傳播數學知識．促進數學教育

Interview with Prof. Rodney J. Baxter

**Interview Editorial Consultant:** Tai-Ping Liu, Yeong-Chuan Kao

**Interviewer:** Chuan-Tsung Chan(CTC)

**Interviewee:** Rodney James Baxter(RJB)

**Date:** November 17th, 2001

**Venue:** Institute of Mathematics, Academia Sinica

Rodney J. Baxter was born on February 8, 1940 in London, England. He received his BS from Trinity College, Cambridge University and PhD from the Australian National University (ANU) in 1964. Dr. Baxter has been a Professor of the Department of Theoretical Physics at ANU since 1970. For his achievements in statistical mechanics and integrable models, Dr. Baxter is elected a Fellow of the Australian Academy of Science and the Royal Society of London.

CTC: Could you tell me about your educational background? Starting from the university.

RJB: Well, I was born in England, northeast of London, I received my undergraduate degree in 1961 from Cambridge, and then I went to Australia to take a PhD scholarship. A lot of talented people from England and USA.

CTC: But in Australia you majored in physics?

RJB: Well, I was in the theoretical physics department.

CTC: Is it more like England, that theoretical physic is actually more of mathematical type?

RJB: Oh, a bit from place to place, but Cambridge has always tried to apply mathematics to physics. In 1968 I went to work with Elliot Lieb at MIT in the math department, Lieb himself has since been a Professor of Mathematics and Physics at Princeton.

CTC: What made you change?

RJB: It's an accident. My supervisor in Canberra Australia, Professor Le Couteur, attempted to get me involved in field theory and S-matrix theory. It was a top premier, it is still popular. But I wasn’t getting too far. Then I picked up a paper I think by Andrew Lenard on the one-dimensional Coulomb gas. He made the statement, he can do the problem on electrons in the uniform charge background in one-dimension. So I look at that problem and did it. That was a specific problem I can look at, so I got involved. But I then got involved in the other 2-dimensional problems with Elliot Lieb, that was the major step, once I did it correctly. I got the result when I got there. I tried to understand some of Elliot's works, and based on that I continued doing it after I was back in Canberra.

CTC: Elliot did it in England?

RJB: We were both in MIT then. (Two Cambridges in my life which confuse people.)

CTC: You consider Elliot to be the greatest influence on you?

RJB: Probably, yes, in academic sense. We only intended to go there for 2 years. But we did leave. We wanted to go back to Australia via England. We wanted to go by ship from England to Australia. There weren't many ships in 1970's, we have two choices---two or five months’ wait in England. My wife Elizabeth urged me to get the five months one, when we got there with complete culture shock, after two years in Vietnam War. It paid off handsomely. Toward the end of five months, I picked up the work I have been doing with Elliot L Lieb had the inspiration to see the eigenvetors of the transfer matrices which should depend on two parameters but in fact only depend on one, and that makes the transfer matrices with the different values of the opposite parameters commute with one another. I then went to see the Bethe Ansatz's result, that actually tells me that there is another matrix which must exist and commute with the transfer matrix, and which has certain properties that matters. This then actually simplified the alternative method to solve six vertex model. From that alternative method, I was able to generalize this alternative method. There was an outstanding problem, at that time, the eight-vertex model, which you could not solve with the original method, but you could if you used the second method, the alternative.

CTC: This is related to the Yang-Baxter relation?

RJB: The Yang-Baxter relation is the reason things can be commutative. What I am saying is that you don’t necessarily make progress by thinking all the time when you should take a sabbatical. OK, after that we went back to Canberra and I worked on the eight-vertex model. Very quickly, I got an invitation from Yang to visit Stony Brook. There was another sort of challenging thing I did with Barry McCoy. In 1980, C.P.Yang, the brother of C.N.Yang, wrote a paper in 1984, he calculated the residue entropy to be 0.33333…. he thought it may be 1/3.

CTC: What method did he use?

