Questions About the Book... Why Was the Discovery of the Hofstadter Butterfly So Important?

Questions about the book...

Why was the discovery of the Hofstadter butterﬂy so important?

From 1930 onwards, many physicists, including some of the pioneers of quantum theory, struggled to solve what may sound like a simple problem: What happens if you immerse a crystal in a magnetic field? What is the spectrum of electron energies, plotted as a function of the magnetic field strength? In 1974, doctoral student Douglas Hofstadter discovered that the answer was a graph resembling a butterfly, consisting of copies of itself nested down infinitely many times. The graph’s infinitely nested structure was a great surprise at the time. There was no precedent in physics for such a shape. As we see in a historic letter (August 11, 1975) from Gregory Wannier (Douglas Hofstadter’s Ph.D. advisor) to Lars Onsager — Nobel Laureate — Wannier candidly admits, “It looks more complicated than I ever imagined it to be.” (The letter is included in the book.) The stunning form of the butterfly graph, when it first appeared in print in 1976, fascinated physicists and mathematicians, as it revealed new aspects of the beauty, simplicity, and harmony of the solid state of matter in the universe. It showed that simple systems can exhibit amazingly complex and subtle behavior. However, underlying this complexity, there is also something mysterious and beautiful. Now the butterfly graph, though visually intriguing, could easily have been forgotten, had it not been for the laboratory realization, in 1985, of systems manifesting the quantum Hall effect — an exotic quantum phenomenon revealing the astonishingly precise quantization of Hall conductivity in materials, independent of their degree of purity. The butterfly graph turned out to be a quantum fractal whose wings encode the integers that are the quantum numbers of Hall conductivity. Some people believe that laboratory observations of the fractal nature of the butterfly could well pave the way for new kinds of materials with exotic properties whose potential is yet to be imagined. However, as is the case for most major discoveries in physics, scientists are still trying to comprehend what role it may play in shaping our scientific world. Cory Dean, an experimental physicist at the City College of New York and one of the first people to catch glimpses of the butterfly in the laboratory, put it this way: “This is a very good example of a fundamental discovery that opens doors that we do not even know about yet. Why go to a distant planet? We

1 go there to discover what’s out there. We do not yet know what this new world will result in and what will emerge out of this...”

Why do you think the Hofstadter butterﬂy has endured, and how is it applicable to physics research today?

Ever since the theoretical butterfly graph was first displayed in the pages of Physical Review some 40 years ago, the goal of actually spotting this astonishing beast, or traces of it, in a physics laboratory has been a holy grail for certain scientific adventurers, who have come up with many creative ideas and thoughts in order to track it down. Many dedicated scientists are still pursuing this goal today. Recently, ingenious new experimental techniques have allowed the fractal behavior of real physical systems to begin to emerge, and these exciting developments have brought new energy into this field. But there is one thing that is absolutely clear in my mind... the Hofstadter butterfly is a generic phenomenon that goes far beyond the specific physical conditions in which it was first conceived. Although it made its first appearance in two-dimensional crystals, where electrons move very slowly compared to speed of light, it has now also been seen in the colorful high-energy world of quarks and antiquarks, described by the laws of quantum chromodynamics (QCD). Here we are talking about the four-dimensional world of space and time... And of course, as one can see in various images in the “Butterfly Gallery” section of the book, the same basic theme as was first found in a square lattice shows up again and again in all sorts of highly diverse crystalline lattices, including not just hexagonal and triangular ones, but also the Kagome lattice and even Penrose’s famous aperiodic tiling of the plane. Each time it arises, it reveals a different kind of beauty, but deep down, all these fractal butterflies are one and the same thing. So you see, the Hofstadter butterfly is a robust, generic phenomenon that has great richness and beauty, and it still holds many mysteries for future explorers. One key question about the butterfly graph is what aspects of it can be realized in a laboratory. At this point, this is not totally clear, and that fact acts as a strong motivator for creative new experimental physics ideas. Experimentalists also hope that researching the butterfly will lead them to the discovery of materials with novel exotic properties that are beyond our present imagination. And there are numerous theoretical questions about the interplay between topology and fractality, and there are universal aspects of the butterfly graph that will certainly occupy

2 theoretical physicists for a long time. Last but not least, as is mentioned at the end of Chapter 10, when we think of the butterﬂy and its relation to the quantum Hall effect, it is important to remember how the underlying theme of that effect has cropped up in a wide variety of seemingly unrelated problems in physics. This suggests that there exists a much broader perspective from which to view the butterﬂy fractal, home of the quantum Hall effect.

Other than the Hofstadter butterﬂy, what has been the most exciting development in physics during the course of your career?

The butterﬂy was discovered before I started my graduate studies in physics. I came across it in the mid-1980s, when I was fascinated by fractals and quasicrystals. Firstly, let me mention a few important discoveries in physics that are directly relevant to the Hofstadter butterﬂy.

