## Periodic Report Summary 1 - 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 phase 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 deconfined quasiparticle excitations—anyons—and their braiding statistics. However, the data encompassing the properties of the anyons does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the symmetry group does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization. During the course of the QTopDev project, we considered the general case of a symmetry group G which permutes the anyon types. By manipulating defects which twist a symmetry with this type of action, one can obtain topological operations akin to non-Abelian statistics. 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 such a symmetry action and the properties of twist defects in these phases.

Despite the lack of a local action of G, our work showed that one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization. Importantly, our work showed how the data describing symmetry fractionalization is encoded in the associativity of fusion rules of the twist defects of the symmetry. Our work gives a construction of a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and which allow for explicit construction of twists defects. Furthermore, we connect these exactly solvable models to our formal framework describing the symmetry fractionalization and the properties of the twist defects.

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; the sensitivity of the device we propose, at these wavelengths, is comparable to the sensitivity of infra-red detectors at conventional wavelength.

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. In our work, we studied in detail the dependence of the photocurrent response of Weyl semi-metals on the properties of the material and the light source properties. Our results suggest that Weyl materials with broken inversion symmetry can be utilized for room temperature detections of mid- and far-infrared radiation.

Another class of devices which exhibit an interplay between topological behavior of electrons and light-matter interactions are systems in which topological properties are induced via illumination with coherent electromagnetic fields. Dynamically controlling properties of materials using time-periodic driving fields provides a versatile tool for exploring topological phases of matter. 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. To realize the promise of such optically controlled topological phenomena, it is therefore essential to develop a clear understanding of the factors that govern the environment’s impact on a driven system and how these factors can be shaped to our advantage.

One of the main goals of the QTopDev project, is to study these questions in the context of electronic systems whose band topology is dynamically altered when subjected to coherent electromagnetic fields. As a first step, we focused on a periodically driven, one-dimensional electronic system. This system serves as a prototype to investigate the steady-state characteristics of Floquet topological insulators (i.e., driven systems that are internally insulating but may conduct along their surfaces). Our work provides an in-depth view into the dissipation mechanisms that determine the steady states of the driven system. Using a kinetic equation for the populations of the Floquet eigenstates, we show that the steady-state distribution can be controlled using the momentum and energy relaxation pathways provided by the coupling to phonon and Fermi reservoirs. In order to utilize the latter, we propose to couple the system and reservoir via an energy filter which suppresses photon-assisted tunneling. Importantly, coupling to these reservoirs yields a steady state resembling a band insulator in the Floquet basis. When coupled to a filtered fermionic reservoir, the system exhibits incompressible behavior, while hosting a small density of excitations. We discuss transport signatures and describe the regimes where insulating behavior is obtained. Our results give encouraging prospects for realizing Floquet topological insulators and observing their topological phenomena. Based on these results, in the next stage of the project we shall focus on the steady states of two dimensional Floquet topological insulators.

The non-equilibrium many-body states of closed, interacting quantum system subjected to a periodic drive were also addressed within the QTopDev project. 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. In the one dimensional system we studied, this separation of time scales arises due to a suppression of scattering between chiral Floquet bands of opposite chirality. 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.

In equilibrium systems, some of the most intriguing topological phases exhibit edge or surface modes with chiral or helical dispersion relations. Such dispersion relations can only arise for modes on the boundary of a higher dimensional system; otherwise a quantum anomaly would be encountered. For similar reasons, the presence of such boundary states implies that the bulk must host modes that are extended throughout the entire system, even when disorder is introduced. 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 with chiral edge modes, for which 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). Such a spectrum is impossible for a time-independent, local Hamiltonian. Our work identifies the topological invariants that distinguish the AFAI from a trivial, fully localized phase, and show that the two phases are separated by a phase transition. The unique characteristics of the AFAI give rise to a new topologically protected non-equilibrium transport phenomenon: quantized, yet non-adiabatic, charge pumping. Crucially, we show that the quantization of the charge pump is protected by disorder and the two dimensional nature of the system. This should be contrasted with adiabatic quantum pumps, where quantization can only be stabilized in the limit of small frequencies.

Is there a bulk observable that can detect the non-trivial topological properties of the AFAI? To answer this question, we next focused on the micromotion that takes place within each driving period, and which is crucial for the topological classification of the AFAI. Our work revealed a physical relationship between the micromotion of a periodically driven system and its topological invariant. This relationship is manifested by a quantization of the time-averaged magnetization density within a region where all states are occupied. The quantized value is given by the topological invariant (the winding number discussed above) which classifies the AFAI. The winding number is thus related to a bulk experimental observable. In addition, our work gives a detailed proposal for a bulk interference measurement which probes this invariant in cold atomic systems.

