s-wave order parameter is an experiment by the group at the University of British Columbia measuring the London penetration depth A(T) at low temperatures [29]. In conventional superconductors the penetration depth saturates exponentially to a constant at low temperatures. In single crystals of YBCO this measurement finds a linear temperature dependence, as expected for all non-swave singlet gaps [30]. Many groups have found a T 2 dependence, but this is an expected result of impurities. In fact, the UBC group has changed the temperature dependence from T to T 2 by doping with zinc. Since there appear to be nodes, how can we go about finding them? In contrast to the case of superfluid 3He where we do not have a direct handle on the anisotropy of the state, in crystals the anisotropy is the crystal giving experimentalists an additional tool. This past year two experiments have addressed the issue of anisotropy directly. Angle-resolved photoemission from Bi2SraCaCu208+ ~ has found a significant anisotropy in the superconducting gap in the a - b plane, with a significant depression along the diagonal consistent with a node [31]. These measurements are extremely difficult because the instrumental resolution (30 meV) is comparable with the size of the effect, and the inferred gap appears to shrink as a function of time arising from surface contamination. Significantly, it appears that the m a x i m u m gap in this anisotropic superconductor can be quite a bit larger than (presumably average) gaps reported earlier.
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Here, the authors conservatively cite gaps as large as 20 meV (2zl(O)/kT c = 6 in contrast to the weak coupling value 3.5). The actual gap maximum could conceivably be a factor of two larger. The other search for the symmetry of the gap, by the group at the University of Illinois, takes the direct route of sampling different directions in k-space by placing junctions on differing faces of a single crystal, completing a loop with superconducting Pb, and measuring the phase shift in the critical current of the resulting DC SQUID as a function of flux applied to the loop [32]. In this experiment, junctions were placed on the a and b faces, through which a 180° phase shift was expected for the dx2_y2 state. This experiment is made difficult by SQUID asymmetries and trapped magnetic flux in the junctions, but after a careful accounting for these effects it appears that the critical current pattern is indeed shifted by 180° as expected. An exciting aspect of this work is that it is a new technique to determine angular anisotropy. The reason the experiment works is that the Josephson supercurrent is sensitive to the phase of the order parameter in a direction normal to the junction because the tunneling probability is strongly peaked in the forward direction. How strongly it is peaked will determine the ultimate angular resolution of this technique. Assuming that all of the current experiments are correct in pointing us toward the dx2_y2 s t a t e as the correct irreducible representation for the cuprate superconductors order parameter, the situation now is analogous in the 3He case to settling on p-wave states: that was the starting point for all of the hard work in identifying the A phase. Fortunately, unlike 3He where the order parameter in the correct irreducible representation had a huge selection of remaining forms to take, in the cuprate superconductors the dx2_y2 representation is one-dimensional. The precise functional form is yet to be determined, but its symmetry is fixed. Of course, as in the case of 3He, the requirement of a single irreducible representation applies only in the limit as T---~ To. As temperature drops toward zero, we ought to expect admixtures of other representations, but if Nature is
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kind (or rather, bland) the symmetry near T~ will remain an adequate description all the way down. Although the issue of a theoretical explanation of the physical origin of the superconducting state is highly controversial, I would like to cite one recent result. For many years, microscopic calculations of a two-dimensional single-band nearly half-filled Hubbard model with no additional interactions (i.e. on-site repulsion only) have consistently found superconductivity to be most favored in the dx2_y2 channel [33]. Recently, Bickers and co-workers at USC [34] have been able to push toward the low-temperature limit and find a limiting value of 2A(O)/kT~ of 10-12 given a hole filling factor of about 12%. This very large result (again compared to the weak coupling value of 3.5) is larger than that cited by any experiment measuring average gaps, but is not inconsistent with the latest photoemission results given above. It would certainly be astonishing if all of the important physics leading to superconductivity in the cuprates were contained within such a simple model.