(1) the integer quantum Hall effect, discovered in 1980 (Nobel Prize in 1985); (2) the fractional quantum Hall effect, discovered in 1982 (Nobel Prize in 1998); (3) the Berry phase, discovered in 1983, as is described in Chapters 8 and 9; (4) quasicrystals, discovered in 1982, as discussed in Chapter 4 (Nobel Prize in 2011); (5) topological insulators, discovered in 2005, like quantum spin Hall systems; (6) graphene, discovered in 2004 (Nobel Prize in 2010), as discussed in Chapter 12.

And then there is the phenomenon of high-temperature superconductivity, which was only discovered in 1986 (Nobel Prize in 1987). There are also some important very recent discoveries, such as the discovery of the Higgs boson and of gravitational waves. Both of these phenomena were theoretically conjectured long ago, and now there is very strong experimental evidence that they are real. However, let me say that although these are some of the important discoveries in recent years, none of them comes close to the great revolutionary discoveries in physics, such as Maxwell’s equations, Einstein’s theories of relativity (both special and general), superconductivity, and quantum mechanics. In my view, there have been no truly revolutionary discoveries in physics since the quantum revolution of the mid-1920s.

3 Though it was important, the discovery of the butterfly was a revelation rather than a revolution. However, the discovery of the butterfly was not just about science; it was also about aesthetics in science. Just look at the color image of the butterfly in the book. It is beautiful; indeed, it is a feast for the eyes... But there is much more than that. If you know a little number theory and if you represent the horizontal edges and the center of any butterfly in this landscape using a pictorial representation of rational numbers by circles that the American mathematician Lester Ford discovered in the 1930s, you will find a configuration of four mutually tangent or “kissing” circles... This geometrical problem had already been studied by the Greek mathematician Apollonius in 300 BC, and much later (17th century A.D.) by the French philosopher Rene´ Descartes, and even later, by chemist Frederick Soddy (20th century). When you see these marvelous circle patterns, you realize that the butterfly fractal is in some sense a kaleidoscope. Suddenly you see mathematical beauty that you didn’t see before and could never have anticipated. Sheer poetry begins to flow out of the butterfly... If you step back into the world of physics, you also learn that the butterfly’s wings are labeled by integers that describe the quantization of resistance. Dig a little deeper, and you learn that the theory underlying these mysterious integers is not that different from the theory underlying the Foucault pendulum. In my view, this shows that the story of the Hofstadter butterfly is a feast for the eyes, for the mind, and for the soul...

Questions about writing an ebook...

How is writing an ebook different from writing a research paper? Do you take a completely different approach, or are there some similarities?

Whatever one produces in science, be it a paper or a book, there is a feeling of creating some magic. But there are some important differences between a paper and a book. In scientific papers, you are constrained. Let me describe how things that would be extremely hard to do in a research article can be done easily in a book... Firstly, in a book, you are free to narrate your story the way you want it. You can mold your central theme however you personally perceive it. You can step outside the confines of the scientific world and you can connect your story with everything else that has meaning for you, and you can bring in anything whatsoever that resonates within you, marinating your scientific ideas

4 with other issues in life. You begin to see a bigger picture where scientific ideas fit inside the huge world that you live in, and you see how scientific challenges resemble those that you face in real life... For example, I end Chapter 9 with a beautiful image of Ribbon Falls in Grand Canyon National Park and with a quotation from Rabindranath Tagore that says, “Love is an endless mystery, for it has nothing else to explain it.” In my view, this pithy phrase relates so well to the dilemmas that I encountered while explaining some beautiful concepts from topology. In fact, as I put it in the book:

“Approaching the mysteries of physics through geometry, a philosophy in which Albert Einstein deeply believed, has inspired some of the greatest minds in physics today. This approach has yielded both great beauty and profound clarity, although of course following this pathway is not always easy or straightforward. That, for better or worse, is the nature of life.”

The earlier quotation from Tagore came to my rescue at one point during the writing of this book, and it helped me explain to my readers that the challenges we face in science are not so very different from the challenges of real life... Understanding love is no easier than understanding the abstractions that underlie elusive topological concepts. One of my favorite parts of the act of book-writing was to add quotations. I am deeply fascinated by them. In just a few words, a remark made by a great thinker can say so much, can be so powerful, can be so inspiring — and of course a quotation can be profoundly poetic... I should also mention that to my great surprise, while I was writing this book, other types of ordinary activities, such as teaching, became almost like torture. I simply could not believe the relief I felt each time after I came out of my classroom, after having taught for an hour. Those brief periods of separation from my book (or the draft of the book) gave rise to intense pain, which caught me by surprise. I had never experienced anything like that before. I have published more than scientiﬁc 100 papers, and I’ve immensely enjoyed writing them. However, the writing of this book was something else entirely. I loved every minute of it. I discovered that in scientiﬁc writing, if you bring in other things, such as poetry and nature and enchanting visual forms, it enriches and broadens your theme, and it adds to the joy of reading a book. And of course it will help make your book appeal to a broader audience. There may even be some ideas you can explain to young children, and other ideas to slightly older science

5 enthusiasts. Furthermore, even for those readers who ﬁnd no fascination in science but who love art, there is something in the butterﬂy graph and its inspiring history that they can relate to. If one is a writer, one can make all of this play an important role in one’s book, and that is of course something that you cannot possibly do in a research article...

