## Periodic Reporting for period 3 - TopFront (Expanding the Topological Frontier in Quantum Matter: from Concepts to Future Applications)

Reporting period: 2018-09-01 to 2020-02-29

A deep and surprising relation between the properties of matter and the mathematical field of topology is playing an increasingly important role in contemporary condensed matter physics. This relation gives rise to phases of matter whose investigation requires a paradigm shift in the methodology used in the field. 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 purpose of the TopFront research program is to expand the scope of possible realizations of topological quantum matter, and to develop methods to detect, control and manipulate them. Two main research directions are considered: the first will focus on utilizing defects to synthesize new non-Abelian systems. The second direction will explore the exciting possibility of inducing topological behavior in non-equilibrium systems.

A fascinating property of many topological phases are a special kind of collective excitations, which can effectively be considered as a new type of particle which emerges on the background of these phases of matter. One of the most striking phenomena related to these particles occurs when two such particles (of the same type) exchange their position. Let us consider for a moment ordinary particles in nature, such as electrons and photons. When two photons exchange their position, their quantum wavefunction is not changed. When electrons exchange their position, their wavefunction changes sign (i.e. from a minus to a plus or vice versa). In fact, for all of the elementary particle in nature, these are the only two possibilities. Strikingly, when collective excitations in topological phases exchange their position, the wavefunction can be multiplied by a complex number, or even undergo more complicated transformations. The most intriguing of these transformations occur in systems which support “non-Abelian” excitations. The low energy states of such systems can encode quantum information in a manner which is immune to noise and decoherence, which pose a major obstacle in the effort to realize a quantum computer. Thus, non-Abelian systems may hold the key for achieving the ability to perform quantum computations, which is perhaps one of the greatest challenges of contemporary physics and engineering.

Despite years of intense research, examples of experimentally accessible topological states with non-Abelian properties remain relatively few and far between. Because of their subtle nature, non-Abelian states of matter tend to be rare and fragile. Recently, a new route for realizing non-Abelian systems has been discovered. This route employs externally-induced ``twist'' defects in topological phases. As an example, consider two layers of a particular topological phase of matter, called the fractional quantum Hall state. The propagating modes at the edge of the bilayer can be removed using two distinct methods: either by coupling them to an external superconductor or by allowing electrons to back-scatter between the layers. We showed that the interface between two regions of the edge that are effected by these two distinct mechanisms hosts a new type of a non-Abelian mode. Consequently, it was realized that many other types of defects in topological phases exhibit non-Abelian statics (which is absent in the host phase).

Defects in topological phases offer several favorable properties, both from a fundamental and applicative perspective. First, these are externally imposed objects that can, in principle, be controlled and manipulated, for example, by applying local voltages and magnetic fields. In contrast, intrinsic quasi-particles are often difficult to control. Second, non-Abelian defects can be induced in robust, experimentally accessible topological phases. Finally, defects can be used as building blocks for realizing even more exotic topological phases of matter. Our objectives in this part of the research program is to explore the possibilities offered by twist defects in topological phases, and answer many basic questions such as: What kinds of non-Abelian statistics can be realized by using defects? How can defects be realized in experimentally accessible systems, and furthermore, how can they be controlled and manipulated for performing quantum information tasks? What kinds of new quantum phases of matter can be constructed by forming networks of such defects? We expect the answers to these questions to serve as the basis for future work on non-Abelian states of matter.

The efforts to realize new topological phases have focused mostly on equilibrium systems at zero temperature. The exciting possibility of realizing topological phenomena in systems which are in a state far from thermal equilibrium has been recently proposed in the context systems subjected to periodic external driving forces. A major advantage of this setup is the wider range of experimental controls it allows, and new types of accessible topological states. An intriguing example is the proposal that topological properties can be induced in an ordinary, non-topological material, by subjecting it to electromagnetic radiation. This approach may open a completely new route to exploring topological phases, which are not accessible in equilibrium setups. Combined with the highly developed technology for controlling low-frequency electromagnetic modes, potential applications such as devices which utilize fast switching of topological properties in materials might become possible.

Dynamically controlling properties of materials using time-periodic driving fields provides a versatile tool for exploring and utilizing topological phases of matter. Our objectives in this part of the research program is to lay out the ground work for this exploration, and to answer basic questions regarding the myriad of possibilities that this approach can yield, such as: What is the nature of topological phenomena in periodically driven systems? What kinds of phases can be realized following this route, and how different are they from the equilibrium ones? How does topology interplay with many-body interactions and the coupling to the environment to determine their properties? In which experimentally accessible systems can they be realized?

