Summary of my Doctoral Thesis for Non-Physicists
This is the English version of the popular summary of my dissertation submitted in 2019. This summary is written for readers without a background in science.
Preliminaries
According to the degree regulations of the University of Stuttgart, a thesis in English is to be accompanied by a summary in German. I doubt that there is a single physicist who would profit from a German summary. Therefore, I decided to dedicate it to non-specialists and readers without scientific background. What follows is a translation of this text.
Publications of popular science — be it books, documentaries, or the summary at hand — cut in both ways. On the one hand, they are essential to bond science and society, to introduce people to the scientific method, and to promote basic research in general. But the communication of complex, highly specialized science to a lay readership is, depending on the desired result, complicated to impossible. There is a virtually unbridgeable disparity between the images cast in the mind of the layman and those in the mind of the scientist. The reason is not a lack of intelligence but a lack of experience. And experience — unlike knowledge — cannot be shared. The difference is years of studying physics, years of getting used to the oddities of quantum mechanics and the peculiarities of relativity. I cannot convey to you the pictures in my mind — my insight. So be aware of these limits, because more dangerous than ignorance is pseudo-knowledge. Be aware that the pictures I will draw in your mind are just caricatures of mine. Do not trust analogies beyond their domain of validity. Quantum mechanics is not easy.
What is this thesis about? Let’s begin at the beginning: It contributes to the field of theoretical physics. While experimental physicists study physical phenomena in the laboratory, theoretical physicists construct abstract, mathematical models (with pencil and paper or on the computer) to explain the observations made by experimental physicists. But this order is not mandatory: Theoreticians have a wild imagination and often they propose theoretical models that are realized only afterwards in the laboratory (if at all; it is surprisingly difficult to make nature do what theoreticians cook up). The thesis at hand is of this kind: It describes theoretical models that — at least on paper — have interesting (and useful) properties. With one exception, none of these models has yet been realized in experiments. That doesn’t sound very encouraging, but this is the rule rather than the exception in theoretical physics.
What is a model anyway? In theoretical physics, a “model” is a mathematically precisely defined framework, with abstract objects describing an idealized, physical system. Mathematical concepts such as vectors and functions are used to describe the state of the system. These are then related by equations to predict the time evolution or the response of the system to perturbations. In other words, models allow for the mathematical description of real processes and phenomena. Models are the bread-and-butter tools of theoretical physicists.
Overview
What are the models studied in this thesis? The thesis at hand belongs to the field of condensed matter physics. This discipline is concerned with the properties and the description of materials that are comparatively “cold” and “dense.” The elementary building blocks of these materials are atoms (or ions, i.e., electrically charged atoms) and electrons. Important branches of this field are solid-state physics that describes, e.g., crystalline materials and the semiconductors used for computers, and the physics of fluids that describes their dynamics and turbulences. Condensed matter physics explains why metals conduct electricity, why semiconductors (e.g., silicon) do so only under certain conditions, and why iron can be magnetized. An important aspect in this context are phase transitions. Phase transitions describe abrupt changes of the properties of a material when external parameters (e.g., temperature and pressure) rise above or fall below certain critical values. The best known phase transitions are the freezing of water at 0 °C and its vaporization at 100 °C. Although the building blocks — the water molecules — are the same in all three phases, ice, water and steam are completely different. Explaining and characterizing such differences is a central topic of condensed matter physics.
The last 100 years of physics were dominated by two influential theories: the theory of relativity by Albert Einstein and quantum mechanics (pioneered by Werner Heisenberg, Max Born, Erwin Schrödinger and others). For condensed matter physics, quantum mechanics is of paramount importance because its goal is the description of many, atomically small particles at very low temperatures. The theory of relativity also plays a role in this context, but we shall not dwell on these issues here. Searching for descriptions of phases and their transitions, the physicists of the 20th century realized that the oddities of quantum mechanics fundamentally influence the properties of matter. Phases that occur at very low temperatures and can only be understood in terms of quantum mechanics are called quantum phases. Probably the most impressive example is the superconducting phase of certain metals (e.g., mercury and aluminum). Above a material-specific critical temperature close to absolute zero (-273 °C), these metals conduct electricity as usual (that is, with a small but non-zero resistance that leads to losses). Below the critical temperature, this resistance disappears completely and current can flow without losses (a phenomenon that is put to use in magnetic resonance scanners). This superconducting phase differs fundamentally from the common metallic phase, and the transition between the two is another example of a phase transition. The successful description of resistance-free charge transport in superconductors is one of the great achievements of quantum mechanics and marks a milestone of condensed matter physics.
