I am sitting alone at the head of a large conference table when an oddly familiar voice greets me: “Hey, you must be Spiros!” I turn around to find Paul Rudd, the Hollywood actor, wearing his famed disarming smile. He is in sweats, on his way back from some type of superhero training.
A few minutes later he and a bunch of other film people are sitting around me. Rudd cuts straight to the chase: “So what kinds of cool things happen when you shrink?” I have been flown in to consult on the physics of Marvel Studios' superhero flick Ant-Man, and now I must deliver. Yet all I really know about shrinking to ant size comes from watching Honey, I Shrunk the Kids! as a nine-year-old. For a moment, I consider telling him that he's got the wrong guy, but there is no way I am going to let this opportunity slip between my fingers. I may not know much about ants, but I know a thing or two about quantum physics. “The concepts of time and space lose their usual meaning when you shrink to the quantum scale,” I reply with confidence. Reading the room, I can tell that this is the last thing they expected to hear. But they are hooked. The floor is mine for the next two hours, as I delve deeper and deeper into the rules and weirdness of quantum mechanics.
A day later one of the producers e-mails me: “Hey, what should we call the place you enter when you shrink to microscopic size?” I type back: “How about the Quantum Realm?” Five years later, in 2019, Marvel's Avengers enter the Quantum Realm and travel back in time to save the universe. All of a sudden, being an expert in quantum physics seems pretty cool.
I was not always into physics or comic-book heroes. In college, I majored in mathematics and computer science, spending my summers trying to predict how one-dimensional DNA sequences folded into three-dimensional proteins. It was not until graduate school that I took my first physics class beyond the basic college requirements. My Ph.D. adviser at the University of California, Davis, had decided to enroll me in graduate-level quantum mechanics, and I had no choice but to go along with it. When on the first day of class we were handed a one-page undergraduate-level assessment test, I returned mine with my name and a smiley face next to it. Still, I persisted, graduating in June 2008 with a doctorate in applied mathematics and an emphasis on mathematical physics and quantum information theory. Three months later I would pack my things and move to Los Alamos, N.M., the birthplace of the atomic bomb, to take a postdoctoral position at Los Alamos National Laboratory. I did not know it at the time, but during the next year I would delve deep within the quantum realm. This is the story of what I discovered there and how I made it back to tell Marvel the story.
Something Interesting
It all began with a simple question.
My adviser at Los Alamos, Matthew Hastings, a rising star and one of the sharpest minds in physics, was sitting across from me at a sushi restaurant when he popped the fateful question: “For your postdoc here at the lab, do you want to start with a warm-up, or do you want to work on something interesting?” Without asking for further clarification, I answered, “I want to work on something interesting.” He seemed pleased with my answer. Later that day he sent me a link to a list of 13 unsolved problems in physics maintained by Michael Aizenman, a professor at Princeton University and a towering figure in mathematical physics. I was to work on the second problem on that list, a question posed by mathematical physicists Joseph Avron and Ruedi Seiler: “Why is the Hall conductance quantized?”
You may wonder what the Hall conductance is or what it means for it to be quantized. I had the same questions back then. No problem on the list besides the third—cryptically titled “Exponents and Dimensions”—had “SOLVED!” next to it. Clicking through, I saw that it was actually only partially solved. Yet one of those partial breakthroughs had led to a Fields Medal, one of the highest honors in mathematics, in 2006, and the other would earn one four years later. In this company, it was clear the problem I was tasked with solving was no ordinary quandary. I considered carefully if I could solve such a question within a year. The reason for the time limit is that a postdoc in math or physics usually lasts two years. At the end of your first year, if you have done great research, you may apply to top universities for a tenure-track professorship. If your research is good but not great, you may apply for a second postdoc or look for a less competitive tenure-track position. If you have nothing to show after your first year, there is always Wall Street.