A fascinating property of many topological phases are a special kind of collective excitations, which can effectively be considered as a new type of particle which emerges on the background of these phases of matter. One of the most striking phenomena related to these particles occurs when two such particles (of the same type) exchange their position. Let us consider for a moment ordinary particles in nature, such as electrons and photons. When two photons exchange their position, their quantum wavefunction is not changed. When electrons exchange their position, their wavefunction changes sign (i.e. from a minus to a plus or vice versa). In fact, for all of the elementary particle in nature, these are the only two possibilities. Strikingly, when collective excitations in topological phases exchange their position, the wavefunction can be multiplied by a complex number, or even undergo more complicated transformations. The most intriguing of these transformations occur in systems which support “non-Abelian” excitations. The low energy states of such systems can encode quantum information in a manner which is immune to noise and decoherence, which pose a major obstacle in the effort to realize a quantum computer. Thus, non-Abelian systems may hold the key for achieving the ability to perform quantum computations, which is perhaps one of the greatest challenges of contemporary physics and engineering.

Despite years of intense research, examples of experimentally accessible topological states with non-Abelian properties remain relatively few and far between. Because of their subtle nature, non-Abelian states of matter tend to be rare and fragile. Recently, a new route for realizing non-Abelian systems has been discovered. This route employs externally-induced ``twist'' defects in topological phases. As an example, consider two layers of a particular topological phase of matter, called the fractional quantum Hall state. The propagating modes at the edge of the bilayer can be removed using two distinct methods: either by coupling them to an external superconductor or by allowing electrons to back-scatter between the layers. We showed that the interface between two regions of the edge that are effected by these two distinct mechanisms hosts a new type of a non-Abelian mode. Consequently, it was realized that many other types of defects in topological phases exhibit non-Abelian statics (which is absent in the host phase).

Defects in topological phases offer several favorable properties, both from a fundamental and applicative perspective. First, these are externally imposed objects that can, in principle, be controlled and manipulated, for example, by applying local voltages and magnetic fields. In contrast, intrinsic quasi-particles are often difficult to control. Second, non-Abelian defects can be induced in robust, experimentally accessible topological phases. Finally, defects can be used as building blocks for realizing even more exotic topological phases of matter. Our objectives in this part of the research program is to explore the possibilities offered by twist defects in topological phases, and answer many basic questions such as: What kinds of non-Abelian statistics can be realized by using defects? How can defects be realized in experimentally accessible systems, and furthermore, how can they be controlled and manipulated for performing quantum information tasks? What kinds of new quantum phases of matter can be constructed by forming networks of such defects? We expect the answers to these questions to serve as the basis for future work on non-Abelian states of matter.

The efforts to realize new topological phases have focused mostly on equilibrium systems at zero temperature. The exciting possibility of realizing topological phenomena in systems which are in a state far from thermal equilibrium has been recently proposed in the context systems subjected to periodic external driving forces. A major advantage of this setup is the wider range of experimental controls it allows, and new types of accessible topological states. An intriguing example is the proposal that topological properties can be induced in an ordinary, non-topological material, by subjecting it to electromagnetic radiation. This approach may open a completely new route to exploring topological phases, which are not accessible in equilibrium setups. Combined with the highly developed technology for controlling low-frequency electromagnetic modes, potential applications such as devices which utilize fast switching of topological properties in materials might become possible.

Dynamically controlling properties of materials using time-periodic driving fields provides a versatile tool for exploring and utilizing topological phases of matter. Our objectives in this part of the research program is to lay out the ground work for this exploration, and to answer basic questions regarding the myriad of possibilities that this approach can yield, such as: What is the nature of topological phenomena in periodically driven systems? What kinds of phases can be realized following this route, and how different are they from the equilibrium ones? How does topology interplay with many-body interactions and the coupling to the environment to determine their properties? In which experimentally accessible systems can they be realized?

Dynamical control over the properties of materials using time-periodic driving fields yields states of matter which are intrinsically non-equilibrium. These states 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 TopFront project, is to study these questions in the context of Floquet Topological Insulators (FTIs): electronic systems whose band topology is dynamically altered when subjected to coherent electromagnetic fields (the term Floquet is derived from the name of Gaston Floquet, a French mathematician). Our work provides an in-depth view into the dissipation mechanisms that determine the steady states of the driven system. The properties of the steady states of driven systems affect all quantities accessible to experimental verification (e.g. transport properties, spectral measurements, etc). The issue of the electronic steady state of the systems is thus of major practical importance for further progress in the field.

