REVIEWS OF MODERN PHYSICS VOLUME 29, NUMBER 2 APRIL, 1957 Superfluidity and Superconductivity R. P. FEYNMAN Norman Bridge Laboratory of Physics, California Institute of Technology, Pasadena, California AM sorry that Professor Landau was unable to come somehow when we rub the atoms together we can I for two personal reasons. The first is that I have understand, more or less, how friction arises. But we do worked on this problem of helium upon which he has not understand, more or less, how superconductivity also done so much, and I would have liked to have works and I would like to address my attention to this spoken with him about it. The other reason is that I problem of understanding it more or less, not of under­ wouldn't have had to give this lecture. standing the details of a lot of special phenomena. In Quantum mechanics was developed in 1926, and in other words, I would like to concentrate here on the the following decade it was rapidly applied to all kinds problem of interpretation from first principles. We of phenomena with an enormous qualitative success. would like to connect the Schrodinger equation directly The theories of metals, other solids, liquids, chemistry, to some experimental facts. There are many interesting etc. came out very well. But as we continued to advance things that happen when one works on a problem in the frontiers of knowledge, we left behind two cities this way. For example, if one makes some approxi­ under siege which were completely surrounded by mations, they cannot be justified by comparing them knowledge although they themselves remained isolated with experiments. It is necessary to justify the approxi­ and unassailable. The first of the two very similar mations directly in terms of arguments from the phenomena for which we still lacked a qualitative equations as a mathematical problem. explanation is the superfluidity of helium. I may remind In the first stage of the development of such explana­ you that the discovery that helium flows without tions there is an enormous amount of phenomenological resistance through very thin tubes was made as far explanation made which is of great value, and I don't back as 1911. The second phenomenon is the super­ mean to say anything against it. That is just not what I conductivity of metals. The fact that electricity flows want to speak about. This phenomenological develop­ through some metals without any resistance at low ment is vital in combining various experiments together temperatures was discovered, I believe, in 1905. Of and is helpful in giving us hints as to what to try to these two phenomena, I think we now understand explain. If we were any good, however, we wouldn't qualitatively the superfluidity of helium, but we do not need the experiments and we wouldn't need the hints. yet understand qualitatively the superconductivity. I We could simply calculate everything directly from the propose to give a brief summary of views on liquid Schrodinger equation. So I will assume some familiarity helium insofar as they may give some clue as to the with the wide range of experimental properties and will kind of thing that is involved in superconductivity. It only make a very brief mention of the salient points. is clear, however, that the solution of the superfluidity First we will discuss the properties of liquid helium. problem does not give us a very good clue because we As a function of temperature, the specific heat appears have solved the first problem but not the second. something like Fig. 1 experimentally. The details near First, I make some semiphilosophical remarks about the transition are unknown. There is a certain tempera­ the kind of problem facing us, and the kind of view that ture at which there is a transition, the so-called X point, I take toward it. I do not want to discuss all of the and the properties of the liquid below the transition aspects of superconductors and of superfluids. I want are very peculiar. First, helium remains a liquid all to discuss only the interesting qualitative features, i.e., the way down to absolute zero, which we know is due the curious problem of how does it work more or less. to the zero point motion of the atoms. The reason for In other words, we would like to make an analogy the existence of the transition is undoubtedly the same between the problem of, say, superconductivity and as for the transition in the corresponding Einstein-Bose the problem of friction. After all, how is friction gas neglecting interaction between the atoms. The explained on the basis of the Schrodinger equation? Einstein-Bose transition for an ideal gas, however, has No one has ever computed the coefficient of friction of quite a different shape. It looks something like Fig. 2, two blocks of copper, but qualitatively we feel that and the problem of why the actual transition looks like FIG. 1. Specific heat of liquid helium as a function c. FIG. 2. Einstein-Bose c. of temperature. transition for an ideal gas. 205 206 R. P. FEYNMAN The energy of the excitations required as a function of FIG. 3. Coefficient of vis­ their momenta to produce the specific heat and general cosity of liquid helium 'l!ersus temperature as deter·­ properties seems to be a curve of the general nature of mined by rotating two Fig. 5. The curve is linear for smaller momentum, concentric cylinders relative meaning that at long wavelengths sound waves are T. to one another. excited. But the states of higher momentum excitations T are of some other kind. At any finite temperature one excites states mainly in two regions; small p near where Fig. 1 when there are forces involved is not yet com­ the curve is nearly straight, and near the bottom of the pletely solved. The detailed beha:vior near the ~nsition, concave part of the curve. The lower states are called however, is a very delicate and difficult propositiOn. One phonons and the higher states are called rotons. is convinced that the transition is due to an Einstein­ The reason why the flow is perfect, from the point Bose condensation, but exactly how interatomic forces of view of Landau, can be seen in the following way. modify the Einstein-Bose specific heat is not yet Put a little ball inside the liquid at absolute zero and thoroughly understood. try to make it move through the material, allowing the To discuss superfluidity, I would like to discuss the liquid to flow around it. How can the ball lose energy properties of helium near 0°K, far from the transition to the liquid? Only by exciting states inside the liquid, region. If one measures the viscosity of the liquid as a and if the aforementioned states are the only ones function of temperature by rotating two concentric available, it is easy to show by the conservation of cylinders relative to one ~nother, the coefficie~t. of energy and momentum that a very high velocity is viscosity behaves as in F1g. 3. Near the transitiOn needed before we have the necessary energy at the temperature nothing in particular happens. given momentum transfer to produce such a state. The On the other hand, if one measures the viscosity by ball must have a velocity v high enough that the straight making the liquid flow through a very narrow tube, line E= pv intersects the curve in Fig. 4. Actually, the the liquid flows through such a tube with an apparent viscosity above the transition temperature, but below the critical temperature it flows with apparently no FIG. 5. Energy of exci­ viscosity. Incidentally, if one measures the r~sistant tations required to produce E specific heat and general force to push the liquid through a tube at different properties of liquid helium velocities then above a certain critical velocity a in the two fluid model as a function of their mo­ resistance does arise but below the critical velocity the mentum. resistance is zero. A relatively thick tube has a lower critical velocity than a thinner tube as is shown approxi­ mately in Fig. 4. The explanation of these things at resistance sets in at a very much lower velocity by a least in a qualitative way, has come essentially from factor of about 100, and this fact has to be explained. Landau. He pointed out that we should imagine that We have to explain not the superfluidity, but why it is the liquid at, say, absolute zero is a perfect fluid for not quite so perfect a superfluid as it should be from some reason (or rather it is a perfect liquid when the this point of view. flow is potential flow and has zero curl). As we heat The essential feature of the liquid is that there are the sample up, the energy goes into local excitations very few states of low energy, a fact directly shown in of some kind, for example, quantized sound waves the specific heat and one which also explains the super­ called phonons and excitations of higher momen~um fluidity ultimately. So the question is, why are there so and energy called rotons. If, as the temperatur~ nses, few states of very low energy? Why in this liquid, for the excitations are few and far between (a fact drrectly example, can't we get a very low energy quantum state, indicated by the very small specific heat), then they say, by letting some portion of liquid slowly rotate? will be localized inside the liquid and bounce around The portion of liquid could have a big moment of within it.