Still, the idea of backing out now, without even trying to attack the problem, was difficult to swallow. For a person growing up in Spata, a small town outside of Athens, Greece, big dreams were unusual. My dad grew up in the same house I did. He played soccer and got into fights. When he eventually dropped out of high school, his dad offered him a position at the local grocery store. My father refused. Despite being a dropout, he had ambition. He interned at the local real-estate agency and learned the ropes of buying and selling land. Later, he went back to school to get his GED at my mother's insistence. Down the line, when my older brother, Nikos, brought home his first-grade report card, my father cried with happiness when he realized that his son was a good student. Nikos and I would go on to compete at the International Mathematical Olympiad, an honor afforded to six high school students from each country every year. Then, one after the other, Nikos, I and my younger brother, Marios, traded high school in Athens for college at the Massachusetts Institute of Technology in Cambridge—a rare accomplishment for any family, let alone one of modest means, and a testament to my parents. I thought that if they could perform miracles, maybe I could, too. So, in the fall of 2008, I began working on problem number two, aiming, as the list put it, to “formulate the theory of the integer quantum Hall effect, which explains the quantization of the Hall conductance, so that it applies also for interacting electrons in the thermodynamic limit.”
The integer quantum Hall effect has a long history. The original Hall effect was discovered in 1879 by Edwin H. Hall, a student at Johns Hopkins University. Young Hall had decided to challenge a claim made by the father of electromagnetism, James Clerk Maxwell. In his 1873 Treatise on Electricity and Magnetism, Maxwell confidently declared that, in the presence of a magnetic field, a conducting material with current flowing through it will bend because of the magnetic force on the material, not on the current. Maxwell concluded that “when a constant magnetic force is made to act on the system ... the distribution of the current will be found to be the same as if no magnetic force were in action.” To test the idea, Hall ran current across a thin leaf of gold placed in a magnetic field perpendicular to its surface and noticed that his galvanometer (an instrument used to detect small currents) registered a current, which implied a voltage (electric potential) in a direction perpendicular to that of the current's original path. He concluded that the magnetic field was pushing the electrons in the current toward one edge of the conductor, permanently changing their distribution on the surface of the material. Maxwell was wrong. This unexpected charge buildup along the conductor's edges became known as the Hall voltage.
The quantum Hall effect was first observed nearly a century later, on February 5, 1980, in Grenoble, France, by German experimental physicist Klaus von Klitzing. His aim was to study the Hall effect more carefully under ultralow temperatures and high magnetic fields. He was looking for small deviations from the expected effect in certain two-dimensional semiconductors, the materials underlying all modern transistors. In particular, he was trying to measure the Hall resistance, a quantity proportional to the Hall voltage. What he observed was astonishing: the Hall resistance was quantized! Let me explain. As the strength of the magnetic field increased, the resistance between the edges of the material would stay exactly the same, until the field got high enough. Then, the resistance would jump to a new value instead of climbing up steadily the way Hall had originally observed—and all known physics at the time predicted. Even more surprisingly, the values of the Hall conductance, the inverse of the Hall resistance, were precise integer multiples of a quantity intimately related to the fine-structure constant, a fundamental constant of nature that describes the strength of the electromagnetic interaction between elementary charged particles. The integer quantum Hall effect was born.
Von Klitzing's discovery was remarkable, not least of all because the fine-structure constant was supposed to describe aspects of the quantum realm that were too fine-grained for any macroscopic phenomenon, such as the Hall conductance, to be able to probe, let alone define with incredible precision. Yet not only did the Hall conductance capture an essential aspect of the microscopic world of quantum physics, it did so with impossible ease. The integer plateaus of the Hall resistance appeared irrespective of variations in the size, the purity or even the particular type of semiconducting material used in the experiment. It was as if a symphony of a trillion trillion electrons maintained their collective quantum tune across vast atomic distances without the need for a master conductor and, even more astonishingly, were impervious to the principles of physics that, for billions of years, had guarded the quantum realm from macroscopic interlopers.
A door to the quantum realm was opened that day—a macroscopic door that many thought did not exist. In 1985, five years after the discovery, von Klitzing was awarded the Nobel Prize in Physics. His finding would lead to further breakthroughs, with three more Nobel Prizes awarded to two experimentalists (Horst Störmer and Daniel Tsui) and a theorist (Robert Laughlin) in 1998, for discovering that electrons acting together in strong magnetic fields can form new types of “particles,” with charges that are mere fractions of electron charges, a phenomenon now known as the fractional quantum Hall effect.