To obtain the steady state, we developed and numerically simulated the kinetic equations in the presence of strong, time-periodic driving fields, as well as external environments. In equilibrium, particle transport in many topological phases of matter proceeds through modes residing at the edge of the system in which the motion is strictly unidirectional. Transport in FTIs depends crucially on the distribution of electrons in the unidirectional modes induced by the periodic drive (note that these modes are only present in the driven system and do not exist in the absence of the drive). Therefore, to carry out this study we needed to develop a simulation which allowed for a spatially inhomogeneous distribution, as well as scattering of particles between the bulk of the system and the unidirectional modes at the edge, allowing us to correctly take into account the interplay between them. The kinetic equations we developed, allowed us to solve for the steady state populations of electrons in the presence of: a periodic drive; coupling to the electromagnetic environment and to a bath of acoustic phonons; and exchange of particles between the system and a reservoir of electrons (coupling to such reservoirs is crucial for investigating transport measurements). The tools developed in this approach are versatile and can be applied to a variety of different systems.

Our results show that when photonic recombination rates are small compared to the phonon scattering rates, the steady state distribution resembles that of a topological phase of matter called a topological insulator. We developed a simplified model which predicts the universal features of the steady state. Importantly, we show that a judicious coupling to external electronic leads yields quantized, topological transport through the system, and propose an experimental setup for observing topological transport in FTIs. In the next stage of our analysis, we extended the Floquet kinetic equation formalism to account for electron-electron collisions. We found the regimes in which the steady state features an insulatorlike filling of the Floquet bands with a low density of additional excitations. To obtain a more detailed understanding of the electronic distribution function in the steady state, we developed a new formalism for energy and charge balance equations.

The non-equilibrium many-body states of closed, interacting quantum system subjected to a periodic drive were also addressed within the TopFront 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 TopFront 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. The mechanism that enables this phenomenon relies on a separation between two different time scales for energy absorption from the driving field. We show how this separation of scales is achieved in systems subjected to low frequency drive. We furthermore show how this separation of scales opens an exponentially long time window in which quasi-steady states with universal properties are stabilized.

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. In the one dimensional system we studied, this separation of time scales arises due to a suppression of scattering between states propagating in opposite directions. 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 arise in a variety of driven quantum systems.

To extend the applicability of our results to higher dimension and different types of topological properties of the periodically driven system, we obtained a mathematical upper bound on the heating rate which controls the decay time of the quasi-steady state. Our derivation of this bound, is important in asserting the existence and wide applicability of universal topological phenomena manifested by quasi-steady states which can be stabilized for long times.

The interplay with disorder is a crucial ingredient in topological phenomena. We studied the effects of disorder on Floquet topological insulators (FTIs) which are induced by a time periodic drive whose frequency matches a transition between the valence and conduction band of the driven material. The two main objectives of this study were to study the stability of the topological properties of the FTI in the presence of disorder; and to investigate whether addition of sufficiently strong disorder can induce the opposite transition, from a trivial to a topological state of the driven system.

To achieve these goals, we performed both numerical and analytical analysis. Using our numerical simulations, we studied the phase transition in which an FTI becomes localized and insulating at sufficiently strong disorder. To study the possibility to induce a topological state in the driven system by adding disorder, we developed an analytical method which generalizes an approach called the Born approximation used extensively in equilibrium systems. This generalization allowed us to calculate how the basic parameters of the system are effected when disorder is added. Examining the form of the change of these basic parameters, allowed us to find a driving protocol which yields a transition by disorder from a trivial to a topological driven system. For static systems, a disorder-induced topological phase is often referred to as a “topological Anderson Insulator”. In the static case, the occurrence of this phase can be traced to a change of the mass parameter of the static system due to the disorder. In contrast, the transition we have found in this work occurs due to a change of the drive parameters due to the disorder.

In equilibrium systems, some of the most intriguing topological phases exhibit edge or surface modes with unidirectional motion. The rules of quantum mechanics restrict such modes to appear only on the boundary of the system (for example, in two dimensions these modes would incur unidirectional motion on the one dimensional boundary). 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 TopFront 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 and disorder. Specifically, our work uncovered the existence of a topological phase with unidirectional (chiral) edge modes, for which all states in the bulk of the system are localized by disorder (by localized, we refer to the fact that they are not spread out over the entire system). 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.

Another important question we addressed during the TopFront project is the possibility to obtain topologically robust transport measurements in periodically driven quantum systems. The standard setups for measuring topological transport of particles, which are used in equilibrium situations, cannot be applied straightforwardly to periodically driven systems. Our investigation led us to uncover a quantized transport phenomenon of a new type, which is exhibited by topological phases unique to periodically driven systems, and 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. Another important new aspect is that the new phenomenon occurs over a wide range of temperatures. This is in contrast to all other equilibrium quantized transport phenomena (such as the conductance in the quantum Hall effect), which only occur at low temperatures. We simulated the currents obtained in this new setup for several different realizations of the AFAI, and showed that the quantization of the current is accompanied by a unique, non-equilibrium charge and current distribution throughout the system. 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.