5. Neutron stars
Following the high-temperature superconductors, I want to discuss the highest-temperature superconductors: neutron stars. This is a field in which quite a few low-temperature theorists have contributed, and I think that it is time for us experimentalists to get involved. Given core temperatures of 107-109 K, it sounds incredible to consider these low temperatures until we compare it in reasonable units (1-100 keV) to typical Fermi energies (10-100 MeV). The Standard Model of a neutron star [35] consists of an outer solid crust, which will not interest us here, an inner crust hosting a degenerate neutron liquid, a liquid interior of neutrons and a few protons, and a core too exotic to consider. Superfluidity of nucleons in neutron stars was first proposed in 1959 [36] and serious attempts to calculate transition temperatures
were published in 1970 [37]. Their estimates for T c are in the range of 0.1-1 MeV, small compared to T v, but large compared to the temperature. The incredible part of these calculations is that they predict specific gap functions in particular irreducible representations, and without a single measurement of specific heat, angle-resolved photoemission, or tunneling, these predictions are still generally accepted today. We experimentalists need to get to work, although given the current difficulties facing the SSC we will probably not get very far proposing an on-site Superfluidity Observatory for Neutron Stars. There are two properties of neutron stars which are observable on Earth and which may be affected by superfluidity. The first is the rate at which a neutron star cools [38]. Experimentally, neutron stars cool faster than predicted by simple gas models without magnetic fields, and superfluidity provides one mechanism to speed cooling. The essential physics of this mechanism is that the transition to superfluidity causes more entropy to be removed at high temperatures (To) than would be removed without a phase transition, slowing the initial cooling, but speeding cooling in later stages. Superfluidity is a plausible, but not a unique, mechanism. The second property depends upon the incredible precision with which pulsars keep time. In the midst of this accurate timekeeping, pulsars suffer 'glitches' every few years which speed the rate of pulsing up on the time scale of seconds to minutes, which then relax back on the time scale of days to years. Dick Packard [39] suggested that the glitches reflect metastable superfluid flow. Since then, a more intricate theoretical picture has developed that explains several aspects of the glitches in terms of vortices in the superfluid interior [40]. The picture, however, is not sensitive to the particular symmetry of the gap function, merely its existence. What this theoretical picture needs is experiments that can guide the theorists in identifying vortex pinning and unpinning mechanisms. Here is where low-temperature experimentalists working in all exotic superconductors can conceivably assist our astrophysics colleagues.
C.M. Gould / Physica B 197 (1994) 472-480
6. Extending the possibilities Finally I want to briefly mention three rather speculative ideas devised to escape the bounds of the traditional generalized BCS theory. First is the odd-parity singlet-pairing (OPSP) state of Scalettar, Singh and Zhang [41] in which instead of pairing in a state of zero total momentum, pairs form with large momentum, on the order of reciprocal lattice vectors. The anomalous average for this case is (ak+Q,~a_k, ~ ). It has long been recognized that as Q initially increases from zero, pairing is suppressed, but whether any interaction can favor a state with large Q is not yet known: Model calculations have not been encouraging [42]. Second is the odd-frequency gap originally proposed by Berezinskii [43] in the context of the A phase of superfluid 3He. This has recently been taken up by Balatsky and Abrahams [44] and others. The central idea is that if the frequency dependence of the anomalous average (which I have ignored up to now) is odd, then the Pauli principle requires that antisymmetric singlet states have antisymmetric o d d angular m o m e n t a , the reverse of the usual rule. Odd-gap superconductors have a Meissner effect, no Hebel-Slichter peak, and other essential properties [45], but in model systems that show susceptibility to such pairing, other channels go superconducting first [34]. Last, given their great theoretical interest, I want to mention anyons.
7. Conclusion In superftuid 3He, it appears likely that experiments will confirm the Standard Model for the symmetry of the order parameter of the A phase. Experiments have shown, however, that we do not understand some important physics of the n o r m a l state of liquid 3He. Saying that spin fluctuations account for the important interactions in the superfluid is wrong. But to be fair to the theorists, this could not have been known before extensive measurements in the superfluid.
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In high-temperature superconductors, experimental constraints appear to be converging. In the highest-temperature superconductors, current experimental constraints leave a great deal to be desired. Table-top experiments would be quite beneficial. Finally, the ingenuity of theorists in devising new ways for matter to self-organize should never be underestimated. Even if the more speculative ideas do not apply today, they may tomorrow, for we should similarly never underestimate the ingenuity of Nature.
Acknowledgements The experimental work at USC cited here has been done in collaboration with Hans Bozler and a series of our post-docs (Ben Crooker, Yi-Hua Tang and Steve Boyd) and graduate students (Duane Bates, Ulf Israelsson and Inseob Hahn). I would particularly like to acknowledge tremendous assistance on the theoretical front on numerous issues in physics from G e n e Bickers. Finally I thank T o n y Leggett for explaining the history of Ref. [8]. This work is supported by the Nation~/l Science Foundation through grant DMR92-16842.