The goal of the QTopDev project is to find methods to engineer devices which can allow access to new topological phase 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 deconfined quasiparticle excitations—anyons—and their braiding statistics. However, the data encompassing the properties of the anyons does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the symmetry group does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization. During the course of the QTopDev project, we considered the general case of a symmetry group G which permutes the anyon types. By manipulating defects which twist a symmetry with this type of action, one can obtain topological operations akin to non-Abelian statistics. 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 such a symmetry action and the properties of twist defects in these phases.

Despite the lack of a local action of G, our work showed that one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization. Importantly, our work showed how the data describing symmetry fractionalization is encoded in the associativity of fusion rules of the twist defects of the symmetry. Our work gives a construction of a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and which allow for explicit construction of twists defects. Furthermore, we connect these exactly solvable models to our formal framework describing the symmetry fractionalization and the properties of the twist defects.

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; the sensitivity of the device we propose, at these wavelengths, is comparable to the sensitivity of infra-red detectors at conventional wavelength.

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. In our work, we studied in detail the dependence of the photocurrent response of Weyl semi-metals on the properties of the material and the light source properties. Our results suggest that Weyl materials with broken inversion symmetry can be utilized for room temperature detections of mid- and far-infrared radiation.

Another class of devices which exhibit an interplay between topological behavior of electrons and light-matter interactions are systems in which topological properties are induced via illumination with coherent electromagnetic fields. Dynamically controlling properties of materials using time-periodic driving fields provides a versatile tool for exploring topological phases of matter. 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. To realize the promise of such optically controlled topological phenomena, it is therefore essential to develop a clear understanding of the factors that govern the environment’s impact on a driven system and how these factors can be shaped to our advantage.

One of the main goals of the QTopDev project, is to study these questions in the context of electronic systems whose band topology is dynamically altered when subjected to coherent electromagnetic fields. As a first step, we focused on a periodically driven, one-dimensional electronic system. This system serves as a prototype to investigate the steady-state characteristics of Floquet topological insulators (i.e., driven systems that are internally insulating but may conduct along their surfaces). Our work provides an in-depth view into the dissipation mechanisms that determine the steady states of the driven system. Using a kinetic equation for the populations of the Floquet eigenstates, we show that the steady-state distribution can be controlled using the momentum and energy relaxation pathways provided by the coupling to phonon and Fermi reservoirs. In order to utilize the latter, we propose to couple the system and reservoir via an energy filter which suppresses photon-assisted tunneling. Importantly, coupling to these reservoirs yields a steady state resembling a band insulator in the Floquet basis. When coupled to a filtered fermionic reservoir, the system exhibits incompressible behavior, while hosting a small density of excitations. We discuss transport signatures and describe the regimes where insulating behavior is obtained. Our results give encouraging prospects for realizing Floquet topological insulators and observing their topological phenomena. Based on these results, in the next stage of the project we shall focus on the steady states of two dimensional Floquet topological insulators.

The non-equilibrium many-body states of closed, interacting quantum system subjected to a periodic drive were also addressed within the QTopDev project. 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. In the one dimensional system we studied, this separation of time scales arises due to a suppression of scattering between chiral Floquet bands of opposite chirality. 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.

In equilibrium systems, some of the most intriguing topological phases exhibit edge or surface modes with chiral or helical dispersion relations. Such dispersion relations can only arise for modes on the boundary of a higher dimensional system; otherwise a quantum anomaly would be encountered. For similar reasons, the presence of such boundary states implies that the bulk must host modes that are extended throughout the entire system, even when disorder is introduced. 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 with chiral edge modes, for which 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). Such a spectrum is impossible for a time-independent, local Hamiltonian. Our work identifies the topological invariants that distinguish the AFAI from a trivial, fully localized phase, and show that the two phases are separated by a phase transition. The unique characteristics of the AFAI give rise to a new topologically protected non-equilibrium transport phenomenon: quantized, yet non-adiabatic, charge pumping. Crucially, we show that the quantization of the charge pump is protected by disorder and the two dimensional nature of the system. This should be contrasted with adiabatic quantum pumps, where quantization can only be stabilized in the limit of small frequencies.

Is there a bulk observable that can detect the non-trivial topological properties of the AFAI? To answer this question, we next focused on the micromotion that takes place within each driving period, and which is crucial for the topological classification of the AFAI. Our work revealed a physical relationship between the micromotion of a periodically driven system and its topological invariant. This relationship is manifested by a quantization of the time-averaged magnetization density within a region where all states are occupied. The quantized value is given by the topological invariant (the winding number discussed above) which classifies the AFAI. The winding number is thus related to a bulk experimental observable. In addition, our work gives a detailed proposal for a bulk interference measurement which probes this invariant in cold atomic systems.