Interview with F. Duncan M. Haldane on 6 December 2016, during the Nobel Week in Stockholm, Sweden.
Duncan Haldane, welcome to Nobel Week in Stockholm
Duncan Haldane: Thank you.
I noticed that you brought an artefact for the museum, what is it?
Duncan Haldane: Well, when I initiated the work, which this prize was given for, one of the two works which I received this prize for took a long time for to be published, because it was actually contradicting conventional wisdom at the time. Perhaps I didn’t explain it clearly enough, but in any case, this paper was rejected by a number of journals. And in fact, the original arguments which I discovered that magnetic chains had what we now know is a topological phase, when the quantum spin with an integer-spin, but not a half-integer-spin, which was … no one had thought that was an important feature. Why I discovered this really accidentally, I mean I, as a consequence of another piece of work I had done earlier. And it turned out that people’s understanding of magnetic chains was confused and that there were two sources of it.
One was kind of that semiclassical picture that thought of spins like little compasses, little arrows, that pointed and wanted to be parallel/antiparallel. And that works very well in high dimensions when long-range order occurs at low temperatures. But it was actually when it had been known mathematically that a one-dimensional chain, a very long, thin chain of atoms couldn’t, it ought to be destroyed by quantum mechanical fluctuations at any finite temperature, but also at zero temperature. On the other hand, there was a mathematically exact solution of a toy model, that Hans Bethe had discovered, before he went on to, you know, do huge work in understanding why the sun, how the sun works, and working on nuclear energy. But his early work on magnetic chains had this interesting solution which he guessed, and it turned out to be correct. And he probably didn’t understand why it worked, because he thought he would apply the same methods to higher dimensional magnets, and he promised that in his original paper. But of course, it turns out that what he’d locked on to, involves some very deep mathematics that took about 50 years to understand.
So, most of the scientists who work in magnetism, they knew of the existence of Bethe’s exact solution for a chain of magnetic atoms where the spin took the smallest possible value of 1/2, and locked superficially exactly like the spin wave theories that worked for, that were basically semiclassical, that worked in higher dimensions. And so, they assumed that ok, there was a little kind of, some kind of technical problem with the spin wave theory, but it had to be morally correct; the detail, this little problem didn’t really matter. But they were completely wrong, because the resemblance was purely superficial, and we now know that Bethe’s solution describes very interesting excitations which you call ‘spinons’ now, that it should carry spin-1/2 why the spin wave would carry spin-1, and they got nothing, it’s just an accidental coincidence of the formulas.
I actually, by covering this from a completely different angle, which was that there had been a, again going back to the 30ies, a remarkable relation between fermions and spins, a spin-1/2 object could take two possible states; up and down. And if I have an orbital which you can put a fermion in, it can either be empty or filled, and there is a kind of mapping between the two. But fermions, they have this fundamental property that if you exchange two of them you get a minus one sign, so fermions are said to … the operators that create them are said to anti-commute well. You get a minus sign if you add or remove them in different orders, while the spins don’t have that. But Jordan and Wigner in the 30ies had discovered a nice little way to map the two by putting in, what’s now called a string, in front of the things in the operator. So, they had a remarkable relation between spins and fermions that worked. So, it was pointed out that you could map the spin half chain problem that Bethe solved into a problem involving fermions. And at some limit these were non-interacting fermions, so you could understand the thing.
Based on that language, a completely different language, which I had followed on some work that Alan Luther and Vic Emery and I guess Luther and Ingo Peschel had done in the early 70ies. This allowed one to calculate things – not exactly – but in an approximate method. While the Bethe solution, the one that gave you the energy levels, no one at that point had ever worked out how to calculate anything else out of these very complicated wavefunctions that Bethe gets. It took another, probably 20 years, before, finally the solution of how to calculate with Bethe’s solution was found. So, using these methods, which a normal physicist, who isn’t a very abstract mathematician, could actually do a calculation. So, in this language of the fermions, I made a kind of theory called ‘Luttinger liquids’, which gave a general scheme for what interacting fermions should do in one dimension, and then applied it to this spin-1/2 model. So, Luther and Peschel had made a large progress in the spin-1/2 model; by treating it as a fermion field theory we could actually do a calculation.
