Interview with Leo P. Kadanoff, part II

II. From correlation functions to scaling

BA

We left off at Copenhagen. When did you leave Copenhagen?

LPK

In the winter of 1961-1962. I was offered an assistant professorship at Urbana, and accepted with pleasure, giving up the last six months of my National Science Foundation postdoc. Of course, Urbana was, at the time, a trememdous center of activity, as it is now, in condensed matter. Fred Seitz was head of the department when I arrived. Bardeen and Pines were my immediate seniors. Schrieffer was about to leave (I hadn't known he was leaving), and Pines was off on a sabbatical. They bequeathed me some of their graduate students, and I was off and running. The research on superconductivity, on condensed phases of matter, continued. Work on polarons and transport theory began, and I dug in and started to learn how to teach and to direct graduate students. The next big milestone in my story, although you people may want to ask different questions, is in 1965 which, when I go off to England, professor Sir Neville Mott is having what he called a theoretical jamboree at Cambridge.

SS

You take a trip to the Soviet Union before then. Could you tell us a little about that?

LPK

My mind is not sharply focused on that. However, let me put things in a little bit different context, since you've asked that question. The world which is for me the known world is Paris and some of England and some of the United States, and of course Copenhagen, in which research in many-body physics is going on. I'm also moderately well factored into some things that are happening in particle physics. I know, probably, with a part of my mind, that in Russia, things are going on which are very much like things that are going on in the United States, but I do not follow that kind of physics. In Copenhagen I met some physicists, but they were not Landau school physicists. The people who were directly relevant to me are Landau school people with some Boguliobov school people also, but mainly Landau school, and I just don't know them until 196-whatever-it- is, until I go to a conference being run by the Landau school and meet the people.

My recollection, for the little that it's worth, is that I don't learn much physics then. The Russians remain inpenetrable, I think, because their talks typically start out, as I recall, with a statement "as is well known," and I don't happen to know that stuff well. Even though if it were explained to me in words of one syllable, I could probably appreciate it, it never does get explained, and so the talks are a little bit incomprehensible. But I meet people, Migdal, Polyakov, Abrikosov, Khalatnikov, whole bunches of people who are doing physics analagous to mine, and who in fact are looking at the world from a point of view which is very much like mine. So it's an important social event for me, and will turn out to be important intellectually. Of course, if you look in terms of ethnology or whatever you call the subject, I basically come from the Ukraine, my family is Eastern European, and these are people whose background is very similar, and whose genes, probably, are very similar to mine.

BA

You make two trips to the Soviet Union?

LPK

I make lots of trips to the Soviet Union, and I don't remember which trip is which. There is an Urbana-connected trip, but I don't remember whether that is the 1966 trip.

BA

I think that's 1963.

LPK

Right, I know my first scientific conference abroad is at Ravello above the Amalfi Bay in about 1964, so probably 1963 is a wrong date for a trip to the Soviet Union. However, there may be a trip to the Soviet Union in 1965, in pre-scaling days. I simply do not know.

SS

The context of why we ask you: we did talk to Patashinski, and he remembers specifically a trip in 1963-64 of the Urbana group, namely, he remembers you coming there, Pines, Bardeen (and also Martin).

LPK

Paul Martin, he said. Well, I'm more clear here about some elements of science, which will turn out to be important. In one of these trips, I learned from, I believe it is Migdal and Polyakov, about a K to the 3/2's kind of theory of correlation functions in critical phenomena. I probably meet Patashinski and Pokrovsky on one of the two trips. Certainly, if they remember it, and they probably tell me about something or other. I don't know what I was told.

BA

Can you tell us, you jumped to 1965, can you tell us some of the work you're doing at Urbana before 1965?

LPK

There is stuff on Boltzmann equation for polarons. I could tell you, but not without a publication list in front of me.

BA

Baym comes a year after you, and presumably you continue to work with Baym. Who else at Urbana do you begin developing a working relationship with?

LPK

Here you can't tell so well from my papers, at least as far as my recollection is concerned. For example, there is no collaborative paper with Slichter, however Slichter offers me good critical advice, and is helpful to me. There's no collaborative paper at all with Bardeen. But Bardeen is a wise man who speaks in short sentences and few of them, but nonetheless is capable of conveying considerable information. Pines and I interact a lot. However, I maintain my individual style of research, and do a lot by myself in this period, I believe. Now, we can find some joint papers with graduate students, I think, but I do not believe that there are so many joint papers with faculty people in this period.

SS

And what do you teach, mostly?

LPK

I teach quantum mechanics, statistical mechanics, mostly advanced graduate courses and occasional senior level undergraduate course.

