# Topological Phenomena in Quantum Dynamics and Disordered Systems (13w5158)

Arriving in Banff, Alberta Sunday, February 3 and departing Friday February 8, 2013

## Organizers

(University of British Columbia)

Netanel Lindner (California Institute of Technology)

(California Institute of Technology)

## Objectives

The discovery of topological insulators revealed a beautiful mathematical structure underlying gapped phases of matter, and triggered the theoretical prediction and experimental verification of myriad of unique physical phenomena. This demonstrates the great potential present in this new frontier of condensed matter and mathematical physics. Topological phenomena in dynamics of driven system, as well as in disordered systems, is still a mostly unexplored frontier, while at the same time holding great promise for both novel fundamental physics and technological applications.The proposed workshop will bring together top researchers in theoretical and experimental physics, mathematics and quantum information theory, interested in topological phenomena in quantum physics. The objectives of these workshop are twofold. First, it is important to emphasize that this is an interdisciplinary research field, wherein progress requires a combination of expertise ranging from mathematics to condensed matter physics. The workshop will therefore provide an opportunity for an intellectual exchanged between researchers from these different disciplines. Researchers will discuss both the mathematical tools necessary to study topological phenomena in dynamical and disordered systems, as well as physical implications and experimental issues related to these phenomena. Second, we believe that such a multidisciplinary environment, which will provide a unique combination of physical insight and mathematical background, could stimulate discovery of new types of topological phenomena in quantum matter. Specifically, we plan to focus on the following topics:1) Theoretical models for topological states in dynamical quantum systems. Recent developments yielded several models which demonstrate how topological phenomena can arise in dynamical quantum systems with an external driving mechanism. Examples are electromagnetically irradiated Graphene, semiconductor quantum wells, and even three dimensional narrow-band semiconductors. These proposals are based on different physical mechanisms - while the former relies on an indirect process which opens a gap at the Dirac node, the latter rely on a direct resonance between energy bands. These models were shown to yield different classes of topological phenomena. Furthermore, these theoretical models demonstrate that it is possible to achieve topological phenomena in systems which are trivial in equilibrium. However, these proposals are still focused around specialized systems which exhibit strong spin-orbit coupling or Dirac nodes. The proposed workshop will stimulate participants in formulating new theoretical models manifesting topological phenomena in dynamical systems, with the purpose to achieve such states in a larger class initial systems.2) Topological classification of dynamical quantum systems. The dynamical systems proposed so far were shown to yield different classes of topological phenomena, depending on the symmetries of the Floquet operators describing their dynamics. However, a systematic mathematical classification of topological states in dynamical quantum systems is still missing. In contrast, recent developments, for example, a powerful mathematical description using K-theory developed by Kitaev, enable a rigorous and complete classification of topological insulators and superconductors of free particles. Time dependent Hamiltonians and non-equilibrium physics yield quantum dynamics with substantially more degrees of freedom, and therefore their topological classification is expected to yield a richer structure. While currently remaining elusive, an indication towards the complexity of this structure is provided by several examples of topological quantum states in dynamical systems which have no analog in equilibrium systems. The proposed workshop will provide a platform which will enable considerable progress both to a mathematical classification and to proposals to realize different classes of topological states in experimentally realizable systems.3) Probing dynamical topological phenomena. Dynamical quantum systems are expected to be out of thermodynamic equilibrium. As a consequence, many of the theoretical and experimental tools used for probing topological phenomena in equilibrium may not be directly applicable to dynamical systems. An important goal is, therefore, to develop new tools for probing topological phenomena in these systems. In order to use transport measurements, electronic steady states need to be taken into account, which entails a careful study of the different dissipation mechanisms coupled to the system. Moreover, new measurement schemes need to be devised in order to eliminate any contributions to transport signals coming from non-topological origins. Another approach is to use methods which measure the electronic spectral function directly, such as scanning tunnelling microscopy (STM) or angular resolved emission spectroscopy (ARPES). The workshop will provide a forum for discussion between theorists and experimentalists, with the purpose of developing the necessary tools to test these new theoretical predictions.4) Topological behavior in Hamiltonians lacking translational symmetry. Understanding the interplay of disorder and topological behavior is extremely interesting from a theoretical perspective, as well as crucial for applications to possible quantum computation or electronic devices. The description of topological behavior often relies on the definition of a gauge structure and topological invariants in momentum space. When the system is disordered, however, such a description becomes problematic. Several new developments infused excitement in this topic. Disorder induced topological behavior in simple HgTe-like semiconductors, as well as in Graphene (using random doping with spin-orbit coupled atoms as In) is one such direction. Concomitantly, mathematical methods developed by Moore and especially Hastings allowed the calculation of a topological invariant for strongly disordered systems. Surprisingly, Hastings' method can be applied also to topological insulators whose spectral gap is demolished by disorder, and it was found that the disorder-averaged index interpolates continuously from 1 to 0. It is not known, however, how to interprete this result. The objectives of the workshop in this context are to elucidate the meaning of the topological invariant in gapless topological matter, explore the possibilities to induce topological order using disorder, and understand the effects of disorder on the unique properties of topological matter.5) Axion response of disordered topological systems. An alternative description of topological insulators in 3 spatial dimensions can be given in terms of their effective electromagnetic response. For ordinary translation-invariant topological insulators the effective electromagnetic Lagrangian is known to contain the characteristic `axion' term of the form $theta{bf E}cdot{bf B}$ with axion angle $theta=pi$. This term leads to several unusual physical phenomena predicted to occur in these materials, such as the topological magneto-electric effect and the Witten effect. More importantly, the non-trivial value of $theta$ can be taken as a unique signature of the topological phase. In a time-reversal invariant system very general arguments dictate that only two values of $theta$ are permitted: $0$ or $pi$. They correspond respectively to the trivial and topological insulator. The above statement is true in systems containing disorder and in interacting system (with the caveat that the ground state is non-degenerate). Therefore, axion angle $theta$ should provide a sharp distinction between trivial and topological phases in the presence of disorder. How to evaluate $theta$ in a system lacking translational invariance is unknown at present. The proposed workshop will provide a platform to discuss both analytical and numerical approaches suitable for evaluation of this quantity in disordered systems. Participants will also reflect upon the possibility of similar characterization of driven topological systems.