Laughlin's Quantum Pump
Laughlin was one of the first physicists to attempt an explanation of the quantum Hall effect. In 1981 he came up with a brilliant thought experiment—an idealized simulation of the original experiment that provided a mathematical metaphor to understand it. Laughlin imagined electrons traveling along a conducting loop with a flat edge, like a wedding band. A magnetic field ran perpendicular to the surface of the band, but Laughlin added a fictitious magnetic field line—called a magnetic flux—threading through the middle of the loop like a finger through the ring. Increasing the fictional flux induced a current running around the loop, thus introducing the longitudinal current present in the classical Hall effect. The process, named Laughlin's quantum pump, would complete one cycle every time the fictional magnetic flux increased by one “flux quantum”—an amount defined as h/e, where h is Planck's constant and e is the electron's charge.
After each cycle, the quantum system would return to its original state as the result of a phenomenon known as gauge invariance. Laughlin argued that this reset implied that the Hall conductance was quantized in whole numbers equal to the number of electrons moved by the quantum pump. Great! Alas, there was an issue. The Hall conductance was experimentally measured (and averaged) over many cycles of the pump. Because Laughlin assumed (correctly) that the system was described by quantum mechanics, there was no guarantee that each cycle would transfer the same number of electrons. As Avron and Seiler would write later with their collaborator Daniel Osadchy: “Only in classical mechanics does an exact reproduction of a prior state guarantee reproduction of the prior measured result. In quantum mechanics, reproducing the state of the system does not necessarily reproduce the measurement outcome. So one cannot conclude from gauge invariance alone that the same number of electrons is transferred in every cycle of the pump.” Physicists needed a new set of ideas to show that the average number of electrons transferred over several cycles was also an integer.
Inspired by Laughlin's argument, the next attempts at explaining the quantization of the Hall conductance relied heavily on the concept of adiabatic evolution. Adiabatic evolution is a physical process that aims to capture the evolution of a system that remains in its lowest-energy state at all times while some external parameter varies. When the system's spectral gap—the energy required for it to jump to an excited state—becomes small, adiabatic evolution slows down to prevent the system from crossing over to an excited state. Laughlin's original argument used this notion to mathematically model the quantum Hall effect as the adiabatic evolution of the electronic state of a quantum Hall system under the increase of a fictitious magnetic flux.
Unbreakable Play-Doh
To study the quantum hall effect more deeply, physicists turned to a branch of mathematics called topology. Topology is a way of thinking about the fundamental essence of shapes—the properties that do not change even as they are continuously deformed. Think of a kind of Play-Doh that is unbreakable and impossible to glue onto itself. You can turn a cube of this substance into a ball by rounding out its sharp edges and corners, but you cannot turn it into a doughnut. The latter transformation would require either poking a hole through the cube or stretching and gluing it onto itself. In that sense, cubes and doughnuts are topologically distinct shapes, but cubes and balls are topologically the same (although they are all geometrically different). Topology was formalized in 1895 but had rarely interacted with physics until the 1950s and 1960s.
The initial efforts to understand the role of topology in the quantum Hall effect were considered so significant, in fact, that in 2016 theoretical physicists David Thouless and F. Duncan M. Haldane won a Nobel Prize for this work. Thouless and his collaborators, in particular, extended Laughlin's argument by showing that the Hall conductance was quantized on average. Because one fictitious flux was not enough to prove quantization, they proposed a second fictitious flux. In the new thought experiment, one flux induced the electric current across a semiconductor, and the other detected changes in the current between pump cycles. This scenario simulated cycles of Laughlin's pump under distinct initial conditions. The adiabatic evolution generated by the extra fictitious flux played the role of averaging over many cycles of Laughlin's pump and showed that the average Hall conductance was quantized.