SS

And 1965 is when you go to Mott?

LPK

To Mott. This is an eight-month visit in England and is supported, I believe, on the English side. I find more culture shock in England than I did in Copenhagen. That's something else. I take with me a product of my Harvard period, which is the problem of critical phenomena. I'm reminded of that problem by Voronel's work on the specific heat of fluids. I believe I listened to Paul Martin say that this is an indication that critical phenomena is a real important problem. In fact, Kurt Gottfried and I had unpublished work on critical phenomena in one of the Harvard periods. We looked at near-critical behavior in superconductors, three-dimensional superconductors of the kind that were standard to look at in those days. We were disconcerted to discover that you wouldn't see anything interesting happen unless you got within one part in ten to the thirteenth of the critical temperature. I won't guarantee it's between thirteen and fifteen, or something like that, but that was sort of what we discovered, which was the right answer, but did not encourage us to publish. We never did. Azlamazov and Larkin later, as I recall, saw that you could get other behavior in film environments, and understood better what was the nature of the problem of superconductors. Nonetheless, the thinking that Gottfried and I did was important in developing stuff that I later on used in my other work on critical phenomena.

SS

With Paul Martin you're in correspondence, or you talk by phone.

KH

You finish up another paper with him, after getting back from Copenhagen you have the hydrodynamic equations and correlation functions paper?

LPK

Yes, I do not recollect how we keep in touch. It is a small world, I can remember finding out while Paul is on the plane coming back from Paris that Paul has had some back troubles in Paris. It is a small world, but I can't recall exactly how we keep that world small. Nonetheless I'm in reasonably good scientific contact with the people at Harvard, maintained excellent scientific, personal, contact with my colleagues, Urbana provides a magnificently supportive atmosphere for young scientists, and have all kinds of encouragement to do good work.

BA

Up to 1965, in your teaching of advanced graduate courses, did you teach on the material you were researching?

LPK

I gave lectures, which were the same lectures that I gave in Copenhagen and Poland, on the Kadanoff-Baym book. I remember, particularly, because Bardeen came to the lectures quite faithfully, and was very pleased by some of them. In particular, he liked the connection between Green's functions and the Boltzmann equation connecting classical ways of looking at the problem, and hence being very pleasing to him, and pleasing to me, of course, I was very flattered by that.

BA

Before we talk about Cambridge, what was your culture shock in England?

LPK

I did not fit into any class of that society. I had the usual situation that only the Australians could talk to me comfortably.

SS

That included Neville Mott?

LPK

That included me! Neville was a wonderfully supportive and charming person. Neville and his wife arrived at our little house a day and half after we arrived. The door opened at 9 o'clock, and there was this huge man saying "I'm Mott. This is Lady Mott," walking in, among our spread-out diapers and general impossibleness. So the lord of the manor arrived to see how the new person was doing. Wonderfully supportive. We didn't know what to make of it. Nor did I know what to make of Mott asking me in 1965, "Kadanoff, what do you think of the spin 1/2 Heisenberg model in two-dimensions?" I suppose it was like any other Heisenberg model in any other dimension. What do I know. Of course, that became an extremely important problem, and he probably knew in 1965 why it was an important problem, but I did not have a clue. So I would not put all of the blame on the Brits for my difficulties, but nonetheless there was culture shock.

Ok, so I arrived and I have plenty of time on my hands. And I arrive with critical phenomena. And I have at some stage, probably at Urbana, learned what people had done with the Onsager theory, including certainly a spinor formulation of that which had been done mostly by Onsager and Kaufmann. This is rather like the spinor formulations that I had struggled with for using the Nambu formalism in superconductivity. Different, more sophisticated, but not so tremendously different. And I spend my time in Cambridge doing another hideous, seat-of-the-pants, hardworking type calculation of spin correlations in the two-dimensional Ising model. And about halfway through I had the asymptotic large-separation form of the correlation function, and I send a paper to J. Math Phys., run at that time by Elliott Montroll, and it is rejected. Exactly why it's rejected is a little less clear to me at this period, because it returns across the Atlantic by boat, and arrives in Cambridge after I have returned to the United States, and hence undergoes another boatride. In the meantime, I'm in the happy position of being the only person in the world, except Michael Fisher, who knows about the asymptotic form of the spin correlations in the two dimensional Ising model. Michael kindly invites me to come up to King's College, and I give a lecture, highly technical lecture, must have lasted forever, must have been totally incomprehensible to the students, but give a lecture about the work which reports the results.

SS

And this is the first time you meet Michael Fisher?

LPK

Indeed, the first time I meet Michael Fisher and his children.

SS

When you go to England, you're already aware of the National Bureau of Standards conference and some of Voronel's work on liquid gas transitions?