But they had missed a detail, which was if the spins like to be in the line plane, like compass needles, you could treat it easily this way, because one of the limits is non-interacting fermions. But the interactions in the fermion picture when the spin is to stand vertically upwards and down, like point towards the North pole or South pole, and they’d missed out why the transition happens at a certain point. From doing numerical calculations, in fact, by applying my general theory to the Bethe “Ansatz” solutions, I saw within them exactly what had been missed. And so, it gave me the understanding of how to properly do a treatment of spins. And then I could actually treat any spin, not just spin-1/2, and immediately I did it. I found that in fact, using arguments based on what Kosterlitz and Thouless had done, that the conventional argument was just wrong. And something quite different happened for the next spin up, spin 1, and all spins which were integer values rather than half-integer values. So, that was it. In fact, this paper states the first lines of stating that, you know, this leads to a very unexpected conclusion; that the spin-1 chains behave completely different from what the Bethe “Ansatz” solution said/did, and it was nothing to do with the spin waves.
So, this was kind of against the orthodoxy … because a lot of people … I mean spin chains were kind of obscure, but in the 70ies a lot of people started working on them because there was some possibility of actually making materials that did this. And, in fact, spin chains had grown a lot of current research on understanding how thermal equilibration happens; people are using spin chains for all kind of things now. But, anyway, this started getting popular, but I was unable to get this published. I sent it to one journal, got turned down with three dismissive reports. So, I just sent it to another one, and I got essentially, I don’t know if it went to the same referees, but essentially, I got the same kind of reports back. So, it took another couple of years, to get the thing published, I mean I, perhaps I didn’t explain it in the language that these people could understand. But in fighting these referees’ reports, I rephrased the arguments in a much more, perhaps, a better way. But the original way I discovered this got completely lost from it. So, this paper was actually, was referred to in the literature because people went on and basically validated the results in here.
There wasn’t an archive at those times, and somehow, I, in moving around, I didn’t have this paper. The people who worked on this problem, and actually validated that I had found what they referred to, that they never had any copies anymore. The institute in Grenoble, the institute in Laue-Langevin, where I did this work, there was an official number of pre-print, but of course they had cleaned out all their cabinets. And I was searching in the boxes in my basement. In fact, the Swedish television people wanted me to meet to show them the boxes, because I have a problem with throwing out papers. So, I have lots of boxes, unopened boxes, which had accompanied me in various moves that was accumulated, and I thought that it might be in there. But in fact, a Hungarian physicist Jenő Sólyom, who worked this problem, eventually came up with the missing pre-print. So, it was actually very interesting cause it actually, it does make a connection that I had forgotten about completely with the work of Kosterlitz and Thouless.
Initially you wouldn’t even think there was a connection because their work was based on, essentially, classical problem of a super fluid film in two dimensions. And Kosterlitz, when he talks about this, he makes it clear that he just deals with classical mechanics, and quantum mechanics introduces too many complications. But, in fact there is a remarkable relation between two special dimensions and one space- and one time-dimension, which is kind of like the relation between relativity in space-time; three space dimensions and one time-, and you get the four-dimensional space-time continuum. Well, a one-dimensional system, where you have a one-dimensional space-time continuum, one plus one dimensional, you have time and space. And there is a remarkable mapping that maps statistic, classical, statistical mechanics of systems at finite temperature described by the Boltzmann factor, with probability weight, and quantum mechanics, which is described by an amplitude for a process to happen. So, the vortices in two dimensions, which if you wonder around the vortex, the spins rotate by two π, turns into, in one space-time dimension, what’s called an “instanton” process, or “tunnelling” process, where, if the spins are all kind of untwisted at one time, and they twist through one turn around a plane and a second plane, and if I do a walk in space time around that, the spin rotates by two π. But the remarkable thing is that quantum mechanics is far richer than the statistical mechanics, cause in the Boltzmann’s formulation of statistical mechanics, the weight factor, the probability factor is always positive. Probabilities, in classical mechanics, are positive, where in quantum mechanics, there are in general complex numbers. But if there is some kind of time-reversal-invariance property, they can be their real numbers, which are plus or minus. But, when things can be both plus or minus, when you combine them, they can cancel, which cannot happen in that.
So, it turned out that the basic process, which should be, the general process was the one I discovered, and the spin-1/2 was a very special case, which was behaving differently, so, the question was not why the spin-1s behave differently from the spin-1/2s, but why the spin-1/2s behaved the way they did. And that’s basically because there was a minus one factor when cancel things. So, this relation to Kosterlitz-Thouless was actually how I realised the thing is a mapping from the classical Kosterlitz-Thouless transition in two dimensions, to the quantum version, in one plus one, which has been incredibly fruitful for all the people working in one-dimensional systems too. So, despite Kosterlitz’s disembowel of quantum mechanics, in his work it has actually been very important also in quantum mechanics. So, this paper has that in it, and as to say, the published version, which is two years later, that connection was completely lost. So, it’s very interesting to remember this and how I discovered things. I was very pleased that someone finally, after I had drawn a blank everywhere, for copies of my own work …
Did you ever doubt yourself; did you think that you must have made a mistake or must be wrong?
