# Topological Phenomena in Non-equilibrium Systems

The efforts to realize new topological phases have focused mostly on equilibrium systems at zero temperature. The exciting possibility of realizing topological phenomena in out-of-equilibrium systems 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 a non-topological material subject to electromagnetic radiation can have properties that mimic those of a topological insulator; this 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 edge state transport might become possible. Following this route, many important questions remain to be explored: 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?

### Selected Work

#### Universal Chiral Quasisteady States in Periodically Driven Many-Body Systems

Many novel experimental systems, such as cold atom and trapped ion systems, allow us to study the non-equilibrium many-body states of closed, interacting quantum systems subjected to a periodic drive. Generically, such systems are expected to absorb energy and heat up rapidly washing away any interesting quantum, and in particular, topological effects. In this work, we show that the tendency of driven quantum systems to heat up can in fact yield a new universal quantum phenomenon, which persists over a long intermediate time window. In the one dimensional prototype system we study, the universality is manifested in a persistent current, whose magnitude depends only on topological properties of the driving protocol and the density of particles. Quick heating of the low-energy degrees of freedom erases the initial details of the system. The high-energy degrees of freedom, meanwhile, remain cold for a long time, during which the system exhibits its topological properties. Our analysis serves as a prototype for a new class of phenomena that can arise in a variety of driven quantum systems.

- Netanel H. Lindner, Erez Berg, and Mark S. Rudner, Universal Chiral Quasisteady States in Periodically Driven Many-Body Systems, Phys. Rev. X 7, 011018, 2017, arXiv:1603.03053

#### Anomalous Floquet-Anderson Insulator as a Nonadiabatic Quantized Charge Pump

Topological phases typically exhibit protected edge or surface modes, which necessarily coexist with propagating bulk modes that are spread throughout the system. The presence of such propagating bulk modes is required in order to avoid a quantum anomaly. This situation remains true even when disorder is present. A periodically driven system, however, is not bound by the same rules. In this paper, we show that the special character of periodically driven systems drastically changes the relationships among topology, disorder, and localization. We introduce a topological phase that has chiral edge modes, despite all of the bulk states of the system being localized by disorder. This topological phase is only achievable in a periodically driven system. We refer to this unique non-equilibrium phase of matter as an anomalous Floquet-Anderson insulator (AFAI). The AFAI phase possesses remarkable properties and is robust over a wide swath of parameter space. Most strikingly, it exhibits non-adiabatic yet precisely quantized charge pumping at a finite driving frequency. This quantized transport is observed when all of the states near one edge of the system are filled with fermions. The quantization of the charge pump is protected by disorder and the two-dimensional nature of the system. This situation is in contrast to adiabatic quantum pumps, proposed by Thouless three decades ago, in which quantization can only be stabilized by going to the zero-frequency (adiabatic) limit. The finite frequency quantized pumping phenomenon of the AFAI may find applications in the development of precision current standards.

- Paraj Titum, Erez Berg, Mark S. Rudner, Gil Refael, Netanel H. Lindner, Anomalous Floquet-Anderson Insulator as a Nonadiabatic Quantized Charge Pump, Phys. Rev. X 6, 021013, 2016, arXiv:1506.00650

#### Disorder-induced Floquet Topological Insulators

We investigate the possibility of realizing a disorder-induced topological Floquet spectrum in two-dimensional periodically-driven systems. Such a state would be a dynamical realization of the topological Anderson insulator. We establish that a disorder-induced trivial-to-topological transition indeed occurs, and characterize it by computing the disorder averaged Bott index, suitably defined for the time-dependent system. The presence of edge states in the topological state is confirmed by exact numerical time-evolution of wavepackets on the edge of the system. We consider the optimal driving regime for experimentally observing the Floquet-Anderson topological insulator, and discuss its possible realization in photonic lattices.