Our work on disordered periodically driven systems containing non-interacting particles showed that they can support a unique topological phase, for which the bulk of the system is fully insulating (localized) while unidirectional modes at the edge of the system remain topologically protected. What happens when interactions between the particles are introduced? We investigated this question and discovered a new non-equilibrium phase of matter which we call the Anomalous Floquet Insulator (AFI). This phase 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. AFIs provide a platform for studying a fundamental open problem - the competition between localization and thermalization. By altering the geometry of a sample of AFI, once can tune the strength/extent of the thermal bath provided to the insulating bulk by the unidirectional modes at the edge of the system.

We established the regime of stability of the AFI phase in the presence of interactions, by developing an analytical approach. Our analytical method can be applied to a large class of simple, yet realizable models of the AFI. In addition to the AFI phase, this approach is broadly applicable and will serve as a tool for probing the stability of other non-equilibrium phases of matter. In addition, we performed extensive numerical simulations of the AFI phase, which despite their limitations, agree well with the theoretical predictions given by our analytical analysis.

One of the main goals of the TopFront project, is to study these questions in the context of Floquet Topological Insulators (FTIs): electronic systems whose band topology is dynamically altered when subjected to coherent electromagnetic fields (the term Floquet is derived from the name of Gaston Floquet, a French mathematician). Our work provides an in-depth view into the dissipation mechanisms that determine the steady states of the driven system. The properties of the steady states of driven systems affect all quantities accessible to experimental verification (e.g. transport properties, spectral measurements, etc). The issue of the electronic steady state of the systems is thus of major practical importance for further progress in the field.

To obtain the steady state, we developed and numerically simulated the kinetic equations in the presence of strong, time-periodic driving fields, as well as external environments. In equilibrium, particle transport in many topological phases of matter proceeds through modes residing at the edge of the system in which the motion is strictly unidirectional. Transport in FTIs depends crucially on the distribution of electrons in the unidirectional modes induced by the periodic drive (note that these modes are only present in the driven system and do not exist in the absence of the drive). Therefore, to carry out this study we needed to develop a simulation which allowed for a spatially inhomogeneous distribution, as well as scattering of particles between the bulk of the system and the unidirectional modes at the edge, allowing us to correctly take into account the interplay between them. The kinetic equations we developed, allowed us to solve for the steady state populations of electrons in the presence of: a periodic drive; coupling to the electromagnetic environment and to a bath of acoustic phonons; and exchange of particles between the system and a reservoir of electrons (coupling to such reservoirs is crucial for investigating transport measurements). The tools developed in this approach are versatile and can be applied to a variety of different systems.

Our results show that when photonic recombination rates are small compared to the phonon scattering rates, the steady state distribution resembles that of a topological phase of matter called a topological insulator. We developed a simplified model which predicts the universal features of the steady state. Importantly, we show that a judicious coupling to external electronic leads yields quantized, topological transport through the system, and propose an experimental setup for observing topological transport in FTIs. In the next stage of our analysis, we extended the Floquet kinetic equation formalism to account for electron-electron collisions. We found the regimes in which the steady state features an insulatorlike filling of the Floquet bands with a low density of additional excitations. To obtain a more detailed understanding of the electronic distribution function in the steady state, we developed a new formalism for energy and charge balance equations.

The non-equilibrium many-body states of closed, interacting quantum system subjected to a periodic drive were also addressed within the TopFront 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 TopFront 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. The mechanism that enables this phenomenon relies on a separation between two different time scales for energy absorption from the driving field. We show how this separation of scales is achieved in systems subjected to low frequency drive. We furthermore show how this separation of scales opens an exponentially long time window in which quasi-steady states with universal properties are stabilized.

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. In the one dimensional system we studied, this separation of time scales arises due to a suppression of scattering between states propagating in opposite directions. 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 arise in a variety of driven quantum systems.

To extend the applicability of our results to higher dimension and different types of topological properties of the periodically driven system, we obtained a mathematical upper bound on the heating rate which controls the decay time of the quasi-steady state. Our derivation of this bound, is important in asserting the existence and wide applicability of universal topological phenomena manifested by quasi-steady states which can be stabilized for long times.

The interplay with disorder is a crucial ingredient in topological phenomena. We studied the effects of disorder on Floquet topological insulators (FTIs) which are induced by a time periodic drive whose frequency matches a transition between the valence and conduction band of the driven material. The two main objectives of this study were to study the stability of the topological properties of the FTI in the presence of disorder; and to investigate whether addition of sufficiently strong disorder can induce the opposite transition, from a trivial to a topological state of the driven system.