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Methods to Many-Body Theory, ed. A.I. Alekseev (Gostatomizdat, Moscow, 1963) p. 94. [9] N.D. Mermin and G. Stare, Phys. Rev. Lett. 30 (1973) 1135; N.D. Mermin and G. Stare, in: Sverkhtekuchestv Geliya-3, ed. I.M. Khalatnikov (Izdatel'stvo Mir, Moscow, 1977) p. 121. [10] D. Rainer and J.W. Serene, Phys. Rev. B 13 (1976) 4745. [11] P.W. Anderson and W.F. Brinkman, Phys. Rev. Lett. 30 (1973) 1108; W.F. Brinkman, J.W. Serene and EW. Anderson, Phys. Rev. A 10 (1974) 2386; J.A. Sauls and J.W. Serene, Phys. Rev. B 24 (1981) 183; Z. Te~anovi6 and O.T. Valls, J. Low Temp. Phys. 52 (1983) 31; M. Pfitzner, J. Low Temp. Phys. 61 (1985) 141. [12] C.M. Gould, Physica B 178 (1992) 266. [13] P.R. Roach, B.M. Abraham, P.D. Roach and J.B. Ketterson, Phys. Rev. Lett. 34 (1975) 715. [14] D.M. Bates, S.N. Ytterboe, C.M. Gould and H.M. Bozler, J. Low Temp. Phys. 62 (1986) 143. [15] J.E. Berthold, R.W. Giannetta, E.N. Smith and J.D. Reppy, Phys. Rev. Lett. 37 (1976) 1138. [16] J.R. Hook, E. Faraj, S.G. Gould and H.E. Hall, J. Low Temp. Phys. 74 (1989) 45. [17] P. Bhattacharyya, T.-L. Ho and N.D. Mermin, Phys. Rev. Lett. 39 (1977) 1290. [18] S. Narasimhan and T.-L. Ho, Instability of Laminar Flow in the Axi-Planar State of Superfluid 3He-A (preprint). [19] D.S. Greywall, Phys. Rev. B 33 (1986) 7520. [20] U.E. Israelsson, B.C. Crooker, H.M. Bozler and C.M. Gould, Phys. Rev. Lett. 53 (1984) 1943; 54 (1985) 254E. [21] Y.H. Tang, I. Hahn, H.M. Bozler and C.M. Gould, Phys. Rev. Lett. 67 (1991) 1775. [22] S.T.P. Boyd, Inseob Hahn, H.M. Bozler and C.M. Gould (preprint). [23] T.R. Mullins, V.V. Dmitriev, A.J. Armstrong, J.R. Hook and H.E. Hall, Physica B 194-196 (1994) 769. [24] M.R. Rand, H.H. Hensley, J.B. Kycia, T.M. Haard, Y. Lee, P.J. Hamot and W.E Halperin, Physica B 194-196 (1994) 805. [25] M. Sigrist and T.M. Rice, Z. Phys. B 68 (1987) 9. [26] M. Takigawa, P.C. Hammel, R.H. Heffner and Z. Fisk, Phys. Rev. B 39 (1989) 7371; S.E. Barrett, D.J.
Durand, C.H. Pennington, C.P. Slichter, T.A. Friedmann, J.P. Rice and D.M. Ginzberg, Phys. Rev. B 41 (1990) 6283. [27] T. Imai, T. Shimizu, H. Yasuoka, Y. Ueda and K. Kosuge, J. Phys. Soc. Japan 57 (1988) 2280. [28] L.C. Hebel and C.P. Siichter, Phys. Rev. 113 (1959) 1504; L.C. Hebel, Phys. Rev. 116 (1959) 79. [29] W.N. Hardy, D.A. Bonn, D.C. Morgan, R. Liang and K. Zhang, Phys. Rev. Lett. 70 (1993) 3999. [30] J. Annett, N. Goldenfeld and S.R. Renn, Phys Rev. B 43 (1991) 2778. [31] Z.-X. Shen, D.S. Dessau, B.O. Wells, D.M. King, W.E. Spicer, A.J. Arko, D. Marshall, L.W. Lombardo, A. Kapitulnik, P. Dickinson, S. Doniach, J. DiCarlo, A.G. Loeser and C.H. Park, Phys. Rev. Lett. 70 (1993) 1553. [32] D.A. WoUman, D.J. Van Harlingen, W.C. Lee, D.M. Ginsberg and A.J. Leggett, Phys. Rev. Lett. 71 (1993) 2134. [33] N.E. Bickers, D.J. Scatapino and R.T. Scalettar, Int. J. Mod. Phys. B 1 (1987) 687; N.E. Bickers, D.J. Scalapino and S.R. White, Phys. Rev. Lett. 62 (1989) 961. [34] Chien-Hua Pao and N.E. Bickers, submitted for publication in Phys. Rev. Lett. [35] G. Baym, R.I. Epstein and B. Link, Physica B 178 (1992) 1. [36] A.B. Migdal, Nucl. Phys. 13 (1959) 655. [37] M. Hoffberg, A.E. Glassgold, R.W. Richardson and M. Ruderman, Phys. Rev. Lett. 24 (1970) 775; R. Tamagaki, Prog. Theor. Phys. 44 (1970) 905. [38] S. Tsuruta, Phys. Rep. 56 (1979) 237. [39] R.E. Packard, Phys. Rev. Lett. 28 (1972) 1080. [40] P.W. Anderson and N. Itoh, Nature 256 (1975) 25; J.A. Sauls, D.L. Stein and J.W. Serene, Phys Rev. D 25 (1982) 967; L. Bildsten and R.I. Epstein, Astrophys. J. 342 (1989) 951. [41] R.T. Scalettar, R.R.P. Singh and S. Zhang, Phys. Rev. Lett. 67 (1991) 370. [42] N.E. Bickers, Phys. Rev. B 45 (1992) 7418. [43] V.L. Berezinskii, JETP Lett. 20 (1974) 287. [44] A. Balatsky and E. Abrahams, Phys. Rev. B 45 (1992) 13125. [45] E. Abrahams, J. Phys. Chem. Solids 53 (1992) 1487.