- Paraj Titum, Netanel H. Lindner, Mikael C. Rechtsman, Gil Refael, Disorder-induced Floquet Topological Insulators, Phys. Rev. Lett. 114, 056801, 2015, arXiv:1403.0592

#### Lighting up topological insulators: large surface photocurrents from magnetic superlattices

The gapless surface states of topological insulators (TI) can potentially be used to detect and harvest low-frequency infrared light. Nonetheless, it was shown that significant surface photocurrents due to light with frequency below the bulk gap are rather hard to produce. Here we demonstrate that a periodic magnetic pattern added to the surface dramatically enhances surface photocurrents in TI’s . The ability to produce substantial photocurrents on TI surfaces from mid-range and far-infrared light could be used in photovoltaic applications, as well as for detection of micrometer wavelength radiation.

- Netanel H. Lindner, Aaron Farrell, Eran Lustig, Gil Refael, Felix von Oppen, Lighting up topological insulators: large surface photocurrents from magnetic superlattices, arXiv:1403.0010

#### Anomalous edge states and the bulk-edge correspondence for periodically-driven two dimensional systems

Recently, several authors have investigated topological phenomena in periodically driven systems of noninteracting particles. These phenomena are identified through analogies between the Floquet spectra of driven systems and the band structures of static Hamiltonians. Intriguingly, these works have revealed phenomena that cannot be characterized by analogy to the topological classification framework for static systems. In particular, in driven systems in two dimensions (2D), robust chiral edge states can appear even though the Chern numbers of all the bulk Floquet bands are zero. Here, we elucidate the crucial distinctions between static and driven 2D systems, and construct a new topological invariant that yields the correct edge-state structure in the driven case. We provide formulations in both the time and frequency domains, which afford additional insight into the origins of the “anomalous” spectra that arise in driven systems. Possibilities for realizing these phenomena in solid-state and cold-atomic systems are discussed.

- M. S. Rudner, N. H. Lindner, E. Berg, M. Levin, Anomalous edge states and the bulk-edge correspondence for periodically-driven two dimensional systems, Phys. Rev. X 3, 031005 (2013); arXiv:1212.3324

#### Floquet topological insulator in semiconductor quantum wells

Topological phase transitions between a conventional insulator and a state of matter with topological properties have been proposed and observed in mercury telluride-cadmium telluride quantum wells. We show that a topological state can be induced in such a device, initially in the trivial phase, by irradiation with microwave frequencies, without closing the gap and crossing the phase transition. We show that the quasi-energy spectrum exhibits a single pair of helical edge states. The velocity of the edge states can be tuned by adjusting the intensity of the microwave radiation. We discuss the necessary experimental parameters for our proposal. This proposal provides an example and a proof of principle of a new non-equilibrium topological state, Floquet topological insulator, introduced in this paper.

- N. H. Lindner, G. Refael, and V. Galitski, Floquet topological insulator in semiconductor quantum wells, Nature Physics 7 , 490-495 (2011); arXiv:1008.1792

#### Topological Floquet Spectrum in Three Dimensions via a Two-Photon Resonance

Three dimensional (3D) topological insulators display an array of unique properties such as single Dirac-cone surface states and a strong magnetoelectric effect. Here we show how a 3D topological spectrum can be induced in a trivial insulator by a periodic drive and, in particular, using electromagnetic radiation. In contrast to the two-dimensional analog, we show that a two-photon resonance is required to transform an initially unremarkable band structure into a topological Floquet spectrum. We provide an intuitive, geometrical picture, alongside a numerical solution of a driven lattice model featuring a single surface Dirac mode. Also, we show that the polarization and frequency of the driving electromagnetic field control the details of the surface modes and particularly the Dirac mass. Specific experimental realizations of the 3D Floquet topological insulator are proposed.

- N. H. Lindner, D. L. Bergman, G. Refeal, and V. Galitski, Topological Floquet Spectrum in Three Dimensions via a Two-Photon Resonance, Phys. Rev. B 87 235131 (2013); arXiv:1111.4518