## Periodic Reporting for period 2 - TopFront (Expanding the Topological Frontier in Quantum Matter: from Concepts to Future Applications)

Reporting period: 2017-03-01 to 2018-08-31

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 ty

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 ty

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, top

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, top

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. Usin

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. Usin