ELSEVIER
The difficulty in identifying exotic superconductors C.M.
Gould
Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA
Abstract
Conventional s-wave superconductors have order parameters with no internal degrees of freedom - they are characterized by an amplitude and a phase. All unconventional superconductors share the property of a nontrivial internal symmetry which makes their behavior more interesting and their identification more difficult. The determination of the order parameter of the longest-studied exotic superconductor, superfluid 3He, remains inconclusive despite the existence of a Standard Model. Using superfluid 3He as a cautionary tale, the experimental evidence for the identification of order parameters of other exotic superconductors is also examined.
1. Introduction
Thirty years ago, superconductivity was simple. T h e BCS theory of superconductivity [1] was still a young theory enjoying success in explaining n u m e r o u s experimental results. The underlying idea was that quasiparticles near the Fermi surface with opposite m o m e n t a find it energetically favorable to pair with each other, with all systems then known pairing in an isotropic (swave) singlet state. This means that in the superconducting state, the anomalous average of two annihilation operators (ak, a ~ + ) develops an isotropically nonzero value. Even strong-coupling systems like lead and mercury were accounted for with this simple pair wavefunction. Just over twenty years ago, however, the discovery of superfluid 3He [2] d e m a n d e d a generalization of this scheme. The discovery of superconductivity in the heavy-fermion system CeCu2Si 2 in 1979 [3], and subsequent discovery of the high-T c cuprate superconductors in 1986 [4] have stimulated a reexamination of the allowed superconducting states.
T h e question I want to focus on is "what are the possibilities?" and then examine the experimental evidence. In the case of 3He, I will try to show that even after all these years, the question is still a legitimate one. As we will see, for all exotic superconductors there is a wide variety of pair wave functions from which Nature can choose. Only when experiment has demonstrated a reason for picking one particular state over all others can we say we have identified it. N o single experiment can do the complete job because in general, several possible pair states will have identical, or nearly identical, responses to any particular probe. Nor is any experiment free from theoretical bias in its interpretation. These problems are the heart of the difficulty in identifying exotic superconductors.
2. What are the possibilities?
The generalization of the BCS isotropic singlet gap function to a direction- and spin-dependent one was given by Balian and W e r t h a m e r in 1963
0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI: 0921-4526(93)E0173-E
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C.M. Gould / Physica B 197 (1994) 472-480 [5]. The gap now becomes a matrix (in spin space) of functions of direction in momentum space:
evidence of this in 3He [6]. The objective in this paper is to identify experimentally the correct irreducible representation at T c, and for multidimensional representations to discover how Nature has lifted the remaining degeneracy.
=
3. Superfluid 3He
_ 0 d0(k )
+
d0(0/¢))
- dx(I¢) + idy(/¢)
dz(/¢)
dz(f¢)
dx(fc)+idy(f¢) J
where we have explicitly broken the matrix into antisymmetric and symmetric parts, which correspond to spin-singlet and -triplet wave functions. Singlet pair wave functions have a single component, while triplets naturally have three. Writing the triplet part as we have done has the advantage that the construct d transforms as a vector under rotations in spin space: A a ( k ) = (d(/¢)-o- ~io-(2) , re" Because the total pair wavefunction must be antisymmetric under interchange of particles, symmetric spin states (triplets) must be coupled with antisymmetric orbital states (L = odd), and vice versa. In 3He spin-orbit coupling (in the form of nuclear dipole-dipole interactions) is small, as it is in the cuprate superconductors. It is not small in the heavy fermion superconductors, or in nuclear matter. We can now classify all possible solutions to the BCS gap equation according to the irreducible representation of the symmetry group of the system into which they fall. In 3He, an isotropic liquid, we get an infinite set labelled by the orbital angular momentum, neatly separating into singlets (l = even) and triplets (l = odd). In crystals we get a finite set of irreducible representations which is different for every crystal structure. The point of this classification scheme is that asymptotically as T - ~ T c the exact pair wavefunction must fall into one, and only one, of these irreducible representations, the one with the highest T c. As the temperature is lowered there may be mixing of different representations, but in general this is small. There is some