RJB: Some trick in the numerical method, I told my student Shiu-Kuen Tsang, to tackle the problem with the corner transfer matrices, and very quickly got twelve digits accuracy 0.333216949, and it's not 1/3. Even that wouldn't be the end of the story, I have already looked very closely at her computer results, realized that the eigenvalues of the transfer matrix she was calculating, a certain product of these eigenvalues, combinations which I expect to be roughly one, based on the low temperature expansion. Well, that is exactly one, and that is just associated eight-vertex model, and that suggest that one may solve the model directly, and I did, it gave another application of the Yang-Baxter Relation, and I promptly got involved in popularizing the Ramanujan identity. And I had a number of other identities which occurred in the calculations at that time. I got some quite helpful response from Michael Hirschhorn, but most of all from George Andrews in Penn State. He wrote back with sixteen identities, the first due to Ramanujan, the third in Lucy Slater's paper, another one in someone's paper, this follows from such and such equations. I think all of them must be from existing identities. He came to Canberra, where I stayed, for a sabbatical and we work together. We came up the Andrews-Baxter-Forrester model. He actually suggested to me that hard hexagon model is a 2-state model, there was a generalization of Ramanujan's identities for it. Basically if you have a k-state model, you have an identity which has 2k+1 modules , so this is what the k=2, Ramanujan identities are all about, and they are about modular functions of period 5. The next one would be 3-state with period 7 repetitions. And my first reaction is, this is a solution in search for a problem. What we find in this Andrews-Baxter-Forrester model is not what we want in mind. It was actually found later simultaneously by Miki Wadati and myself. But we find another model with other look-alike identities, and it became an industry of bosonic and fermionic identities, solid on solid models, and let’s play the game and then there is a thing I have been working on and talked about yesterday and today here, the Chiral Potts model. I guess it goes back to 30 years ago by now, there are still things we don't know about it, understanding free energy on certain range, but we certainly don't understand it. Order parameter remains as a subject today, I tend to analyze proper, simple conjectures, but hard to prove it.

CTC: So in your early stage of research, you kind of like these well-defined questions and try to solve them in a mathematical way.

RJB: It is a challenge, like C.P. Yang suggested that the number may in fact be 1/3, which I found suspicious, the entropy of the logarithm of partition function, the key proof is, I see no source of curious number comes in solvable model.

CTC: When did you decide to write the book (Exactly Solved Models in Statistical Mechanics)?

RJB: The book came out at '82, I think the progress starts at 1979.

CTC: You did it alone, all by yourself? I guess it is part of lecture note you used for your classes.

RJB: I have used it for a few of the courses.

CTC: So what you consider to be your best work? Eight-vertex model and hard hexagonal model?

RJB: Yes, and then I think of corner transfer matrices. These old papers have a long history, they began with approximate works on dimer-monomer systems. Well, I just tried to guess it for the wave function, and tried to seriously seek increasingly, and more accurate approximation. In 1975 I was in Edinburg, the idea of corner transfer matrices started and you applied them to eight-vertex model with beautiful properties.

CTC: So you interact more with mathematicians?

RJB: Mathematical physicists.

CTC: So what do you think are the most important or interesting questions in statistical models and statistical mechanics in general?

RJB: Very good and thoughtful question, and I find……

CTC: The Chiral Potts model?

RJB: Maybe. I have been looking recently at dichromatic polynomial, revisiting stuffs, a model I did a long while ago with Ian Enting, the ordinary Potts model, you can ..., well, the mathematician of William Tutte, of Waterloo in Canada, years ago in the early 70's wrote down equations for dichromatic polynomial on random graphs, it seems connected with what’s now in statistical mechanical theory, random model on random lattices. The paper I should read 15 years ago. I could probably anticipate these stuffs, that is the interesting thing. I also revisit very recently the Bethe Ansatz, the question whether it is complete or not. It appears that again somebody claims the Bethe Ansatz is not complete, or appears to claim that the Bethe Ansatz is not complete. My feeling is that this one is always complete. I have a variety of papers to justify my view. One trouble is that you make, people said things, look, that is not true, they said you just assert that, you have no evidence for it.