An important motive of physics is generalization. Physicists learn phenomena on the basis of special cases, and then try to explain them with general principles. Such generalizations often lead to new theories and can shape the mindset of entire generations of physicists (then called paradigms). Both the freezing of water and the transition to the superconducting state are phase transitions. The obvious question to ask is whether there are universal principles that govern all phase transitions. Is there a general ordering principle that describes what distinguishes different phases? Such an ordering principle was proposed and developed by the Soviet physicist Lev Landau (Nobel Prize for Physics 1962) in the 1930s. The basic idea is quite simple: Different phases differ in their symmetries. A symmetry of a physical system is a transformation that does not change its appearance. For instance, a perfect sphere is rotationally symmetric: When rotated, it always looks the same. Landau’s ordering principle states that phases can be characterized by their symmetries. The example of freezing water nicely illustrates the rationale: Liquid water always looks the same under arbitrary rotations. A statement that is no longer true for water that is frozen to crystals. In a generalized form, this idea can be applied to many phases and phase transitions — including superconductors (there, however, with a more abstract symmetry). Landau’s ordering principle became known as “spontaneous symmetry breaking” because phase transitions are characterized by the fact that certain symmetries of one phase are “spontaneously broken” at the transition to another phase (for example, ice crystals break the rotation symmetry of liquid water “spontaneously” as they grow along randomly selected directions). The theory of spontaneous symmetry breaking was so successful that until the 1960s, physicists were convinced that they had basically understood everything there is to know about (quantum)phases and their transitions.
A novel type of quantum phase
But in the early 1970s, there were signs that things could be a bit more complicated. Theoretical models were developed that exhibit phases with exactly the same symmetries while being separated by phase transitions. In 1980, Klaus von Klitzing (Nobel Prize for Physics 1985) observed that a special form of the conductivity of two-dimensional semiconductors only changes in precisely defined steps when a strong magnetic field is applied and varied. The measured step size is practically independent of the material (even for samples with impurities) and is directly related to fundamental constants of nature. This phenomenon is known as the quantum Hall effect and marks a milestone of modern physics. (The quantum Hall effect describes the quantized version of the classical Hall effect, named after Edwin Hall.) It is completely opaque how measurements on a material with natural impurities can yield perfect values of certain constants of nature. At that time, physicists had not yet come across such “robustness” in real systems. To make matters worse, the phases realized in the quantum Hall effect all appeared to have the same symmetries. Thus it was undeniable that Landau’s ordering principle cannot account for all quantum phases.
Soon after these discoveries, theoretical physicists pointed out that a discipline of mathematics rarely used in physics is needed to understand these phenomena: Topology. Topology (as opposed to geometry) deals with deformation-resistant features of objects. Imagine you are given a string and your goal is to use it to remember a number (say 5). One solution would be to mimic the shape of “5” with the string; in this case, the number is encoded in its geometry. This approach is viable but not very robust: The shape of the string is easily destroyed by unintentionally touching it (and then the encoded information is lost). A much more sophisticated method would be to wrap the string five times around your forearm and knot it. As long as the string does not break, the number of turns cannot change; even if it is deformed by moving around. The number of turns is therefore a topological property as it is robust against geometric deformations. It is such a topological winding number that is responsible for the robust conductivities of the quantum Hall effect (only in this case, the role of strings is played by abstract, mathematical objects that are used for the quantum mechanical description of the system). These winding numbers of quantum phases are called topological indices; they can usually not be measured directly, but have measurable effects nonetheless (such as the discrete conductivities). The winding numbers also solve the mystery of seemingly indistinguishable phases: They have simply different winding numbers. Since these numbers are not directly accessible (they are “hidden” in the quantum mechanical structure of a material), these phases seem to be the same, even though they are not. When you try to get from one to the other, a phase transition occurs where seemingly nothing changes; however, there the winding number jumps from one integer to the next.
