## Final Report Summary - QTOPDEV (Topological Quantum Devices)

A deep and unexpected relation between the properties of matter and the mathematical field of topology has been playing an increasingly important role in contemporary condensed matter physics. This relation gives rise to phases of matter which require a paradigm shift. Traditionally, physicists relied on the notion of symmetry breaking in order to draw a distinction between different phases of matter. An example for such a breaking of symmetry is the arbitrary direction of the magnetic field formed when a magnet is cooled below its critical temperature. Topological phases of matter, on the other hand, cannot be classified in this manner. To classify them, one cannot rely on information which can be acquired locally (such as the direction of the magnetic field), but rather, global (and hence, topological) information on the quantum state of the system is required. The new paradigm brought out by topological phases of matter is accompanied by a myriad of extraordinary properties. These make them not only scientifically stimulating, but also appealing for ground-breaking future applications, from quantum computing to novel photo-electronics.

The goal of the QTopDev project is to find methods to engineer devices which can allow access to new topological quantum phases of matter, as well as to harness them for technological applications. The project shall focus on two types of applications. The first part of the project deals with methods to engineer novel devices for topological quantum computation, perhaps one of the most intriguing applications of topological phase of matter. The second part will deal with devices in which both photonics and topology play an important role. Such systems are interesting both as a playground for exploring new non-equilibrium phenomena, and can potentially carry important technological implications, such as room temperature detection of millimeter wave electromagnetic radiation.

Our work on devices for topological quantum computing focused on topological phases containing twist defects. Topological order in two-dimensions can be described in terms of excitations called anyons and their braiding statistics. However, the data encompassing the properties of the anyons do not fully describe the topological phase of matterin the presence of a global symmetry. During the course of the QTopDev project, we considered the case of a symmetry whose action exchanges anyon types. By manipulating one dimensional defects along which such a symmetry is “twisted”, one can store and process quantum information. Therefore, such defects can be used as platforms for performing topological quantum computation. In the course of the QTopDev project, we studied the classification of topological phases exhibiting symmetries and the properties of twist defects in these phases.

Our work showed how to classify such defects and their prospects for topological quantum information processing. Our work gives a construction of a wide class of exactly-solvable models which allow for explicit construction of twists defects. Furthermore, we constructed a mathematical framework describing the properties of the twist defects and showed how this framework is manifested in our exactly solvable models.

The second part of the QTopDev project deals with topological devices which are based on light-matter interactions. One of the objectives of the QTopDev projects was to study photocurrent generation in topological materials. A prominent example is photocurrent generation using the gapless surface states of topological insulators (TI’s). These surface states can potentially be used to detect and harvest low-frequency infrared light. Our work showed that a periodic magnetic pattern added to the surface dramatically enhances surface photocurrents in TI's. Moreover, the sensitivity of this set-up to the wavelength of the incident light can be optimized by tuning the geometry of the magnetic pattern. The ability to produce substantial photocurrents on TI surfaces from mid-range and far-infrared light could be used for detection of micrometer wavelength radiation. According to our work, a detector based on the device we propose can serve as a room temperature detector for wavelength which are beyond reach for devices based on current technologies. Continuing this line of work, we considered generation of photocurrents in Weyl semimetals. We showed that Weyl semimetals can generically support significant photocurrents when the material exhibits a combination of inversion symmetry breaking and finite tilts of the Weyl dispersion.

The interplay between topological behavior of electrons and light-matter interactions is most striking in Floquet Topological Insulators, in which the topological properties are induced via illumination with coherent electromagnetic fields. The intrinsically non-equilibrium states that result, however, are challenging to describe and control in the harsh world of real materials where electrons are awash in crystal vibrations and electromagnetic radiation.

One of the main goals of the QTopDev project, is to develop a clear understanding of the environment’s impact on the driven system and the resulting steady states. As a first step, we focused on a periodically driven, one-dimensional electronic system. Our work provided an in-depth view into the dissipation mechanisms that determine the steady states of the driven system, and showed that the steady state of such system can indeed resemble a band insulator in the Floquet basis.

Based on these results, in the next stage of the project we studied the steady states of two dimensional Floquet topological insulators. Topological transport in FTIs depends crucially on the distribution of electrons in the unidirectional modes induced by the periodic drive. In this work, we found the regime in which the steady state distribution resembles that of a topological phase of matter called a topological insulator. Importantly, we showed that a judicious coupling to external electronic leads yields quantized, topological transport through the system, and proposed an experimental setup for observing topological transport in FTIs.

The non-equilibrium many-body states of closed, interacting quantum system subjected to a periodic drive were also addressed within the QTopDev project (here, closed refers to a system which is not coupled to an external environment). Generically, such systems are expected to absorb energy and heat up rapidly. After a short time, any interesting quantum, and in particular, topological effects would are lost in this situation. During the course of the QTopDev project, we searched for regimes in which interesting long-lived states can be stabilized in such systems. Our work showed that the tendency of driven quantum systems to heat up can in fact bring about the emergence of a new universal quantum phenomenon, which persists over a long intermediate time window. We studied a one dimensional system which serves as a prototype system for this phenomenon. In this system, the universality is manifested in a persistent current, whose magnitude is insensitive to any microscopic details of the system, but rather depends only on topological properties of the driving protocol and the density of particles.