At around the same time, Barry Simon, a mathematical physicist at the California Institute of Technology, noticed that adiabatic evolution formed a mathematical bridge between the Hall conductance and the local curvature of the two-dimensional phase space generated by the two fictitious magnetic fluxes. This local curvature is called Berry curvature after its discoverer, mathematical physicist Michael Berry. In particular, Simon showed that the Hall conductance was equal to h/2π times the local curvature at the origin of that phase space. This was a big deal. A famous mathematical result from 1848—the Gauss-Bonnet theorem—declared that the total curvature of a geometric shape was a topological feature, not a geometric one. In other words, the sum of all the local curvatures of a three-dimensional shape is the same for all topologically equivalent shapes with the same surface area. Even more exciting, the total curvature is simply given by 2π(2 − 2g), where g is the number of holes in the shape.
Most important for us, a modern generalization of Gauss-Bonnet by geometer Shiing-shen Chern showed that the same result applied for the total Berry curvature of our two-dimensional phase space describing the quantum Hall effect. The Berry curvature of that space was now given by 2πC, with C denoting an integer known as the first Chern number. To show that the Hall conductance was quantized, Simon and his collaborators looked at the average of the conductance over the whole phase space, which is given by h/2π times (total curvature) divided by (surface area). Plugging in 2πC for the total curvature and (h/e)2 for the surface area yielded C × e2/h. Et voilà. The average Hall conductance was an integer multiple of e2/h, as Thouless had shown. But for the first time ever, the integer in front of e2/h was identified with a “topological invariant”—a property that does not change if you rotate or deform a shape—and therefore the result was impervious to small perturbations and imperfections in the physical setup of the quantum Hall effect. This was a breakthrough insight.
Unfortunately, the beauty of the preceding arguments by Thouless and Simon was marred by a serious issue: the Hall conductance that experimentalists measured corresponded to the local curvature at the origin of the two-dimensional phase space, not the average curvature over the whole space. To see why the local curvature of an arbitrary shape is almost never equal to its average curvature, consider a torus. Gauss-Bonnet implies that the average curvature of a torus, and of any shape with a single hole in it, is zero. But the local curvature of a torus is obviously nonzero along most points on the surface and can take both positive and negative values. Thouless and his collaborators actually tried to address this issue, yet the question remained: Why was the Hall conductance quantized, if one was not allowed to average over all possible initial conditions of Laughlin's pump? Indeed, that was the question I had to answer.
A Sense of Despair
My first steps into the mystery of the quantum Hall effect were supposed to be illuminated by a book written by Thouless himself: Topological Quantum Numbers in Nonrelativistic Physics. A couple of weeks after receiving the book from Matt, I determined that I did not have the background required to understand any of the physics within. I locked the book inside my desk drawer and put the key away. Yet the book's simple existence gave me a sense of despair. How could I make any progress in solving the problem if I could not understand the contents of that book? Back then, I was a blank slate.
Of course, I had the option of going to Matt for help. He could teach me what I needed to know. Heck, we could even work closely on the problem together. But about a month or two after I arrived at Los Alamos, Matt told me he was leaving the lab. With job interviews now taking up most of his time, I barely saw him. A few months later, when he was offered a position at Microsoft's Station Q in Santa Barbara, Calif., my interactions with him all but ended. The few times we did meet, I became convinced that Matt had made a serious mistake in giving me a postdoc at Los Alamos. He would speak, and all I could retain were a few word combinations here and there. One of the phrases he repeated was “quasi-adiabatic continuation,” a notion I was unfamiliar with. To my further dismay, this term did not seem to appear anywhere in the immense literature devoted to the quantum Hall effect up to that point.
Without much else to go on, I did what every young scientist of my generation would do and googled “quantum Hall effect” and “quasi-adiabatic continuation” (QAC). The first phrase returned hundreds of research papers, but I had as much luck reading through any of them as with the book by Thouless. The one thing I did get out of that search, however, was a word that kept coming up in relation to the quantum Hall effect: topological. When I added that word to my search, the first thing that popped up was an article by Avron, Osadchy and Seiler entitled “A Topological Look at the Quantum Hall Effect.” The piece, which appeared in Physics Today in August 2003, was meant for nonexpert physicists. This article was so clearly written that it formed the foundation on which I would build my understanding of the quantum Hall effect.