To achieve these goals, we performed both numerical and analytical analysis. Using our numerical simulations, we studied the phase transition in which an FTI becomes localized and insulating at sufficiently strong disorder. To study the possibility to induce a topological state in the driven system by adding disorder, we developed an analytical method which generalizes an approach called the Born approximation used extensively in equilibrium systems. This generalization allowed us to calculate how the basic parameters of the system are effected when disorder is added. Examining the form of the change of these basic parameters, allowed us to find a driving protocol which yields a transition by disorder from a trivial to a topological driven system. For static systems, a disorder-induced topological phase is often referred to as a “topological Anderson Insulator”. In the static case, the occurrence of this phase can be traced to a change of the mass parameter of the static system due to the disorder. In contrast, the transition we have found in this work occurs due to a change of the drive parameters due to the disorder.

In equilibrium systems, some of the most intriguing topological phases exhibit edge or surface modes with unidirectional motion. The rules of quantum mechanics restrict such modes to appear only on the boundary of the system (for example, in two dimensions these modes would incur unidirectional motion on the one dimensional boundary). 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 TopFront 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 and disorder. Specifically, our work uncovered the existence of a topological phase with unidirectional (chiral) edge modes, for which all states in the bulk of the system are localized by disorder (by localized, we refer to the fact that they are not spread out over the entire system). 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.

Another important question we addressed during the TopFront project is the possibility to obtain topologically robust transport measurements in periodically driven quantum systems. The standard setups for measuring topological transport of particles, which are used in equilibrium situations, cannot be applied straightforwardly to periodically driven systems. Our investigation led us to uncover a quantized transport phenomenon of a new type, which is exhibited by topological phases unique to periodically driven systems, and 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. Another important new aspect is that the new phenomenon occurs over a wide range of temperatures. This is in contrast to all other equilibrium quantized transport phenomena (such as the conductance in the quantum Hall effect), which only occur at low temperatures. We simulated the currents obtained in this new setup for several different realizations of the AFAI, and showed that the quantization of the current is accompanied by a unique, non-equilibrium charge and current distribution throughout the system. 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.

Our work on disordered periodically driven systems containing non-interacting particles showed that they can support a unique topological phase, for which the bulk of the system is fully insulating (localized) while unidirectional modes at the edge of the system remain topologically protected. What happens when interactions between the particles are introduced? We investigated this question and discovered a new non-equilibrium phase of matter which we call the Anomalous Floquet Insulator (AFI). This phase 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. AFIs provide a platform for studying a fundamental open problem - the competition between localization and thermalization. By altering the geometry of a sample of AFI, once can tune the strength/extent of the thermal bath provided to the insulating bulk by the unidirectional modes at the edge of the system.

We established the regime of stability of the AFI phase in the presence of interactions, by developing an analytical approach. Our analytical method can be applied to a large class of simple, yet realizable models of the AFI. In addition to the AFI phase, this approach is broadly applicable and will serve as a tool for probing the stability of other non-equilibrium phases of matter. In addition, we performed extensive numerical simulations of the AFI phase, which despite their limitations, agree well with the theoretical predictions given by our analytical analysis.

The results and methods obtained so far by the TopFront project contain several important advancements beyond the state of the art in the field of topological phases of matter. In our study of the steady state of Floquet Topological Insulators, we have developed methods to analyze a spatially inhomogeneous kinetic equation for the electronic population of the Floquet bulk and topological edge states (which only exist in the presence of the drive). Importantly, our kinetic equations include momentum relaxation due to electron-phonon scattering. Using this analysis, we obtained the steady states of Floquet Topological Insulators in the presence of the natural environments in a solid state system, i.e. coupling to phonons, the electromagnetic environment, and external electronic reservoirs. Subsequently, we have also generalized our method to include the effect of inter-particle interactions. The properties of the steady states of a periodically driven system affect all quantities accessible to experimental verification (e.g. transport properties, spectral measurements, etc.) Our success in obtaining the steady state of topological electronic Floquet-Bloch systems is thus of major practical importance for further progress in the field.

During the TopFront project, we have a discovered a unique topological phase of matter which can only be realized in periodically driven systems. The phase, which we called the Anomalous Floquet-Anderson Insualtor (AFAI), is characterized by a fully localized and insulating bulk coexisting with topologically protected unidirectional (chiral) edge modes. Such a coexistence is impossible for a time-independent, local system. The discovery of this unique topological phase of matter is important for several reasons. First, it demonstrates that the topological classification of phases in periodically driven systems is fundamentally different from the one relevant for static, equilibrium systems. In particular, it shows that some of the fundamental properties of topological phases in equilibrium, such as the bulk-boundary correspondence, have to be revisited when considering topological phenomena in periodically driven systems. Our discovery have propelled a large body of work searching for new unique topological phases of matter in periodically driven systems. Second, during the TopFront project we have discovered several new types of responses exhibiting topologically protected quantization, which are uniquely exhibited by topological phases which appear only in periodically driven systems. These quantized responses include the magnetization density and large bias currents, whose discovery will propel future studies searching for novel quantized responses in periodically driven systems. Finally, the coexistence of topologically protected edge modes with a fully localized bulk implies that the unique topological phenomena exhibited by the AFAI and its generalizations may be stable even in the presence of inter-particle interactions, avoiding the high-entropy fate of generic periodically driven quantum many body systems.