CTC: This is more like, I will call , in a more democratic situation where you do not have a major difficult problem everybody wants to solve, but rather you have different views on this field.

RJB: Well yes, I am suggesting, there are too many things while we are busy working on Chiral Potts model, and let's hope that many works in solvable model are true. But they are very interesting because you do actually solve all models you can solve, you do get exact algorithm, these model have continuous phase transition tricritical ... and usually these models are special cases of the more general model. Knowing what it does is very significant, one example, I always like to quote, is the Ising model itself. One would love to solve the 2-d Ising model with a magnetic field, all we have is the solution in zero field, for instance, and you have to know why it is so, it is extremely important because that is the solution that will tell you where the critical point is and what the critical behavior is, and we would..., it is certainly nice to know the full scaling function. That is an interesting thing you can get out of the solution of the Ising model in a field. For one thing, away from the critical point, you get just quite easily to get 12 digit accuracy numerically. So, you want more than that, probably in the vicinity of the critical point, still any challenge in it would be nice.

CTC: For example, do you consider the 3-dimensional Ising model to be an interesting problem?

RJB: Of course, but I don't know what to do, ha!

CTC: Most of the technique in the integrable models seems restricted to 2-dim.

RJB: Yes, there are some 3-dim models discovered by Zamolodchihov. These models are actually critical and are critical even for finite number of layers, as you increase the number of layers, you will have unusual persistence, so it is not a typical model at all.

CTC: So, it would be nice to have a typical model?

RJB: There is one sheer conjecture : in 1-dim, you can solve the Ising model in a field, in 2-dim, you can only solve it in zero field but at all temperature, you may wonder if in 3-dim, you can solve it in zero field at critical point, which is still interesting, but I have no idea.

CTC: What do you feel about other major fields in theoretical physics, like high energy physics, and condense matter physics?

RJB: I really cannot correctly answer these questions .

CTC: Do you know any of your works has unexpected impacts to other fields, besides mathematics.

RJB: Well, I suppose it has had spinoffs on some fields, besides mathematics, conformal field theory , certainly conformal field theory loves to be tested on solvable model,…, and in knot theory relies on so much of these technique, relies on using Yang-Baxter relation, and of course, you get ...the Quantum Group.

CTC: Are you also involved in the knot theory research?

RJB: I have looked at it.

CTC: Do you have any regret in your academic life?

RJB: No, no.

CTC: Great. If you are given another chance to choose the field you want to major, what things you want to do?

RJB: I have lots of things I want to do. I like to do other things, there are other things I am interested in doing.

CTC: Outside physics?

RJB: Outside science.

CTC: How many graduate students do you have now?

RJB: None.

CTC: In the past?

RJB: Not a great many. Shiu-Kuen Tsang was one, she is my first, she came from Hong-Kong, and is now in Canberra doing very good job in running computers, and Peter Forrester, he was one of my best students, and Alexander Owczarek, and a few others.

CTC: When they decide to major in this field, do you have any suggestion for them?

RJB: Well, I tended to supervise my student very closely except for poor Alexander Owczarek who become my student. I had an operation, so at first I did not attempt to supervise, because I am leaving. Peter Forrester needed no supervision. He already wrote at least one paper when he was an undergraduate student.

CTC: You just say you want to be more closer to them or……

RJB: I think when I started it, I would probably feel I should be...

CTC: So you would actually work with them?

RJB: Yes, I should and I try to do it, but I am not so sure if it is necessary, or even if it is desirable, they may do better being given more freedom to do what they want to do.

CTC: But in general you will find problems for them to solve.

RJB: Yes.

CTC: Do you have any suggestions for young students who are interested in this field?

RJB: Do what you are interested in doing, I suppose I don't have good one-liners to give. It is impossible for you to spend some time doing something, and you find out it is boring. So you are interested and work out that. If you are interested in integrable models, there will be a demand on deep mathematics.

- Yeong-Chuan Kao is a faculty member at the Department of Physics, National Taiwan University. Chuan-Tsung Chan is a faculty member at the Department of Applied Physics, Tunghai University.