In contrast to the hundreds of articles on the quantum Hall effect, my search on quasi-adiabatic continuation returned just two results, both by Matt. The first paper, co-authored with theoretical physicist Xiao-Gang Wen, was an introduction to QAC. The second paper contained, among other applications, a brief section on using QAC to compute a version of the Berry curvature relevant to the fractional quantum Hall effect. This was the first and only published attempt to apply QAC to any type of Berry curvature. I was excited to study Matt's argument inside and out. But I still needed to understand what QAC was about and how it was connected to adiabatic evolution. So I delved into the first paper, and after a month of poring over it, I felt that I had a good grasp of the technique. QAC was proposed as an evolution of a quantum system designed to preserve certain topological properties of its quantum state. In contrast, adiabatic evolution was better suited for local, geometric properties, such as the Berry curvature mentioned earlier.
The next task was to figure out how to compute the Berry curvature using QAC. To my dismay, I could not parse Matt's brief argument on how the two concepts could be bridged. I decided to re-create that bridge (or my version of it, at least) from scratch. The idea was to follow Simon's argument connecting adiabatic evolution to Berry curvature, while sneaking in QAC in place of adiabatic evolution. Substituting one evolution for the other worked out beautifully for one simple reason: I could show that QAC was exactly the same as adiabatic evolution under the following special condition: throughout the evolution of the system, the gap in energy between the ground state and the first excited state had to remain above a fixed positive value, independent of the size of the system. As luck would have it, this special condition was satisfied precisely near the origin of the 2-D phase space. In fact, if that condition was violated, I could show that the Hall conductance was not quantized.
After going through the exercise of connecting QAC to the Berry curvature and, hence, to the Hall conductance, I turned my sights toward the next big hurdle: re-creating Simon's argument, which computed the averaged Hall conductance as an unchanging topological quantity that yields the first Chern number. This was no small feat. As I have mentioned, to get over the initial problem of simulating adiabatic evolution with QAC, I took advantage of the fact that QAC tracked adiabatic evolution exactly, as long as there was a big enough spectral gap between the ground and excited states of the system. Unfortunately, this assumption about the spectral gap went out the window the moment I started exploring deeper into the 2-D phase space, whose total curvature I needed to compute. In fact, this assumption was so powerful that all attempts to quantize Hall conductance up to that point had used it. In other words, nobody thought it was possible to prove quantization without making that extra assumption. And neither did I. When I finally reached out to Matt in late spring of 2009 with a solution that made use of that key assumption, he said to me: “Nice job. But I think you should be able to prove quantization without it.” Matt pointed me toward a seemingly unrelated paper of his entitled “Lieb-Schultz-Mattis in Higher Dimensions” (LSM), where he had laid the foundations for removing this assumption.
As I began to read through LSM, I had the same sinking feeling as when I had tried to parse Matt's attempt at connecting QAC to the Berry curvature. Deciphering it in isolation was going to be a long and arduous journey. But in a second twist of fate, my Ph.D. adviser, Bruno Nachtergaele, working with one of his postdocs at the time, Robert Sims, had published what some considered a mathematically rigorous version of Matt's LSM paper. Although most of the brilliant insights were already in Matt's original paper, Bruno's version was so well written and thorough that within a month I had a clear view of how to proceed. I now knew how to adapt elements of the LSM argument to overcome the second hurdle: to show that the averaged Hall conductance computed using QAC, instead of adiabatic evolution, was still an integer multiple of e2/h.
The original Laughlin's pump argument, which used adiabatic evolution and gauge invariance to deduce a return to the original state of the system after one cycle, did not work with QAC. The main problem was that under QAC, after a flux quantum was inserted, there was no longer any guarantee that the system would end up in the same quantum state at the end of a cycle. Adiabatic evolution accomplished such a feat by forbidding the lowest-energy state of the system from ever getting excited. QAC, on the other hand, had a mind of its own. If the spectral gap ever dropped below a critical value as scientists inserted more and more magnetic flux, QAC would happily allow the system to jump to a new, excited quantum state, leaving behind its low-energy past. Unfortunately for me, that meant that at the end of a Laughlin cycle, even though the dynamics describing the system returned to their original state, the quantum state of the system itself may have changed significantly. If that were the case, then a key element of Laughlin's and Thouless's arguments would go up in smoke.