Having found all of the possible states, we now eliminate the wrong ones. In 3He, finding the correct irreducible representation is easy; removing the residual degeneracy is not. The first experiment to reveal that the phase transition in 3He was in the liquid [7], looked at the change in the NMR resonance frequency relative to the normal liquid. The fact that the superfluid is in this sense magnetically active means that it cannot be a spin singlet state. Half of all possible representations just got eliminated. Among the triplets, the simplest, p-wave, was considered first. Balian and Werthamer in their 1963 paper also solved the weak-coupling limit for p-wave superconductors and found the lowest energy state. (Earlier, Vdovin [8] showed this state was the solution of a different formulation of the BCS problem which, unfortunately, is not ill general correct.) This BW state has surprising and nontrivial properties that match all known properties of the B phase of superfluid 3He. There are no competing alternatives. This is powerful evidence that we have identified the correct irreducible representation. Since T c for the A and B phases is the same, their representations must also be the same. Identifying the order parameter of the A ~hase is more difficult. Fortunately, superfluid He is a good mean field system, so near Tc we can apply a simple Landau expansion, suitably generalized for our multicomponent order parameter. Our gap parameter d(/~) can be rewritten in terms of a 3 by 3 matrix of complex constants: d(k) = A k =
k, kz
C.M. Gould I Physica B 197 (1994) 472-480
474
which explicitly exhibits the 18 degrees of freedom. In forming the second- and fourth-order terms in the Landau expansion, the fact that the rows and columns of the A matrix transform separately as spin and orbital vectors constrain the ways we can combine four of these matrices. There are precisely five ways they can be combined [9], and in the Landau expansion these five terms each have a separate material dependent coefficient, labelled/31,... ,/35. Here is the means by which microscopic theory makes contact with experiment. Rainer and Serene showed [10] that within Fermi liquid theory, knowledge of the quasiparticle scattering amplitudes in the normal fluid is sufficient to uniquely determine these /3-parameters. If only we understood the normal fluid well enough, we could calculate the/3-parameters and predict the superfluid state at each pressure. What gap parameters are consistent with the A phase? Mermin and Stare [9] showed that in comparing the free energies of A phase states, for purposes of finding the lowest, one need not search a five-dimensional parameter space, but instead only a two-dimensional space of ratios and combinations of the/3-parameters shown in Fig. 1. They also showed that given the static
[33 I ~,[
3
axial 2
axi-planar /~weak- coupl;ng
1~ planar 0 polar "~ -1 0 1 IL+I3,
Fig. 1. Phase diagram of candidate states for the A phase, given the physically relevant condition /31 < 0. Solid lines denote a first-order phase transition, dashed lines a secondorder one. Note the weak-coupling (no interactions) point at the critical intersection of three states with different symmetries.
magnetic properties of the A phase, only six states could ever be extrema of the free energy. NMR frequencies (at all pressures) exclude all but three states: the axial, the axi-planar and the planar. Thermodynamic studies exclude the planar state by showing that the phase transition between the A and B phases in a magnetic field is always first order. Thus, we are down to only two candidates. A word about this phase diagram is now in order. At any specific pressure the/3-parameters take on a particular set of values, and will be projected from the five-dimensional/3-parameter space onto this two-dimensional Mermin-Stare diagram. As we change the pressure, the /3parameters will change. Since interactions are expected to increase monotonically with pressure (varying as the ratio of Tc/TF), we expect to find the physical point approaching the weak-coupling point as pressure is reduced to zero, although how close it can come is not known a priori. At this point, theory took a decisive hand. Spin fluctuations are known to be important in explaining the properties of the normal liquid. Models incorporating spin fluctuations were constructed, modelling all normal state properties, and used to predict /3-parameters [11]. Despite their differences in detail, all of these calculations agree that the/3-parameters lie exclusively in the region favoring the axial state. Experiments of the time supported this conclusion, and the Standard Model was established. Unfortunately, thermodynamic data checking the values of/3-parameters directly was not then available. Now that we have such data it is clear that there is a serious problem with the theoretical understanding of the normal liquid. As we turn on the interactions in 3He, the directions that theory and experiment take through/3-space are nearly orthogonal [12]. Theory cannot, therefore, be a reliable guide to selecting between the axial and axi-planar states. There are several probes that have been used as evidence in favor of the axial state. Curiously, much of this evidence also supports the axiplanar state, for the simple reason that they can be continuously deformed into each other, with an energy cost fixed by the/3-parameters. Never-
C.M. Gould / Physica B 197 (1994) 472-480
theless, the two phases can be distinguished with appropriate probes. The best tool presently is probably NMR. The A phase of the superfluid exhibits both a longitudinal resonance at a frequency that depends only upon temperature, OL(T), and a transverse resonance at a frequency to given (in large fields) by to
2
2 2 = ( T H o ) 2 + R T ~ Q L ( T 1.