Indeed, in the absence of a coupling to an external bath, periodically driven quantum many body systems generically tend to heat up, absorbing energy from the drive and tending towards a high-entropy state which does not support any interesting correlations. This is perhaps the most significant challenge in the pursuit for stabilizing topological phenomena in periodically driven systems. During the TopFront project, we considered two possible routes for circumventing this problem and observing topological phenomena in periodically driven quantum many-body systems. The first route directly utilizes disorder to induce localization even in the presence of inter-particle interactions, thus suppressing the energy absorption from the drive. We have developed a new analytical method to analyze the stability of periodically driven systems in the presence of inter-particle interactions. Using this method, we have discovered that a large class of unique topological phases, which we called Anomalous Floquet Insulators, are indeed stable phases of matter which can be found in periodically driven, interacting quantum many-body systems. Topological phases in this class exhibit a fully insulating bulk, coexisting with topologically protected propagating edge modes, a unique combination of properties which sets the stage for future investigation of the interplay between localization and thermalization. The analytical method we developed can be utilized in many future works investigating the stability of matter under time-periodic driving. In parallel, we have also investigated the stability of many-body localization (MBL) under a specific important driving protocol which is ubiquitous in cold atom systems. We discovered not only that MBL is stable under this form of driving, it can be in fact induced by this form of a periodic drive. This discovery will be important when developing proposals for experimentally realizing topological phases of matter in periodically driven systems.

The second route for avoiding the high entropy fate and observing topological phenomena in periodically driven systems, which we developed during the TopFront project, relies on the existence of long lived, quasi-steady states with universal topological properties. We have discovered that such a regime is possible under low-frequency driving, and have demonstrated the universality of the obtained quasi-steady state for the case of one-dimensional systems. In addition, we have been able to derive a mathematically rigorous bound on the heating rate in such a setup, which shows that such a regime can be found across a large variety of slowly driven quantum many body systems. This demonstrates that this route is applicable to a wide range of topological phenomena and for different spatial dimensions, and in particular, to phenomena that cannot be stabilized using the first route described above.

As discussed above, the interplay between disorder and topology was studied extensively under the TopFront project. Another major advancement yielded by the TopFront project concerns the possibility to induce a topological phase transition from a topologically-trivial periodically driven system to a topological one, solely by increasing the strength of the disorder. The possibility to obtain such a phase transition is surprising, since disorder is intuitively thought to inhibit transport and not to enable it. The TopFront project yielded two important results regarding this transition: First, in a theoretical study we have discovered a new mechanism for obtaining this transition, which occurs in Floquet insulators induced by a drive whose frequency is resonant with a transition between the valence and conduction band. The second result is a joint experimental and theoretical effort, in which the transition was observed in a photonic realization of a Floquet insulator which is based on a lattice of evanescently coupled waveguides. The experiment gives the first experimental demonstration of this transition, which is extremely challenging to observe in a solid state system.

The TopFront project also significantly advanced the study of photocurrents induced by illuminating topological materials. We showed that three-dimensional Weyl semimetals can generically support significant photocurrents due to the combination of inversion symmetry breaking and finite tilts of the Weyl dispersions. We explored the dependencies of this photovoltaic effect on symmetry properties, chirality and tilt of the Weyl node, and properties of the light source. Our results suggest that Weyl materials which lack inversion symmetry can be advantageously applied to room temperature detections of mid- and far-infrared radiations.

During the TopFront we also performed inter-disciplinary research that yielded advancement beyond the state art both in condensed matter physics and the inter-related fields. In a recent work, we developed a framework for recovering a generic local Hamiltonian using only polynomial time and measurements. Such a recovery is important for quantum information processing applications as it will serve as necessary step for certifying quantum simulators and devices containing many qubits. In particular, we show that the Hamiltonian can be recovered by time-evolving an arbitrary state and we show that the recovery quality improves as a power-law with the evolution time. We also show that a time-dependent Hamiltonian can be recovered, assuming that the functional form of its time-dependence is known, thus allowing us to recover the Hamiltonian of a periodically driven system using only local measurements. Another recent work performed under the TopFront project deals with the possibility to obtain optical skyrmions in evanescent electromagnetic fields using surface plasmon polaritons (skyrmions are topological defects in two dimensional configurations of a three-component vector field). This discovery could propel realizations and applications of skyrmions in new physical platforms: using light matter interactions, optical skyrmions could induce skyrmionic configurations in a wide variety of material, atomic and molecular systems. The skyrmion configuration of the electromagnetic field may serve as a basis for obtaining periodically driven quantum many–body system in which the driving field itself has a topological character. Furthermore, by utilizing the wide range of nonlinearities known in optics, new physical phenomena involving skyrmions can be investigated.