To overcome this obstacle, I needed to introduce two more fictitious magnetic fluxes in addition to the original two (for a grand total of four), which allowed me to transform the evolution under QAC into one that guaranteed a safe return to the original ground state at the end of a cycle. This trick, borrowed from Matt's LSM paper, forced the state of the system to maintain the exact same energy throughout the modified evolution around the boundary of the 2-D phase space, even when that energy no longer corresponded to the lowest possible energy of the system. In other words, to guarantee the return of the system to its initial state, all one ever needed to know was that the two states had the same energy. The fact that the ground state of the system was uniquely specified by that energy took care of the rest. Adiabatic evolution's insistence on keeping the system in its lowest-energy state throughout the evolution was overkill. More important, as I came to realize later, the insistence on using adiabatic evolution to quantize the Hall conductance was also the reason progress had stalled for nearly two decades.
By now I felt exhausted. But the main hurdle was finally in view. Everything I had accomplished up to this point was a fancy way of showing what Thouless, Simon and their collaborators had already proved: that the averaged Hall conductance was indeed quantized in integer multiples of e2/h. It would seem that I had made no progress in removing the averaging assumption plaguing every effort to explain the mystery of the integer quantum Hall effect. Except for one minor detail: the two-dimensional phase space generated by QAC had near-perfect uniform Berry curvature. In other words, the real Hall conductance, the one corresponding to the Berry curvature of a tiny patch near the origin of the 2-D phase space, was equal to the average curvature over the total flux space. Because the latter was famously quantized, it followed that the actual Hall conductance was also quantized. Quod erat demonstrandum—QED.
This final theoretical hurdle took many months of restless days and sleepless nights to cross over. I nearly gave up several times before reaching my goal. During a particularly dark time, I told my mom that I was not sure I wanted to wake up the next morning. In typical Greek fashion, she responded, “If you do anything stupid, I will fly out there and strangle you with my own two hands.” Lost in a world of hyperanalytical thinking, I needed such an absurd statement to snap me out of it. I finished the proof in November 2009, shared it with Matt, who quickly added a section on how the result could be extended to also explain the fractional quantum Hall effect, and then posted it online. It would take us five more years before getting the result published and another four years before the mathematical physics community could fully digest it. On February 25, 2018, I opened an e-mail from Michael Aizenman—a letter I had waited for eight years to receive. It read:
Dear Matt and Spiros,
The Open Problems in Mathematical Physics web page was now updated with the statement that the IQHE question, which was posted by Yosi Avron and Ruedi Seiler, was solved in your joint work.
I thank you here for your contribution, and also congratulate you on it. It is a pleasure to note that in each of the two problems on which progress is reported there, the advance came through deep novel insights and new tools. The list of solvers is a veritable honor roll.
The fundamental mystery we started with was the question of why a microscopic, quantum phenomenon was showing up on a macroscopic scale. Instead what we found was that one of the most fundamental constants of nature was the reflection of global order beyond our finite grasp—the infinite communing with the infinitesimal. And although we have focused on the theory behind the quantum Hall effect, the experimental efforts it has inspired over the past three decades have been equally, if not more, exciting. Research on topological phases of matter beyond two-dimensional quantum Hall systems is paving the way toward technologies such as large-scale, fault-tolerant quantum computing. Impressive results coming out of labs such as Ana Maria Rey's at the University of Colorado Boulder are even tackling fundamental questions about the very nature of time.
This experience also taught me a valuable lesson: my self-worth is not tied to my success in life. The fateful call with my mom took place three months before I put the finishing touches on the solution. I did not turn into a mathematical genius within the span of a few months. But I made progress by breaking the problem down into simple parts I could understand. To do that, I needed to be okay with feeling incompetent most of the time. Without the faith of my parents in me as a person, whether I was good enough to solve the problem or not, I would have given up right before the finish line. Had I done that, the problem may still be unsolved and Marvel's Avengers would have had to find an even more scientifically implausible way to save the universe than to jump into the quantum realm via a macroscopic portal.
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