# Topological excitonics

When photons are absorbed by an organic light-harvesting material, they transfer their energy to chromophores by creating dipolar excitations known as (molecular) excitons. In organic solar cell applications, excitons are the middlemen: they transfer energy from large uncharged photons to the motion through a wire of small charged electrons. A good organic solar cell channels excitons efficiently from the molecular absorbers to an interface with electrons. Like photons, excitons have no charge, but they are much smaller (10–100nm in diameter), like electrons. Furthermore, unlike photons (which barely interact with other photons), excitons interact with other excitons efficiently. In fact, exciton wires are often mentioned as promising elements for all-optical circuits, as they could in principle feature the fast switching times of photons, but at the same time, be integrated into small devices, just like electronic components.^{1}

However, controlling the motion of excitons is not an easy task. As they move around, they lose energy as heat and often become trapped by structural defects. An exciton located at a particular chromophore cannot move if the neighboring molecules lack resonant energy levels. Instead, after a period of about a nanosecond, the leftover energy will be re-emitted as light. Such energy traps occur due to intrinsic defects in the material that cause fluctuations in the chromophore energies, causing so-called Anderson localization and hence limited transport of excitons. By applying concepts developed in the context of electronic systems, in particular the quantum Hall effect (QHE), colleagues and I have theoretically designed excitonic materials that circumvent the problem of localization.

We have constructed a first theoretical model that anticipates topological exciton transport and is largely immune to defects.^{2} The model consists of porphyrin molecules in a 2D array that transports excitons robustly at its edges. The setup relies on using magnetic fields to create one-way topological excitons, that is, to force the excitons to move in a specific direction.

To understand this phenomenon a bit better, we can draw analogies to the original QHE. Consider a 2D electron gas under a strong magnetic field. In the simplest physical picture, the electrons are distributed across the sample, precessing in small circles (the stronger the magnetic field, the smaller the radius of cyclotron motion, and the faster the speed of precession): see Figure 1. The precession direction is dictated by the Lorentz force (right-hand rule). A quantum mechanical effect of this precession is that when the electron completes a full circular trajectory, it acquires a phase χ (called a Berry phase^{3}) that is proportional to the magnetic flux it enclosed.

A particularly remarkable situation happens at the edges. The electrons can no longer complete a full circle and, instead, move in skipping orbits: see Figure 1. Hence, the QHE setup is one where the bulk is an insulator but the edges are perfect conducting cables! Notice that the edge states have opposite directions, like two counterpropagating lanes in a highway. These edge states are known to be particularly immune to backscattering and robust to disorder. To see this, notice that for an electron in one edge to backscatter, it needs to switch to the opposite edge, but the latter is located a relatively long distance away, hence exponentially suppressing the possibility of this event. Many other phenomena related to this original instance of the QHE have been discovered during the last 30 years. Some of the most remarkable ones are so-called topological insulators,^{4, 5} where the role of the magnetic field is substituted by the presence of strong electron spin-orbit interactions.

In our work, the excitons are effectively chargeless particles, so at a first glance there is no reason to expect them to behave like electrons in the QHE. Yet, we can cross-dress them as electrons, at least for our purposes. That is, we can recreate conditions where the excitons follow effective equations of motion that are similar to those of electrons in the QHE. In our work, excitons move under a real homogeneous magnetic field in the same way as if they were electrons interacting with an effective staggered magnetic field: see Figure 2(a) and (b). In particular, the excitons acquire phases χ and −χ, resulting from the effective staggered magnetic field, compared to Figure 1, which represents a homogeneous magnetic field situation. The staggered-field case corresponds to the well-known Haldane lattice model,^{6} a prescription that generalizes the original QHE setup to inhomogeneous magnetic fields. Curiously, at its time of inception, the Haldane model was largely deemed a mathematical curiosity due to its seemingly forbidding experimental implementation. Its redemption would come much later, as it turns out to be the natural model for many topological insulators, as well as for our current model for topological excitons, which effectively (not literally) features this staggered magnetic field.

Our proposal relies on several ingredients. Porphyrins are ring-like molecules that support two degenerate excitons corresponding to clockwise or counterclockwise electron orbital angular momentum within each ring. In the presence of a magnetic field, these states are Zeeman split in energy, as one of them is favored and the other penalized by the Lorentz force on the electrons: see Figure 2(c). The motion of these chiral excitons is governed, to a good approximation, by classical dipolar interactions. Together, the anisotropy of these interactions and the Zeeman effect produce an effect where, as the excitons move in a closed loop, they acquire a phase, in very much the same way as when electrons move in cyclotron orbits in the presence of a magnetic field.

In summary, our model is the first realization of a nontrivial topological phase in an exciton system and offers a novel conceptual way to think about materials design for light-harvesting and energy transport. Since our work relies on coherence (well-defined phases), it is important to recognize effects of inelastic scattering caused by vibrations. We note that, just like the QHE analogy, what matters most is to have coherence between small loops involving nearest neighbor porphyrins, rather than coherence throughout the entire film. However, a detailed study of these effects is necessary, and is underway. So far, our simulations show that exciton transport in our system is robust to randomization of the parameters in our model, indicating a robust phenomena that can be detected in the laboratory: see Figure 3. Hence, we expect many interesting developments based on topological excitonics, complementing the already prolific fields of topological insulators and topological photonics. Our next steps are to explore topological phenomena in new materials, as well as in different energy and length scales than the ones typically associated with electronic^{4, 5} and photonic systems.^{7}

Massachusetts Institute of Technology (MIT)

Joel Yuen-Zhou currently holds the Robert J. Silbey postdoctoral fellowship at MIT. In July 2015, he will become an assistant professor in the Department of Chemistry at the University of California, San Diego. His research interests center primarily in theoretical chemical physics and include quantum information and spectroscopy.

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