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Slippery Wave Functions

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Abstract

Superfluids and superconductors show a very surprising behavior at low temperatures. As their temperature is reduced, materials of both kinds can abruptly fall into a state in which they will support a persistent, essentially immortal, flow of particles. Unlike anything in classical physics, these flows produce neither friction nor resistance. A major accomplishment of Twentieth Century physics was the development of an understanding of this very surprising behavior via the construction of partially microscopic and partially macroscopic quantum theories of superfluid helium and superconducting metals. Such theories come in two parts: a theory of the motion of particle-like excitations, called quasiparticles, and of the persistent flows itself via a huge coherent excitation, called a condensate. Two people, above all others, were responsible for the construction of the quasiparticle side of the theories of these very special low-temperature behaviors: Lev Landau and John Bardeen. Curiously enough they both partially ignored and partially downplayed the importance of the condensate. In both cases, this neglect of the actual superfluid or superconducting flow interfered with their ability to understand the implications of the theory they had created. They then had difficulty assessing the important advances that occurred immediately after their own great work.

Some speculations are offered about the source of this unevenness in the judgments of these two leading scientists.

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Notes

  1. A good general reference to the development of this part of science is [41, 42]. A predecessor that carefully assesses the helium work is Sébastien Balibar’s The Discovery of Superfluidity [8]. When he and I overlap, our conclusions are much the same and were reached independently. The outline for much of this paper was suggested by the late Allan Griffin [37] in his introduction to a Varenna volume on Bose–Einstein condensation. Mistakes and errors of judgment are my own rather than Allan’s, of course.

  2. Helium II is the name for the state of helium below the temperature for its transition to superfluidity.

  3. The technical name for the magnetic response with this kind of rigidity is diamagnetism. So London is saying that a superconductor is like a big diamagnetic atom.

  4. The interaction between Tisza and Fritz London in the period 1937–1939 is described in detail in [8, pp. 454–456].

  5. See for example [7, 38] for general references to condensate behavior.

  6. In this paper, I discuss the condensate as if it were always one and only one mode of oscillation, macroscopically occupied. This picture applies in three or higher dimensions. In two dimensions, however, the condensate is spread out over a whole collection of modes. The key descriptors of the condensate are the quasi-particle energy of the condensate and its wave function Ψ(r). The wave function is only relevant above two dimensions and will only be used to describe the condensate after it is invented by de Broglie and Schrödinger in about 1926.

  7. An quasiparticle is a long-lived, particle-like, excitation in a many-body system. Some people would reserve the word for a fermionic mode and call a bosonic mode a collective excitation. Since the theory of the two kinds of excitations are quite similar, I here use quasiparticle to refer to both.

  8. Recently G. Baym and C. Pethick [17] have argued that the Landau criterion is neither necessary nor sufficient for superfluidity by pointing to counterexamples of both types. They assert that the criterion does work to describe the possible reduction of superfluidity via the loss of momentum in collisions, but that this loss can have the modest result of converting a fraction of the superfluid component of the liquid into the normal fluid rather than the dramatic result of destroying the superfluidity.

  9. To complete the argument by describing the magnetic perturbation that might exist at non-zero temperatures, one would envision multiplying both sides of a quantum density matrix by factors like that in Eq. (2). This step is taken in a later paper by Ginzburg and Landau [63] described in Sect. 4.1 below.

  10. This strategy of taking the noninteracting system to be the template for the construction of a quasiparticle theory, the latter being modified to include a more general energy momentum relation, is exactly the one later followed by Landau in his later construction of a quasiparticle theory of 3He [59, 60].

  11. Part of Landau’s trouble with Tisza seems to be contained in the word “normal” as in “normal fluid”. They appear to use the word differently so that what Tisza is saying is hard for Landau to interpret.

  12. BCS could have reminded the reader that the 1946 paper of N.N. Bogoliubov [20] contained a very similar form of excitation. They did not do so. This failure pushed the reader away from looking at the analogy between superfluids and superconductors.

  13. Note the historical continuity in which Δ is the symbol used by both Landau [53–55, 61] and BCS to describe the gap in the spectrum, which then plays a crucial role in the Landau criterion for superfluidity.

  14. It is important to the BCS arguments that the transition not be first order in nature. First order phase transitions permit and entail jumps in behavior. Second order ones introduce new behavior at the phase transition, but do so gradually. BCS expect continuity at the phase transition.

  15. In unpublished work, Kurt Gottfried and I showed that the region of applicability of mean field theory does not include a small range of temperatures near the critical temperature for onset of superconductivity. However, the range of non-applicability is so narrow as to be irrelevant in almost all studies of bulk properties of simple superconducting metals.

  16. In my papers [45, 46], I incorrectly assumed that the 1935-7 Landau work made provision for the spatial dependence of the order parameter by including a term in ∇2 applied to that parameter. It did not.

  17. This paragraph and this misprint was pointed out to me by Pierre Hohenberg. The definition of the wave function is given in terms of the quantum density matrix \(\rho(\mathbf{r}_{1},\mathbf{r}_{2}, \ldots, \mathbf {r}_{N}; \mathbf{r}'_{1},\mathbf{r}'_{2}, \ldots, \mathbf{r}'_{N} ) \). The corrected definition is

    $$\varPsi(\mathbf{R}) \varPsi\bigl(\mathbf{R}'\bigr) \sim\int d \mathbf {r}_2,d\mathbf{r}_3, \ldots, d \mathbf{r}_N \rho\bigl(\mathbf {R},\mathbf{r}_2, \ldots, \mathbf{r}_N; \mathbf{R'},\mathbf{r}_2, \ldots, \mathbf{r}_N \bigr) $$

    for the limiting case in which the distance between R and R′ becomes very large.

  18. This tone might be explained by the fact that in the absence of the idea of pairing, one could not see how the fermions (i.e. electrons) involved in superconductivity might exhibit ODLRO.

  19. Specifically, BCS had constructed a theory of weak-coupling electronic superconductivity in situations with time-reversal invariance. This theory must be modified for other cases. For example, it fails in the presence of magnetic impurities and also for the subsequently discovered high temperature superconductors. Nobody knows how to do the analog of BCS for these high T c materials.

  20. Feynman’s vortex predictions [27] were preceded by Lars Onsager’s [77] and followed by estimates of the size of interaction effects in helium II by Penrose and Onsager [81].

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Acknowledgements

This work was partially supported by the University of Chicago NSF-MRSEC under grant number DMR-0820054. I have had instructive conversations on the topics of this paper with David Pines, Silvan Schweber, Gloria Lubkin, Gordon Baym, Edward Jurkowitz, Margaret Morrison, Joel Lebowitz, Roy Glauber, Sébastien Balibar, Humphrey Maris, Pierre Hohenberg, William Irvine, and Paul Martin.

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  1. James Franck Institute, University of Chicago, Chicago, IL, USA

    Leo P. Kadanoff

  2. Perimeter Institute, Waterloo, Ontario, Canada

    Leo P. Kadanoff

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Kadanoff, L.P. Slippery Wave Functions. J Stat Phys 152, 805–823 (2013). https://doi.org/10.1007/s10955-013-0795-8

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