The factor R 2 can distinguish the axial and axiplanar states. In the axial state it must be identically 1, whereas in the axi-planar state it may range from 1 to 1/4 depending upon /3parameters. At the melting pressure this relation has been carefully checked and found to be equal to 1. At lower pressures it has not been done as well. We can try to look at the angular anisotropy required for both states. Ultrasonic attenuation has been studied as a function of the angle between the sound direction and the magnetic field. Unfortunately, the A phase orbital anisotropy axis does not lie along the magnetic field, but rather is perpendicular to it in most experiments. Measurements at a few discrete angles have been used to compare to theory [13], but detailed angular studies do not exhibit the expected anisotropy [14]. This is not evidence against the axial state, but rather evidence that the probe is inadequate. The angular anisotropy of the A phase also shows up in its superfluid density tensor. The ratio of the magnitude of the anisotropy to the maximum magnitude of the superfluid density must be identically 1/2 in the axial state, while in the axi-planar state it is always less than or equal to 1/2. Again because we do not have a handle acting directly on the anisotropy, but only on something perpendicular to it, these studies are difficult. The ratio has long been known [15] to be approximately 1/2, but the best determination was made only a few years ago by the Manchester group [16] who discovered that the ratio is less than 1/2 by a statistically significant margin. Before leaping to any conclusions, however, I note that this measurement is exceedingly
475
difficult, was done at only one pressure, and has not yet been repeated. The major difference between the axial and axi-planar states is the symmetry of the orbital state. This ought to show up in flow properties. Bhattacharyya, Ho and Mermin [17] discovered that the axial state is, in fact, superfluid in the sense of being able to support dissipationless flow, but only barely so. Were it not for the support of the weak nuclear dipole-dipole interactions, the axial state would not be superfluid in the classical sense. Experiments have probed the limits of this stability and have found them much greater than expected from the axial state [14]. Recently, Narasimhan and Ho [18] have performed the first calculation of the stability of the axi-planar state and find that it is significantly greater than that of the axial state, though still less so than experiment indicates. Given that direct measurements have not answered our question, an indirect approach is indicated. If we measure the specific heat jump at the normal-to-superfluid transition, that number will not by itself tell us anything about the symmetry of the order parameter. What it does give us is one combination of the five/3-parameters. If we are sufficiently clever in finding five thermodynamically independent quantities that can be measured in the limit as T---* T~-, we will determine our location in the Mermin-Stare diagram and thereby identify the superfluid. Presently there are four independent quantities that have been measured: (1) the specific heat jump at the normal-to-superfluid transition [19], (2) the ratio of temperature dependences of the A 1 and A 2 phase transitions in very large magnetic fields [20], (3) the magnitude of the quadratic suppression of the B phase relative to the A phase in small magnetic fields [21], and (4) the magnitude of the magnetization discontinuity at the A - B transition in small magnetic fields [22]. Sadly, I do not now have a fifth experiment which would satisfy our needs. Nonetheless, with four experiments we can constrain the/3-parameters at any one pressure to a single line through five-dimensional /3-space. When this line at any one specific pressure is projected onto the Mer-
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m i n - S t a r e diagram, it passes through the heart of both candidate states. At an earlier stage in this quest, I offered the Principle of Least Astonishment [12]. This Principle is just a common sense notion recognizing that as interactions are turned off (as pressure decreases) the /3parameters must approach their weak-coupling values. You should therefore be astonished if Nature selected a point in /3-space that was not approximately as close to the weak-coupling point as possible, subject to the four experimental constraints we found above. The application of this appealing, but unjustified, Principle discovers a series of points that hug the boundary between the axial and axi-planar states, and so, curiously, makes no prediction for the solution. Before leaving superfluid 3He I want to mention two experiments which will be announced at this conference. The first is from the Manchester group which, stimulated by their superfluid density anisotropy results, has tried to check the relationship between the longitudinal and transverse N M R frequencies. Their results, at pressures above the polycritical point, appear to rule out any significant distortion away from the axial state [23]. The second is from Bill Halperin's group at Northwestern which has measured the transverse N M R shift with extreme precision in the A phase at low pressures [24]. These values are compared to measurements done elsewhere at the same pressure, temperature and field in the B phase. These two measurements, after suitable normalization, must be equal if the A phase is the axial state. The fact that the data do agree to within a few percent is strong support for the axial state at low pressures. These two measurements from Manchester and Northwestern are the first good evidence uniquely supporting the axial state at pressures below the melting curve.