What do you think the beneﬁts of an ebook are for readers and students compared to a traditional print publication?

I personally do not like ebooks. But in our modern era, where we move around loaded with iPhones, iPads and Apple watches, I guess it is helpful to have ebooks...

Questions about yourself...

What motivated you to pursue a career in physics?

I just love abstract things, and believe it or not, that is what drove me to physics — and to quantum physics in particular. My father, who inﬂuenced me a great deal, was a man of deep artistic and literary interests. But I, as a youngster, found artistry also in science and mathematics. I saw beauty and poetry in scientiﬁc equations, and I loved simple mathematical manipulations. Even today, I always spend a great deal of energy in order to cast my results in some nice, simple, and mathematically beautiful form. If the end result looks ugly, I have no choice but to conclude that I must have made a mistake somewhere. And I have rarely been wrong in this viewpoint.... In other words, nature’s secrets can be encoded in simple mathematical formulas, and as a youngster, I saw mathematical beauty in those formulas, and that is what resonated inside me very early on, and it continues to do so even today.

You have had a distinguished and successful career. What advice would you give to a student who wanted to shape her or his future with a career in science?

I would tell people to follow their passion in deciding on the direction of their career. I would also tell them that in pursuing science, there is a pretty decent chance of ﬁnding a good job. Moreover, science opens a new window through which you can see the world and explore

6 the secrets of nature that would otherwise remain invisible. Most people, unfortunately, have no interest in science, and in my view they are missing out on something very important — namely, the hidden power and beauty of nature. Since just one life is given to us, it is important to see all that one can, and for that, one needs science... One more thing I would tell a student: Imagination plays a key role in science. You do not need to know a great deal. If you are curious about nature, just look around you, and you may realize that nature ought to behave in a certain way!

What are your expectations and hopes for your book?

I am not fearful of experts pointing out blunders in the book, although some errors will surely turn up. As poet William Blake once wrote, “To be an Error and to be Cast out is a part of God’s design.” The key question in my mind is: Can this first book of mine, in which the water running out of the faucets may be a bit rusty, nonetheless satisfy some thirsty souls? In other words, what matters the most to me is this: How is the entire book perceived? Does it have aesthetic value? Is there something unique, something captivating, something that may resonate with some readers, something that may deeply engage some readers? Is there something in the book that some readers will find so fascinating that they will turn its pages one after the other, perhaps not even being able to put the book down? To borrow a phrase from Freeman Dyson: “Can it open windows to let scientific experts inside the temple of science to see outside and to let the ordinary citizens from outside see in?” I must say that I am particularly fond of the section of my book that I called “Poetic Math&Science”. Ever since my early childhood, poetry has been an essential part of my life. Even today, it is my savior! I have seen poetry’s tremendous power — it has come to my rescue in some of the most difficult moments of my life, and it continues to help me in times of need. Poetry is a faithful companion that I can always count on. So if the poems in the book can enchant at least a few readers, I will feel that I have accomplished a lot... Those are my greatest hopes and expectations — and in the days ahead, I hope to learn if these things, which have meant so much to me over my entire lifetime, are meaningful to anyone else. And let me add that I even look forward to criticism of the book, provided it is given in a constructive manner.

7 In writing this book, did you experience any moments of anguish?

Yes, there were certainly some moments of anguish, although they were quite minor compared to the overall joyful and adventurous journey. There was, for example, the delicate issue of how to give credit for certain discoveries of great importance in the the butterfly story. Several dilemmas of this sort needed to be settled. One especially hard one was amusingly nicknamed “The Ten Martini Problem”. Figuring out the proper credit for that was very tricky, and so I finally decided to contact the experts and see what they would say, instead of simply giving my own opinion, since I myself just could not decide. Even after many email exchanges, I still think the question of credit for that particular discovery remains somewhat blurry, but one thing for certain is that I have done my very best to tell the story in a well-rounded and fair manner, and I am at peace with how I present the matter in Chapter 1. There were also numerous sensitive issues about which articles to cite. Since this is a book aimed roughly at the level of Physics Today articles, I did not intend to cite every article, nor could I have. But I experienced considerable anguish in deciding which articles and which authors to include. For example, while citing a review paper, should I cite someone who is a colleague or someone else whose review is a better review? I finally settled for the second option. During the course of this writing, I was really surprised to discover how greatly people care about their work being cited — even highly established scientists! The problem is, if you cite one such paper, you can’t help feeling that you should cite all other papers related to it, unless there is a very special reason to single out that work alone... This kind of issue caused me some anguish, but I hope to have handled it reasonably well. Another type of anguish arose because some chapters — especially Chapters 2 and 3 — just never seemed to converge, and in the end I simply had to settle for the best that I could deliver in a reasonable amount of time. Last but not least, Chapters 9 and 10, on topological issues, were very challenging chapters to write, as I was trying to keep in mind the competing goals of telling an accurate story involving many sophisticated concepts while not completely losing some inquisitive young science enthusiasts.