In the remaining period of the TopFront project, we expect to continue ongoing work and obtain significant results on novel non-Abelian systems. In particular, we expect the TopFront project to yield a characterization of defects in non-Abelian topological phases in terms of zero modes. This characterization will include an algebraic description that will allow us to construct new quantum many-body models based on arrays of these zero modes. In addition, we are currently investigating microscopic models describing setups for realizing non-Abelian zero modes. This study is expected to yield an in-depth understanding of the detailed physical mechanisms required for realizing these zero modes, and thus lead to concrete proposals for experimentally obtaining them. We expect to have significant results on optimization of protocols for controlling non-Abelian anyons and zero modes, along with proposals for their experimental implementation. Our work on probes for novel non-Abelian systems is expected to yield methods for detecting unambiguous signatures of novel topological states in experiments. Furthermore, we have initiated a research direction which bridges the two main directions of the TopFront project, and investigates the possibility of stabilizing fractionalized, non-Abelian zero modes using a time-periodic drive.

We expect our on-going work on topological phenomena in periodically driven system to yield several new important results. We are investigating new topologically protected quantized responses unique to interacting periodically driven systems. Furthermore, we plan to perform an in-depth study of the effect of disorder on the universal topological phenomena exhibited by quasi-steady states of slowly driven systems. We are currently investigating the possibility to stabilize, over long time periods, interacting periodically driven systems exhibiting non-Abelian anyons. We also expect to have significant results on phase transitions between steady states obtained in solid state systems subjected to time-periodic driving fields, and in other periodically driven coupled to dissipative environments.

During the TopFront project, we have a discovered a unique topological phase of matter which can only be realized in periodically driven systems. The phase, which we called the Anomalous Floquet-Anderson Insualtor (AFAI), is characterized by a fully localized and insulating bulk coexisting with topologically protected unidirectional (chiral) edge modes. Such a coexistence is impossible for a time-independent, local system. The discovery of this unique topological phase of matter is important for several reasons. First, it demonstrates that the topological classification of phases in periodically driven systems is fundamentally different from the one relevant for static, equilibrium systems. In particular, it shows that some of the fundamental properties of topological phases in equilibrium, such as the bulk-boundary correspondence, have to be revisited when considering topological phenomena in periodically driven systems. Our discovery have propelled a large body of work searching for new unique topological phases of matter in periodically driven systems. Second, during the TopFront project we have discovered several new types of responses exhibiting topologically protected quantization, which are uniquely exhibited by topological phases which appear only in periodically driven systems. These quantized responses include the magnetization density and large bias currents, whose discovery will propel future studies searching for novel quantized responses in periodically driven systems. Finally, the coexistence of topologically protected edge modes with a fully localized bulk implies that the unique topological phenomena exhibited by the AFAI and its generalizations may be stable even in the presence of inter-particle interactions, avoiding the high-entropy fate of generic periodically driven quantum many body systems.

Indeed, in the absence of a coupling to an external bath, periodically driven quantum many body systems generically tend to heat up, absorbing energy from the drive and tending towards a high-entropy state which does not support any interesting correlations. This is perhaps the most significant challenge in the pursuit for stabilizing topological phenomena in periodically driven systems. During the TopFront project, we considered two possible routes for circumventing this problem and observing topological phenomena in periodically driven quantum many-body systems. The first route directly utilizes disorder to induce localization even in the presence of inter-particle interactions, thus suppressing the energy absorption from the drive. We have developed a new analytical method to analyze the stability of periodically driven systems in the presence of inter-particle interactions. Using this method, we have discovered that a large class of unique topological phases, which we called Anomalous Floquet Insulators, are indeed stable phases of matter which can be found in periodically driven, interacting quantum many-body systems. Topological phases in this class exhibit a fully insulating bulk, coexisting with topologically protected propagating edge modes, a unique combination of properties which sets the stage for future investigation of the interplay between localization and thermalization. The analytical method we developed can be utilized in many future works investigating the stability of matter under time-periodic driving. In parallel, we have also investigated the stability of many-body localization (MBL) under a specific important driving protocol which is ubiquitous in cold atom systems. We discovered not only that MBL is stable under this form of driving, it can be in fact induced by this form of a periodic drive. This discovery will be important when developing proposals for experimentally realizing topological phases of matter in periodically driven systems.