4. High-temperature superconductors H e r e there is no space to catalogue all of the materials that have been discovered to date. Instead, for present purposes I will make use of another unjustified (but appealing) assumption
that the symmetry of the order parameter of all systems in this class is the same, and use experimental evidence from whichever system is convenient. The cuprate superconductors are orthorhombic, but since the distortion from tetragonal is exceedingly small, these systems are usually treated as tetragonal. Because the important physics occurs on the CuO 2 planes, many theoretical models focus on only a square two-dimensional lattice. The irreducible representations for this symmetry may be tabulated [25] again cleanly separating into spin-singlet and -triplet groups. Unlike 3He, here the difficult task is to identify which representation is manifested, because it now appears that the correct representation is one-dimensional leaving no residual degeneracy. If this discussion was presented several years ago, the conclusion would have been that conventional s-wave pairing is adequate to explain all experiments. Today there is growing evidence that the dx2_y2 s t a t e is the correct answer, though controversy remains. The danger in the linear approach I will now take in reaching an answer is that it diminishes the significance of other conflicting evidence. For every experiment that I will cite below (save one) there are other experiments, albeit usually earlier, with conflicting results. We will just have to wait to see whether the present evidence will hold up over time. First, we distinguish between the triplet and singlet order parameters. The central evidence against triplet pairing is the zero-temperature magnetic susceptibility as inferred from Knight shift measurements in Y B C O [26]. It is important to recognize, however, that the susceptibility in those experiments is not directly measured. Instead, it is inferred from a subtraction of terms which leads to the spin susceptibility only if a series of assumptions is satisfied, including the neglect of spin-orbit coupling, and the isotropy of the spin susceptibility in the superconductor. The latter assumption is known to be incorrect for all p-wave triplet states proposed for the A phase of superfluid 3He. It is also incorrect for all triplet states I am aware of, which could appear in a square lattice. It could be that the spin susceptibility is isotropic in the
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high-T~ materials, but that is not tested by experiments. Assuming that the order parameter is a spin singlet, there are still several irreducible representations to consider. Two experiments argue against a conventional s-wave order parameter. First there are measurements of the nuclear spinlattice relaxation rate in YBCO [27]. In conventional superconductors at temperatures immediately below T~, the relaxation rate increases to a peak (the Hebel-Slichter peak [28]) before falling to zero as T---~0. In contrast to this, in YBCO the relaxation rate decreases monotonically below T~. A second argument against an s-wave order parameter is an experiment by the group at the University of British Columbia measuring the London penetration depth A(T) at low temperatures [29]. In conventional superconductors the penetration depth saturates exponentially to a constant at low temperatures. In single crystals of YBCO this measurement finds a linear temperature dependence, as expected for all non-swave singlet gaps [30]. Many groups have found a T 2 dependence, but this is an expected result of impurities. In fact, the UBC group has changed the temperature dependence from T to T 2 by doping with zinc. Since there appear to be nodes, how can we go about finding them? In contrast to the case of superfluid 3He where we do not have a direct handle on the anisotropy of the state, in crystals the anisotropy is the crystal giving experimentalists an additional tool. This past year two experiments have addressed the issue of anisotropy directly. Angle-resolved photoemission from Bi2SraCaCu208+ ~ has found a significant anisotropy in the superconducting gap in the a - b plane, with a significant depression along the diagonal consistent with a node [31]. These measurements are extremely difficult because the instrumental resolution (30 meV) is comparable with the size of the effect, and the inferred gap appears to shrink as a function of time arising from surface contamination. Significantly, it appears that the m a x i m u m gap in this anisotropic superconductor can be quite a bit larger than (presumably average) gaps reported earlier.
477
Here, the authors conservatively cite gaps as large as 20 meV (2zl(O)/kT c = 6 in contrast to the weak coupling value 3.5). The actual gap maximum could conceivably be a factor of two larger. The other search for the symmetry of the gap, by the group at the University of Illinois, takes the direct route of sampling different directions in k-space by placing junctions on differing faces of a single crystal, completing a loop with superconducting Pb, and measuring the phase shift in the critical current of the resulting DC SQUID as a function of flux applied to the loop [32]. In this experiment, junctions were placed on the a and b faces, through which a 180° phase shift was expected for the dx2_y2 state. This experiment is made difficult by SQUID asymmetries and trapped magnetic flux in the junctions, but after a careful accounting for these effects it appears that the critical current pattern is indeed shifted by 180° as expected. An exciting aspect of this work is that it is a new technique to determine angular anisotropy. The reason the experiment works is that the Josephson supercurrent is sensitive to the phase of the order parameter in a direction normal to the junction because the tunneling probability is strongly peaked in the forward direction. How strongly it is peaked will determine the ultimate angular resolution of this technique. Assuming that all of the current experiments are correct in pointing us toward the dx2_y2 s t a t e as the correct irreducible representation for the cuprate superconductors order parameter, the situation now is analogous in the 3He case to settling on p-wave states: that was the starting point for all of the hard work in identifying the A phase. Fortunately, unlike 3He where the order parameter in the correct irreducible representation had a huge selection of remaining forms to take, in the cuprate superconductors the dx2_y2 representation is one-dimensional. The precise functional form is yet to be determined, but its symmetry is fixed. Of course, as in the case of 3He, the requirement of a single irreducible representation applies only in the limit as T---~ To. As temperature drops toward zero, we ought to expect admixtures of other representations, but if Nature is
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kind (or rather, bland) the symmetry near T~ will remain an adequate description all the way down. Although the issue of a theoretical explanation of the physical origin of the superconducting state is highly controversial, I would like to cite one recent result. For many years, microscopic calculations of a two-dimensional single-band nearly half-filled Hubbard model with no additional interactions (i.e. on-site repulsion only) have consistently found superconductivity to be most favored in the dx2_y2 channel [33]. Recently, Bickers and co-workers at USC [34] have been able to push toward the low-temperature limit and find a limiting value of 2A(O)/kT~ of 10-12 given a hole filling factor of about 12%. This very large result (again compared to the weak coupling value of 3.5) is larger than that cited by any experiment measuring average gaps, but is not inconsistent with the latest photoemission results given above. It would certainly be astonishing if all of the important physics leading to superconductivity in the cuprates were contained within such a simple model.