What are your favorite parts of the book?

Whenever I pick up the book (or bring it up on my screen), my gaze automatically falls on

8 the quotations, like those from Rabindranath Tagore, Katherine Hepburn, and even my father. I also love looking at some of the photos. My favorite photo, taken by Leonard Schiff in 1953, shows physicists Felix Bloch and Robert Hofstadter relaxing atop a high peak in California’s Sierra Nevada mountain range. Some other favorites include a picture of Doug Hofstadter and Francisco Claro at the piano, and Doug meeting his friend Robert Boeninger in 1974 on a platform of the Gottingen¨ train station. And then there are photos of the mathematician Georg Cantor and the physicist P. G. Harper — and also one of Michael Berry being knighted by Queen Elizabeth. I ﬁnd it deeply pleasurable to gaze at these visual records of history. I also derive great satisfaction from reading and rereading the poems in the book. Of course, reading the “Divertimento”, which is all about fascinating people, is a real feast for me. I also enjoy rereading certain sections of Chapters 0, 1, and 5, as well as the chapters on topology. And it is delightful to read the Preface, Prologue, and Prelude, and I am happy to reread my statement called “Gratitude”, because once again it is about people.

What was it like working with Douglas Hofstadter, and what are his major contributions to this book?

Working with Doug Hofstadter brought some real magic to the book-writing process. It was so much fun communicating with him — our email exchanges were delicious. Our co-authored poem “Salute” is one of the most cherished and memorable parts of this collaboration for me. Of course we both have great passion for the ideas that we were writing about. Paradoxically enough, our best moments were when we did not agree. Some genuine improvement always came out of those disagreements. I should also mention that I was highly impressed by the extent to which Doug still knows physics, 40 years after his doctorate, and by his undying passion for physics, in spite of his having pursued a career outside of physics. Working with Doug, I sometimes wondered if I could live up to his standards. I did not want to let him down. Also, I wanted to make sure that I kept the excitement of the ideas alive, but luckily, that desire was something that Doug was keenly aware of, and he shared it fully. In fact, he helped me make some of the more elusive ideas even more alive. However, although he was constantly in the trenches working very hard side by side with me, I had no fear of his dominating my book. He was always open, forthcoming, respectful, and accommodating. (I might add the

9 amusing fact that in a few spots, Doug gently toned down words of high praise that I had lavished on his old discovery, as he felt that my words were a bit over the top.) In short, it was a great privilege to work with Doug. I want my readers to know and appreciate the warm partnership between the two of us, which gave rise to the final form of this book. To be sure, it was I who conceived this book and who brought it to life, but after I sent the manuscript out of the blue to Doug (whom I had never met, and who had never heard of me), I was very pleased when he volunteered to do some editing work on it, and I accepted his offer. As a result, Doug and I then nurtured this book together for a full year. Thanks to all of that very hard labor, Doug’s fingerprints are on every paragraph, and there are many brand-new paragraphs that he added. In addition to writing the Prologue and his guest chapter, he also designed an ambigram at my request. These contributions, plus the historical documents that he unearthed in his old files, are real gems, and they have provided a very special glow to the book. Doug Hofstadter worked extremely hard and with great passion to help make this book as good as possible. There is this rare and absolutely beautiful poetic piece in the butterfly story that will stand out as something only Douglas Hofstadter can contribute and that is the “Onegin stanza” that appears at the end of the Prologue . I wonder, whether this indirect and abstract way of linking this butterfly to this Russian masterpiece tells us that deep down, in some abstract and subconscious way, Doug sees butterfly story as some kind of novel in verse, composed by nature. And there is one last by-product that Doug’s intense involvement will hopefully bring to my book. Just imagine the book is on display in a library or bookstore. The mere presence of the name “Hofstadter” on its cover may bring certain potential readers to pick up the book, to flip through its pages, and maybe even to start reading it. That will help to bring the magic and the beauty of quantum science and of this magical fractal object to a wider audience.