The second route for avoiding the high entropy fate and observing topological phenomena in periodically driven systems, which we developed during the TopFront project, relies on the existence of long lived, quasi-steady states with universal topological properties. We have discovered that such a regime is possible under low-frequency driving, and have demonstrated the universality of the obtained quasi-steady state for the case of one-dimensional systems. In addition, we have been able to derive a mathematically rigorous bound on the heating rate in such a setup, which shows that such a regime can be found across a large variety of slowly driven quantum many body systems. This demonstrates that this route is applicable to a wide range of topological phenomena and for different spatial dimensions, and in particular, to phenomena that cannot be stabilized using the first route described above.

As discussed above, the interplay between disorder and topology was studied extensively under the TopFront project. Another major advancement yielded by the TopFront project concerns the possibility to induce a topological phase transition from a topologically-trivial periodically driven system to a topological one, solely by increasing the strength of the disorder. The possibility to obtain such a phase transition is surprising, since disorder is intuitively thought to inhibit transport and not to enable it. The TopFront project yielded two important results regarding this transition: First, in a theoretical study we have discovered a new mechanism for obtaining this transition, which occurs in Floquet insulators induced by a drive whose frequency is resonant with a transition between the valence and conduction band. The second result is a joint experimental and theoretical effort, in which the transition was observed in a photonic realization of a Floquet insulator which is based on a lattice of evanescently coupled waveguides. The experiment gives the first experimental demonstration of this transition, which is extremely challenging to observe in a solid state system.

The TopFront project also significantly advanced the study of photocurrents induced by illuminating topological materials. We showed that three-dimensional Weyl semimetals can generically support significant photocurrents due to the combination of inversion symmetry breaking and finite tilts of the Weyl dispersions. We explored the dependencies of this photovoltaic effect on symmetry properties, chirality and tilt of the Weyl node, and properties of the light source. Our results suggest that Weyl materials which lack inversion symmetry can be advantageously applied to room temperature detections of mid- and far-infrared radiations.

During the TopFront we also performed inter-disciplinary research that yielded advancement beyond the state art both in condensed matter physics and the inter-related fields. In a recent work, we developed a framework for recovering a generic local Hamiltonian using only polynomial time and measurements. Such a recovery is important for quantum information processing applications as it will serve as necessary step for certifying quantum simulators and devices containing many qubits. In particular, we show that the Hamiltonian can be recovered by time-evolving an arbitrary state and we show that the recovery quality improves as a power-law with the evolution time. We also show that a time-dependent Hamiltonian can be recovered, assuming that the functional form of its time-dependence is known, thus allowing us to recover the Hamiltonian of a periodically driven system using only local measurements. Another recent work performed under the TopFront project deals with the possibility to obtain optical skyrmions in evanescent electromagnetic fields using surface plasmon polaritons (skyrmions are topological defects in two dimensional configurations of a three-component vector field). This discovery could propel realizations and applications of skyrmions in new physical platforms: using light matter interactions, optical skyrmions could induce skyrmionic configurations in a wide variety of material, atomic and molecular systems. The skyrmion configuration of the electromagnetic field may serve as a basis for obtaining periodically driven quantum many–body system in which the driving field itself has a topological character. Furthermore, by utilizing the wide range of nonlinearities known in optics, new physical phenomena involving skyrmions can be investigated.

In the remaining period of the TopFront project, we expect to continue ongoing work and obtain significant results on novel non-Abelian systems. In particular, we expect the TopFront project to yield a characterization of defects in non-Abelian topological phases in terms of zero modes. This characterization will include an algebraic description that will allow us to construct new quantum many-body models based on arrays of these zero modes. In addition, we are currently investigating microscopic models describing setups for realizing non-Abelian zero modes. This study is expected to yield an in-depth understanding of the detailed physical mechanisms required for realizing these zero modes, and thus lead to concrete proposals for experimentally obtaining them. We expect to have significant results on optimization of protocols for controlling non-Abelian anyons and zero modes, along with proposals for their experimental implementation. Our work on probes for novel non-Abelian systems is expected to yield methods for detecting unambiguous signatures of novel topological states in experiments. Furthermore, we have initiated a research direction which bridges the two main directions of the TopFront project, and investigates the possibility of stabilizing fractionalized, non-Abelian zero modes using a time-periodic drive.

We expect our on-going work on topological phenomena in periodically driven system to yield several new important results. We are investigating new topologically protected quantized responses unique to interacting periodically driven systems. Furthermore, we plan to perform an in-depth study of the effect of disorder on the universal topological phenomena exhibited by quasi-steady states of slowly driven systems. We are currently investigating the possibility to stabilize, over long time periods, interacting periodically driven systems exhibiting non-Abelian anyons. We also expect to have significant results on phase transitions between steady states obtained in solid state systems subjected to time-periodic driving fields, and in other periodically driven coupled to dissipative environments.