The mechanism that enables this phenomenon relies on a separation between two different time scales for energy absorption from the driving field. This separation of scales opens an exponentially long time window in which quasi-steady states with universal properties are stabilized. Our analysis of this specific system applies directly to recently developed systems of cold atoms in driven optical lattices. Moreover, our analysis serves as a prototype for a new class of non-equilibrium topological phenomena that can arises in a variety of driven quantum systems.

During the course of the QTopDev project, we searched for unique topological phenomena occurring in periodically driven systems. Our work showed that the special topological characteristics of periodically driven systems drastically change the relationships between topology, disorder, and localization. Specifically, our work uncovered the existence of a topological phase which is unique to driven systems. The phase exhibits chiral edge modes, while all states in the bulk of the system are localized by disorder. We refer to this unique non-equilibrium phase of matter as an Anomalous Floquet-Anderson Insulator (AFAI). Next, we showed that this phase is stable in the presence of interactions between the particles. The resulting phase of matter hosts an intriguing combination of physical properties, which fundamentally cannot coexist in equilibrium. Specifically, while the bulk of the sample cannot thermalize, at the edge of the system the motion of the unidirectional modes cannot be halted, which necessarily leads to thermalization. Therefore, this unique phase provide a platform for studying a fundamental open problem - the competition between localization and thermalization.

Another goal of the QTopDev project was to find experimental probes for topological Floquet systems. Our work revealed a physical relationship between the micromotion of a periodically driven system and its topological properties. This relationship is manifested by a quantization of the time-averaged magnetization density within a region where all states are occupied. In this work, we give a detailed proposal for a bulk interference measurement which probes this invariant in cold atomic systems

Next, we investigated the possibility to obtain topologically robust transport measurements in periodically driven quantum systems. We uncovered a quantized transport phenomenon of a new type, which can only occur in a non-equilibrium setting. Rather than occurring at infinitesimal voltage difference between the source and the drain, which is the usual setup in equilibrium, this new phenomenon occurs in the limit of a large source-drain voltage difference. In contrast to the quantized linear conductance (the derivative of the current with respect to the voltage) which is ubiquitous in equilibrium, the quantization in this new setup is of the current itself. The quantized value of the current is independent of the precise value of the bias in the large-bias limit. Our work gives the first example of a steady-state quantized transport phenomenon obtained in a ballistic open system which is periodically driven at finite frequency.

To summarize, the QTopDev project investigated devices and physical setups which give access to novel topological phenomena in quantum condensed matter systems. The results of the project include discovery of new topological phases of matter and physical phenomena; development of methods to probe these phenomena; and proposals for novel technological applications based on topological quantum matter.

The project website can be found at: https://phsites.technion.ac.il/lindner/

The goal of the QTopDev project is to find methods to engineer devices which can allow access to new topological quantum phases of matter, as well as to harness them for technological applications. The project shall focus on two types of applications. The first part of the project deals with methods to engineer novel devices for topological quantum computation, perhaps one of the most intriguing applications of topological phase of matter. The second part will deal with devices in which both photonics and topology play an important role. Such systems are interesting both as a playground for exploring new non-equilibrium phenomena, and can potentially carry important technological implications, such as room temperature detection of millimeter wave electromagnetic radiation.

Our work on devices for topological quantum computing focused on topological phases containing twist defects. Topological order in two-dimensions can be described in terms of excitations called anyons and their braiding statistics. However, the data encompassing the properties of the anyons do not fully describe the topological phase of matterin the presence of a global symmetry. During the course of the QTopDev project, we considered the case of a symmetry whose action exchanges anyon types. By manipulating one dimensional defects along which such a symmetry is “twisted”, one can store and process quantum information. Therefore, such defects can be used as platforms for performing topological quantum computation. In the course of the QTopDev project, we studied the classification of topological phases exhibiting symmetries and the properties of twist defects in these phases.

Our work showed how to classify such defects and their prospects for topological quantum information processing. Our work gives a construction of a wide class of exactly-solvable models which allow for explicit construction of twists defects. Furthermore, we constructed a mathematical framework describing the properties of the twist defects and showed how this framework is manifested in our exactly solvable models.

The second part of the QTopDev project deals with topological devices which are based on light-matter interactions. One of the objectives of the QTopDev projects was to study photocurrent generation in topological materials. A prominent example is photocurrent generation using the gapless surface states of topological insulators (TI’s). These surface states can potentially be used to detect and harvest low-frequency infrared light. Our work showed that a periodic magnetic pattern added to the surface dramatically enhances surface photocurrents in TI's. Moreover, the sensitivity of this set-up to the wavelength of the incident light can be optimized by tuning the geometry of the magnetic pattern. The ability to produce substantial photocurrents on TI surfaces from mid-range and far-infrared light could be used for detection of micrometer wavelength radiation. According to our work, a detector based on the device we propose can serve as a room temperature detector for wavelength which are beyond reach for devices based on current technologies. Continuing this line of work, we considered generation of photocurrents in Weyl semimetals. We showed that Weyl semimetals can generically support significant photocurrents when the material exhibits a combination of inversion symmetry breaking and finite tilts of the Weyl dispersion.