5. Neutron stars
Following the high-temperature superconductors, I want to discuss the highest-temperature superconductors: neutron stars. This is a field in which quite a few low-temperature theorists have contributed, and I think that it is time for us experimentalists to get involved. Given core temperatures of 107-109 K, it sounds incredible to consider these low temperatures until we compare it in reasonable units (1-100 keV) to typical Fermi energies (10-100 MeV). The Standard Model of a neutron star [35] consists of an outer solid crust, which will not interest us here, an inner crust hosting a degenerate neutron liquid, a liquid interior of neutrons and a few protons, and a core too exotic to consider. Superfluidity of nucleons in neutron stars was first proposed in 1959 [36] and serious attempts to calculate transition temperatures
were published in 1970 [37]. Their estimates for T c are in the range of 0.1-1 MeV, small compared to T v, but large compared to the temperature. The incredible part of these calculations is that they predict specific gap functions in particular irreducible representations, and without a single measurement of specific heat, angle-resolved photoemission, or tunneling, these predictions are still generally accepted today. We experimentalists need to get to work, although given the current difficulties facing the SSC we will probably not get very far proposing an on-site Superfluidity Observatory for Neutron Stars. There are two properties of neutron stars which are observable on Earth and which may be affected by superfluidity. The first is the rate at which a neutron star cools [38]. Experimentally, neutron stars cool faster than predicted by simple gas models without magnetic fields, and superfluidity provides one mechanism to speed cooling. The essential physics of this mechanism is that the transition to superfluidity causes more entropy to be removed at high temperatures (To) than would be removed without a phase transition, slowing the initial cooling, but speeding cooling in later stages. Superfluidity is a plausible, but not a unique, mechanism. The second property depends upon the incredible precision with which pulsars keep time. In the midst of this accurate timekeeping, pulsars suffer 'glitches' every few years which speed the rate of pulsing up on the time scale of seconds to minutes, which then relax back on the time scale of days to years. Dick Packard [39] suggested that the glitches reflect metastable superfluid flow. Since then, a more intricate theoretical picture has developed that explains several aspects of the glitches in terms of vortices in the superfluid interior [40]. The picture, however, is not sensitive to the particular symmetry of the gap function, merely its existence. What this theoretical picture needs is experiments that can guide the theorists in identifying vortex pinning and unpinning mechanisms. Here is where low-temperature experimentalists working in all exotic superconductors can conceivably assist our astrophysics colleagues.
C.M. Gould / Physica B 197 (1994) 472-480
6. Extending the possibilities Finally I want to briefly mention three rather speculative ideas devised to escape the bounds of the traditional generalized BCS theory. First is the odd-parity singlet-pairing (OPSP) state of Scalettar, Singh and Zhang [41] in which instead of pairing in a state of zero total momentum, pairs form with large momentum, on the order of reciprocal lattice vectors. The anomalous average for this case is (ak+Q,~a_k, ~ ). It has long been recognized that as Q initially increases from zero, pairing is suppressed, but whether any interaction can favor a state with large Q is not yet known: Model calculations have not been encouraging [42]. Second is the odd-frequency gap originally proposed by Berezinskii [43] in the context of the A phase of superfluid 3He. This has recently been taken up by Balatsky and Abrahams [44] and others. The central idea is that if the frequency dependence of the anomalous average (which I have ignored up to now) is odd, then the Pauli principle requires that antisymmetric singlet states have antisymmetric o d d angular m o m e n t a , the reverse of the usual rule. Odd-gap superconductors have a Meissner effect, no Hebel-Slichter peak, and other essential properties [45], but in model systems that show susceptibility to such pairing, other channels go superconducting first [34]. Last, given their great theoretical interest, I want to mention anyons.
7. Conclusion In superftuid 3He, it appears likely that experiments will confirm the Standard Model for the symmetry of the order parameter of the A phase. Experiments have shown, however, that we do not understand some important physics of the n o r m a l state of liquid 3He. Saying that spin fluctuations account for the important interactions in the superfluid is wrong. But to be fair to the theorists, this could not have been known before extensive measurements in the superfluid.
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In high-temperature superconductors, experimental constraints appear to be converging. In the highest-temperature superconductors, current experimental constraints leave a great deal to be desired. Table-top experiments would be quite beneficial. Finally, the ingenuity of theorists in devising new ways for matter to self-organize should never be underestimated. Even if the more speculative ideas do not apply today, they may tomorrow, for we should similarly never underestimate the ingenuity of Nature.
Acknowledgements The experimental work at USC cited here has been done in collaboration with Hans Bozler and a series of our post-docs (Ben Crooker, Yi-Hua Tang and Steve Boyd) and graduate students (Duane Bates, Ulf Israelsson and Inseob Hahn). I would particularly like to acknowledge tremendous assistance on the theoretical front on numerous issues in physics from G e n e Bickers. Finally I thank T o n y Leggett for explaining the history of Ref. [8]. This work is supported by the Nation~/l Science Foundation through grant DMR92-16842.
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