Duncan Haldane: No, I was always confident. I mean people were kind of telling me things like this is a … some referee reports were saying this was obviously wrong and in complete contradiction to fundamental principles of physics, such as continuity, or something like that. About that time … and other people called this the ‘Haldane conjecture’, I never, actually I looked in the paper and I used the word ‘conjecture’ about some other tiny detail, which is not the main thing. Somehow this got built as, not a prediction or a theory, but a conjecture – as if it was a guess, a speculation. And of course, it wasn’t. But sometime around that time, the Bethe’s method was applied to a modified spin-1 chain. In fact, two authors in the Soviet Union seemed to have found this spin-1 chain solution about the same time. So, there was two versions of this, and I was the referee of both, actually. And I guess they, of course, found something that looked very much like the exact solution of the spin-1/2 chain. But it wasn’t solving the regular problem, there was a slightly modified problem; an extra piece was added on. And, I suppose I might have had about ten minutes worth of soul-searching when I got the first of these to referee.
But finally, I realised that they were doing a slightly modified problem that was special. And in fact, that turned out to be the case; that if you add this extra piece to the problem, and vary in strength; if you switch it to zero – nothing, it’s the same as what I predicted, but as you switch it up, there’s a critical point at which a phase transition to another kind of state, called the dimary state happens. The solvable, the models aren’t exactly solved by Bethe’s method; they’re usually special, and this one was exactly on a critical point. Now, it turned out that these models were extremely interesting for conformal field theories, which were turned out. So, lot of developments happened in this whole business. And those were very interesting models, but they weren’t generic case, so, I suppose I had ten minutes worth of doubt. But I quickly recovered. It was so clear to me that I couldn’t quite understand why people wouldn’t appreciate my arguments. But maybe we feel that we are clearer than we really are …
Is there any person that you have worked with that has been a huge inspiration for you?
Duncan Haldane: I was very fortunate to work with Philip Anderson, who was a 1977 Laureate in Physics, or one of the three. In fact, he got the Nobel Prize with his graduate advisor, Van Vleck. So, I guess that puts a big pressure on my students to make sure they don’t break the chain, right? So, he had a very unorthodox way of thinking, and maybe that rubbed off on me in some way. I mean, I was extremely inspired by the way he thought about things. I think obviously that was a big influence on the way I developed as a physicist that I had a good chance to interact with Philip Anderson, who has done an amazing amount of different things. With Philip, the Nobel Prize he got wasn’t for the thing he did which was the best; a bit like Einstein, who got the Nobel Prize for the photoelectric effect and not for the gravitational theories.
What do you feel, do you feel that you’ve gotten the prize for the thing you are most proud of?
Duncan Haldane: Well, I’m proud of a number of things. I guess I, this was interesting; it’s probably only in the magnetic chain stuff I always felt was ok, it was interesting, but it didn’t have any obvious applications. It was maybe changing what you thought about things, but more of a kind of theorist thing, right? But it was interesting to a special … some set of people. But a lot of people were very impressed by it. I suppose, partly because … if a lot of people had post it, they had to say, well, it was either, if it was obvious – it meant they were stupid. So, I had to be brilliant instead, to have discovered it, I suppose. But no, I’ve done a number of things, but, the general theme I think, has been this topological matter. And, the other piece of work I did, which was to find, to realise, that you could have a quantum Hall effect without a magnetic field, just due to some kind of magnetic interactions in the system. It is potentially a very practical, useful thing. But it has taken some time to actually be creative in real materials. You know, it doesn’t really matter to me what they gave it for I suppose. It’s kind of a great honour, and it is great for our field, actually.
What I think has happened is that, when I look back and see what, the way I was taught about physics, in the 70ies, and the way that condensed matter was feud by people. I think, my advisor Phil Anderson, were one of the few people who took a very different view point. But the way we think about it has just changed. The things that went in the textbooks, that people thought were important, they’re kind of just rather boring details. And all the new stuff is absolutely absent. So, it’s been a, you know, great honour I think to be involved in laying the groundwork for this complete change of the way we look at things. So, I suppose that’s probably why this prize has happened. And partly because of what happened about ten years ago; some of this work was generalised to so called “time-reversal-invariant topological insulators”. It suddenly turned out that there were real topological materials that had been sitting on people’s shelves for many years without being noticed. And, just the power of theoretical ideas to, reveal something, very worthy, big calculators and experimentalism that’s just not noticed, it’s amazing, right?