LPK

I'm aware of it, but I'm also very aware of airfares. And at that time one did not travel across the Atlantic lightly. And so I do not go to the conference. It is important, we'll come back to it, the story. It is of course a place where people find out a lot about what's going on in critical phenomena. You may be pointing out to me an error in the story I have told, which is that I implied or said that the work I did on critical phenomena was inspired before the correlation function work by Voronel's experiment. I do not know when Voronel's experiment was reported to me. It may have been reported to me after the conference in Gainesburg, the National Bureau of Standards conference, and it may have been reported to me beforehand. Honestly, how does one know?

BA

So before you meet Fisher, where are you, Fisher, and Domb, and the work of that school?

LPK

I probably don't know. On the other hand, I might. It turns out to be very important to the story that these guys had been working showing that the Landau mean field theory of critical phenomena is not asymptotically correct. Another piece is that they showed that all lattices are sort of the same as all other lattices, and I make use of this later on. I can't tell you when I know about it. I do, of course, know about it as part of the ambience, the atmosphere around me. For all I know, someone may have told me about it in the Soviet Union, one of the visits. I simply don't know when I found out.

SS

But in Cambridge you certainly knew of Domb and what these guys were doing in terms of high-temperature expansion.

LPK

Cambridge is not the best place in the world to find out about King's College, London, as we know from other aspects of the history of science. In fact, I could not get good directions from Cambridge as to how to get to King's College, London. I've now completed this work on the correlation functions, the two-dimensional Ising model. I've returned to Urbana with the work done but unpublished.

SS

And you're fully aware, in fact, that at that stage your calculations on the Ising model don't agree with mean field theory.

LPK

I may. Again, it's a question of when one knows it. Of course I know. I'm aware in some sense of the Keller experiment referred to as Buckingham, Fairbank, and Keller, the Keller experiment on the lambda point of helium. I'm aware of the fact that the Onsager theory does not agree with the mean field theory, and since now one's talking about the scheme of beliefs in which I'm working, that scheme of beliefs is that there are equilibrium properties, low-order Green's functions, which then determine to some extent the nature of the transport properties' higher-order Green's functions. So some of my view of the calculation I have just done is that Onsager had figured out the low-order Green's function, so to speak, the zero-order Green's function, free-energy, and that Yang had done the calculation of the half-order and first-order Green's function, and that I had just calculated the second-order Green's function. And I expected that the Yang result and the Onsager result, whose implications I knew something about, namely that mean field theory was wrong, I expected that to be reflected in my result. Hence I did not think that my result was new, only that it was interesting. Not deeply new in the sense that Onsager's work was deeply new.

However, there was a piece of it, which, again, in my philosophy, we're talking about philosophical issues, was important. I had for the first time determined the correlation length and that as part of a viewpoint that somehow had gotten built up in me or around me, connected with correlation functions, that was important new data which could have been derived from Onsager's work on the free-energy, since that work involved an integration of a one-particle Green's function. However, no one had done that, and in my youthful impetuosity I thought they were pretty foolish for not understanding that the correlation length was in there. So I had new information, and also thought that the world was pretty foolish for my information was new, since it really was in the old stuff.

BA

Can you remember if you learned something new from your talk in Fisher's group? And the reaction to questions?

LPK

I do not know. I certainly learned that Fisher was an intelligent and with-it member of the community in which I visualized myself as working, and hence knew that I would have the opportunity to work in parallel with Fisher as long as I continued in critical phenomena.

SS

You remember when you hear of Widom.

LPK

I remember very precisely. Some place in a conversation with Paul Martin, this is after the 1965 paper that got published in the journal Physics got written, but before it was published, I had a discussion with Paul Martin in connection with that paper, or maybe not in connection with that paper, no maybe it was before the paper even. But I had discussion with Paul Martin sometime in that period, unspecified if it was before or after that paper, in which Martin reported that he had heard at the Bureau of Standards conference some magic results. I stored that away in my mind. After I had written the paper, I came back to that, I believe I asked Martin again about it. Or maybe I had the information stored away. I looked through all the journals I had ever heard of, not so many, and it wasn't in Physical Review or Nuovo Cimento or here or there. But then I remembered it was done by a chemist, so I looked at a journal that I had never had much contact with, which was J. Chem. Phys., and discovered that Widom had anticipated me. I'd had, previously, discussions with Bardeen about the obligations of authors to represent correctly the previous literature. I understood my discussions with Bardeen, and did the right thing, and rewrote the introduction saying that the paper was based upon the work of Widom. This is the paper from Physics, 1965, probably published in '66, but was done in 1965.