This concept forms the basis for one of the most active research areas of physics at the time and has radically changed our view on possible quantum phases (and related materials). The theoretical and experimental exploration of these topological quantum phases is not nearly complete. By now, the concept has become so influential in condensed matter physics (and beyond) that the theoretical masterminds of this discipline — David Thouless, Duncan Haldane and Michael Kosterlitz — were awarded the Nobel Prize for Physics in 2016. The transition from Landau’s principle of spontaneous symmetry breaking to the much more diverse concept of topological quantum phases can be compared to the transition from monochrome to color television: The landscape of quantum phases, which until the second half of the 20th century was gray-in-gray, has now turned into a rainbow-colored painting riddled with mysteries. To gauge the scope of this rather novel field of research, let me point out that in 2018 alone, more than 1300 publications with the keyword “topological” in their title are listed online; that is roughly 3 to 4 scientific articles per day. These figures alone should make it clear to the reader that the picture drawn above cannot reflect the true complexity of this topic.
What is my dissertation about?
This doctoral thesis is a contribution to the research field of topological phases (and at least two of the publications it is based on belong to the 1300). Let me briefly comment on the structure of this document: While early doctoral theses in physics often had a monographic character, treating a single problem with a guiding thread throughout the thesis, nowadays, theses that are based on a collection of papers become more and more common. In the course of such a doctorate, several, mostly independent research projects are conducted (often in collaboration with others) and published as independent articles. The doctoral thesis summarizes these previously published results in a single, comprehensive document and integrates them into a common framework. The thesis at hand is of this form: Chapter 1 introduces the concept of topological phases discussed above on a level accessible to students with an undergraduate degree in physics. Each of the following Chapters 2, 3, and 4 builds on these foundations and treats a single project. Calculations that were omitted in the original publications are presented in detail. Finally, in Chapter 5, a few side projects are presented, some of which are precursors or descendants of the three main projects.
Let me now use your new knowledge of topological phases for a brief outline of these projects:
In Chapter 2, I define and study a new model of a topological phase in one spatial dimension. (In solid-state physics, models with less than three dimensions are not uncommon. It is possible to create one-dimensional “wires” of artificial quantum materials in the laboratory.) The building blocks of the investigated model are strongly interacting fermions (think of electrons). Its properties are examined with exact mathematical methods and cross-checked with numerical simulations. The topological properties of this model manifest in a robust “ground state degeneracy”: A physical system described by this model at very low temperatures can be in different states that are hard to distinguish. This indistinguishability can be exploited to manipulate qubits, the “quantum bits” of a hypothetical quantum computer. Hence, the model is not only interesting from an academic point of view, but demonstrates how to harness the robustness of topological phases for the manipulation of quantum information. (An idea that has been around for a while and already spawned a dedicated branch of research concerned with so called topological quantum computers.)
In Chapter 3, I use the model of a well-known topological phase (again in one dimension) and translate it into a completely different context. The original model was introduced to understand the conductivity of a particular polymer (polyacetylene). In this context, it describes the movement of weakly bound electrons along a hydrocarbon molecule. After my “translation,” it describes photons (i.e., particles of light) in artificial networks of resonators. These networks “inherit” the topological properties of the original model. The goal is to construct a robust mechanism for the transmission of quantum information (i.e., a method for the non-destructive transport of qubits between two sites on the chip of a quantum computer). In the studied approach, topology helps to decouple the properties used for transport from possible manufacturing tolerances; similar to the conductivity of the quantum Hall effect that, due to its topological origin, is oblivious to disorder in the system. In summary, this project demonstrates an application of topological phases for the transfer of quantum information.
In Chapter 4, I examine another one-dimensional topological phase of fermions with the goal of constructing a scalable quantum memory. This phase is related to the model of Chapter 2, but easier to describe theoretically. It is also easier to realize it experimentally. Its simplicity makes it one of the few candidates for topological phases that can (hopefully) be implemented in the near future in artificial, scalable structures of semiconductors and superconductors. The topological robustness of this phase makes it a possible building block for a quantum memory. The random access memory (RAM) of a classical computer only works because it corrects errors actively (this is why the data is lost in case of a power failure). The topological quantum memory studied in Chapter 4 faces the same problem: Without active error correction, it “forgets” the stored qubit. The goal of the project was the design of an error correction mechanism that takes the peculiarities of the topological phase into account and, at the same time, remains scalable (the performance of the memory increases with its size; here we are talking about lengths in the range of microns). To achieve this goal, I borrowed the concept of so called cellular automata from the field of computer science. In conclusion, this project demonstrates another possible application of topological phases in quantum information technology; in this case, the storing of quantum information.