The interplay between topological behavior of electrons and light-matter interactions is most striking in Floquet Topological Insulators, in which the topological properties are induced via illumination with coherent electromagnetic fields. The intrinsically non-equilibrium states that result, however, are challenging to describe and control in the harsh world of real materials where electrons are awash in crystal vibrations and electromagnetic radiation.

One of the main goals of the QTopDev project, is to develop a clear understanding of the environment’s impact on the driven system and the resulting steady states. As a first step, we focused on a periodically driven, one-dimensional electronic system. Our work provided an in-depth view into the dissipation mechanisms that determine the steady states of the driven system, and showed that the steady state of such system can indeed resemble a band insulator in the Floquet basis.

Based on these results, in the next stage of the project we studied the steady states of two dimensional Floquet topological insulators. Topological transport in FTIs depends crucially on the distribution of electrons in the unidirectional modes induced by the periodic drive. In this work, we found the regime in which the steady state distribution resembles that of a topological phase of matter called a topological insulator. Importantly, we showed that a judicious coupling to external electronic leads yields quantized, topological transport through the system, and proposed an experimental setup for observing topological transport in FTIs.

The non-equilibrium many-body states of closed, interacting quantum system subjected to a periodic drive were also addressed within the QTopDev project (here, closed refers to a system which is not coupled to an external environment). Generically, such systems are expected to absorb energy and heat up rapidly. After a short time, any interesting quantum, and in particular, topological effects would are lost in this situation. During the course of the QTopDev project, we searched for regimes in which interesting long-lived states can be stabilized in such systems. Our work showed that the tendency of driven quantum systems to heat up can in fact bring about the emergence of a new universal quantum phenomenon, which persists over a long intermediate time window. We studied a one dimensional system which serves as a prototype system for this phenomenon. In this system, the universality is manifested in a persistent current, whose magnitude is insensitive to any microscopic details of the system, but rather depends only on topological properties of the driving protocol and the density of particles.

The mechanism that enables this phenomenon relies on a separation between two different time scales for energy absorption from the driving field. This separation of scales opens an exponentially long time window in which quasi-steady states with universal properties are stabilized. Our analysis of this specific system applies directly to recently developed systems of cold atoms in driven optical lattices. Moreover, our analysis serves as a prototype for a new class of non-equilibrium topological phenomena that can arises in a variety of driven quantum systems.

During the course of the QTopDev project, we searched for unique topological phenomena occurring in periodically driven systems. Our work showed that the special topological characteristics of periodically driven systems drastically change the relationships between topology, disorder, and localization. Specifically, our work uncovered the existence of a topological phase which is unique to driven systems. The phase exhibits chiral edge modes, while all states in the bulk of the system are localized by disorder. We refer to this unique non-equilibrium phase of matter as an Anomalous Floquet-Anderson Insulator (AFAI). Next, we showed that this phase is stable in the presence of interactions between the particles. The resulting phase of matter hosts an intriguing combination of physical properties, which fundamentally cannot coexist in equilibrium. Specifically, while the bulk of the sample cannot thermalize, at the edge of the system the motion of the unidirectional modes cannot be halted, which necessarily leads to thermalization. Therefore, this unique phase provide a platform for studying a fundamental open problem - the competition between localization and thermalization.

Another goal of the QTopDev project was to find experimental probes for topological Floquet systems. Our work revealed a physical relationship between the micromotion of a periodically driven system and its topological properties. This relationship is manifested by a quantization of the time-averaged magnetization density within a region where all states are occupied. In this work, we give a detailed proposal for a bulk interference measurement which probes this invariant in cold atomic systems

Next, we investigated the possibility to obtain topologically robust transport measurements in periodically driven quantum systems. We uncovered a quantized transport phenomenon of a new type, which can only occur in a non-equilibrium setting. Rather than occurring at infinitesimal voltage difference between the source and the drain, which is the usual setup in equilibrium, this new phenomenon occurs in the limit of a large source-drain voltage difference. In contrast to the quantized linear conductance (the derivative of the current with respect to the voltage) which is ubiquitous in equilibrium, the quantization in this new setup is of the current itself. The quantized value of the current is independent of the precise value of the bias in the large-bias limit. Our work gives the first example of a steady-state quantized transport phenomenon obtained in a ballistic open system which is periodically driven at finite frequency.

To summarize, the QTopDev project investigated devices and physical setups which give access to novel topological phenomena in quantum condensed matter systems. The results of the project include discovery of new topological phases of matter and physical phenomena; development of methods to probe these phenomena; and proposals for novel technological applications based on topological quantum matter.

The project website can be found at: https://phsites.technion.ac.il/lindner/