KH

How was it that you decided to submit this to the journal Physics? This is an interestingly collaborative journal. It has a very international editorial board, at least.

LPK

Probably I'd heard about it from Phil Anderson. Phil is a person who hasn't entered the story before. I met Phil in Ravello in the first scientific conference I attended. I remember, and I think Phil still remembers, Phil's driving me down the mountainside in his car, impressing me greatly with his derring-do. I think he was trying to do so. Phil was in other ways, in addition, a very impressive figure. And I believe I had heard that he was putting together this journal which was supposed to contain great papers in physics. I thought this was a pretty good paper, so I thought may I could get it into the journal.

SS

The first issue has Gell-Mann and has Bell in it, and things like that.

LPK

It's almost as good as the Physical Review from 1947, but you know, it was a pretty good journal.

BA

So the paper that Bardeen sat on, this is the spin-spin correlation in 2-D Ising?

LPK

No. The paper that Bardeen sat on was the thesis paper which I recall and I think we've referred to as Martin-Schwinger 2. We left a piece of the story dangling. So in the summer, I believe it is summer, I return to Urbana, and after a couple trips across the Atlantic, finally understood the rejection from J. Math. Phys., and submitted this paper again to Nuovo Cimento. In those days (it's not true now, I think), Nuovo Cimento was a broad-based journal that would include all kinds of stuff. I thought it was likely to get published there. But if I had chosen a method of hiding the paper from the world, I could not have figured out a better method than submitting to J. Math. Phys. and then Nuovo Cimento. So for a relatively long period, I was the only person who had this information about correlation functions, and I was at Urbana working away at this. And I had in front of me some of the results of the conference in Maryland, and was trying to remember all of those damn critical indices.

By this time I must have been very well aware of the King's College school, since I had in front of me zillions of critical indices. And I couldn't keep them all straight so I developed a mnemonic based upon derivatives of the free-energy to remember how the magnetization and the exponent called beta, and the susceptibility exponent called gamma, and the specific heat index called alpha, and the other indices that were there were related to one another. Some of these were well known, the relation between alpha, beta, and gamma was due, I think, to Fisher and Hunter, but in any case is a well-known relation. Others were not so well-known in those days. And were part of this magic stuff that Martin had remembered but had not been fully conveyed to me.

But then the magic sunk in, and in some week around Christmastime, in some year that you guys can identify better than I, maybe it was 1965, I believe it was probably Christmas of 1965. Some week around Christmastime I figured that you could have a running coupling constant. I'd never never heard of the work of Stueckelberg and Gell-Mann and Low and those people, Boguliubov, I'm not so sure about the Boguliubov-Shirkov volume, but I had never heard of the Western stuff. So I was completely naive of renormalization group concepts, but knew that here was a way of getting an effective coupling constant which was changing with scale. I presented it to my colleagues in Urbana soon after and they loved it, and it was presented soon after at Cornell, and they loved it, and I loved it.

SS

And Cornell would now include Fisher?

LPK

Cornell would probably be, although I'm not absolutely sure, another one of Fisher's three-hour-long seminars.

SS

Ken Wilson was there, or you don't remember?

LPK

I'm not sure I'm remembering accurately. Maybe it was Harvard, but there was another place that really liked it. I'm not sure whether I presented it or not, but there was another place that really liked it. Maybe it was reported to me that it was reported there. I'm not remembering accurately. It is certainly true that Wilson hadn't completely disappeared from my radar screen, probably nor me from his. I don't know. So the work was reported and well-received, and then I did something of which I've always been rather proud. I decided that the right way of making this known to the world was to make contact with the experiments. I'd been involved with a thesis which was done a tiny bit slowly, and there were previous publications by other people, so the student had a problem. But the student's thesis was well-received in the end, because he had referred to all the experimentalists who worked in the field. And I absorbed that information. And I decided that I would run a seminar which would include all the experimental work I could find.

The materials research lab had just been set up at Urbana, and my colleagues in the lab kindly, or maybe occasionally not-so-kindly, allowed me to commandeer their graduate students and post-docs, to give lectures in the seminar, and in the end to put together a review paper. This is something we did over a course of six or eight months. We reviewed, I believe, every experiment that we could reasonably find involving critical phenomena. And managed to fit them all into some picture which included this new scaling point of view, based, of course, on the phenomenology that had been developed by Widom. Based upon the phenomenology which had also in parallel, and earlier than my work, I believe, been developed by Patashinski and Pokrovsky. You guys could check the stuff, but although I was probably unaware of the work, the work, I believe, was previous to mine. And maybe even previous to Widom's. You all will know when you look it up. There's a Russian version and an English version.