So, it’s a question of really, I think we need the imagination to see what quantum mechanics can do, and we need to understand all about quantum mechanics, it’s laws, but we don’t really understand this potential. In the past things were studied by basically hitting them with a hammer, but now we have arrived at where we are actually able to try and tweak or nudging things around, and understand how quantum mechanics influences what happens. And I think my basic line on this, is that quantum mechanics does stuff much cooler than we could’ve even imagined, and part of this emerges, part of this work. And much more, cooler, things have been emerging since then. And people have dreams of quantum computers and all kinds of quantum information technologies and things. And I am not sure what’s going to happen, but something’s going to happen, because so many people now are looking at this, and it has really become … I mean, once you’ve got real things, then people start taking it seriously. And it has become incredibly inspirational and exciting to lots of young people. And everywhere one sees, an experimentalist gives a talk and he views a little video of a coffee cup turning into a doughnut and backwards, a physicist, and you can find this on YouTube and …
The idea that topology has something to do with this material – it just, makes such a big impact on people. It’s just incredible how this thing has taken off. So, these ideas were kind of a sleeper for a long time, and it’s really the final thing where you’ve taken – it’s got three mathematical backgrounds to it, which, you know, I don’t really understand. But, I understand some of it, but just knowing the abstract mathematics doesn’t lead to something either. The mathematics was actually around for a very long time, since the 40ies, and it was only when some mathematician realised, the mathematical physicist realised, that, the formulas for example that David Thouless, and Kohmoto, Nightingale and den Nijs had found, were actually, they found that this topological, this number that didn’t change, and it was recognised, oh this is Chern’s integral of a curvature of a manifold, right?
Once you kind of name the mathematics, then of course there’s lots of tools around. But to actually find it, you’ll have actually be able to do, a concrete little, what they call a toy model calculation. So, the mathematics is often too abstract to actually pin down. But in what I have done, and what Thouless did, is to be able to do very simple calculations, that you can actually do, preverbally on the back on an envelope. Perhaps you’ll need a computer a little bit to do something simple. But to actually see what a model does you need to strip all the irrelevant details out of something and go down to the simplest, possible model, that contains the physics that you’re interested in, right. And I take this line in seminars that it is almost like contract law. Because if you have a contract, very often it’s got a little phrase in it that says that “anything that’s not written in this contract cannot be considered to be part of it” and the contract is the entire document, right? Once you’ve got your model down to be very simple, and it still exhibits the physics you are after, then it’s not omissible to discuss, to attempt to use anything if you have already thrown away as part of the explanation of the thing. So, this is a technique to getting down to the cleanest, simplest example of things. Then, if you can actually do a calculation, then you can be concrete. And then maybe you can see how it works with the deep and beautiful mathematics that gets exhibited in the actual solution.
But the final thing you need, on top of that, is once you have actually shown it can be made in a model, in a toy model, it’s remarkable that the material science has got to the point where, if you think it can be made, someone’s going to make it. Before the toy models were attempts to make, you know, text-book thing for modelling complicated real matter, and the large school of ab initio calculation, to get everything right, all the details right. And of course, the people who worked on that, often had a poor opinion of toy models. But it turns out they missed all these fundamentally, simple and beautiful things. So, toy models are great, but once someone actually makes it, then you got the three ingredients; you got the mathematics, calculation that a kind of ‘simple working physicist’ can understand, and then, an actual material. Once that started to happen about in the last decade, once these three things came together, this field has just taken off, and who knows where it’s going to go.
It’s really obvious, talking to you, that you find this really engaging, and that you still find this immensely interesting. How have you been able to keep your enthusiasm up within this field during your career?
Duncan Haldane: I think we’ve just kept on, things just got cooler. I think in this whole field, I mean, it’s been so fruitful. It’s been actually great for young people at every stage in their career in this field. Other fields, like high-temperature superconductivity has really been the graveyard of many people, because it has been unclear, it’s complicated, and there’s lots of rival theories which fights this and that, and there’s people knifing each other… But in this field, every time a problem came up it got solved and something very interesting and new came up, and people said “fine”. But it’s renewed, it has kept on; new things have kept coming on, and in fact, it’s just been a sequence of progressively more and more cool things coming out. So definitely it keeps one’s interest up. And of course, you try and understand the big picture; the big picture is kind of gradually being assembled. So, it is still an exciting field and it is still going to continue to be an exciting field.
Thank you so much
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