Entanglement and topology of time-reversal symmetric fractional topological insulators

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Traditionally, physics has advanced via a reductionist approach, explaining the world we see in terms of its smaller and smaller components - until reaching the level of particles that we believe to be elementary. However, even as we know the precise nature of components of matter and how they interact with each other, there is still a lot of room for surprises. It turns out that when many particles interact with each other, they often display genuinely new and different behaviour. In this sense, the old saying that the whole is more than just the sum of its parts turns out to be true!In this proposal, we investigate the consequences of interacting many-particle systems in a setting that is particularly conducive to observing strong quantum effects. One frequently exploited route for enhancing quantum effects is the application of strong magnetic fields: in this setting, the kinetic energy of particles becomes unimportant, and all of the physics is driven by the interactions. This is the foundation of the fractional quantum Hall effect, which is a macroscopic quantum phenomenon observed when electrons are confined in a thin layer of material pierced by a very strong magnetic field.Surprisingly, it turns out that the effect of the magnetic field can be emulated spontaneously by certain types of materials that have recently been discovered, including strained graphene, or possibly thin films of materials involving heavy elements like iridium. Furthermore, the fields generated by such materials can be 10 - 100 times higher than magnetic fields which can otherwise be experimentally generated in the laboratory. Still, there is a twist to this story: the direction of the synthetic magnetic fields generated in these materials depends on the orientation of the spin, i.e., the axis around which individual electrons turn.The combination of strong magnetic fields, along with the added complexity of the opposite field directions for opposite spins leads to the new area that will be investigated in our project. Naively, this is a very difficult task, however, as the numerical methods that are usually used for this purpose are overwhelmed when the extra spin degree of freedom is included. In addition to new opportunities from materials physics, we will thus exploit another timely development and adopt a new way of looking at many-body quantum systems by characterising the respective wave functions from a quantum information point of view. Rather than giving the wave function globally for the entire system, we will use an efficient approach to encode how individual degrees of freedom depend on those of their immediate neighbours. In more technical terms, we describe the system in terms of its entanglement properties, where entanglement quantifies to which extent the quantum state of a particle at a given location is conditional on the quantum states of other particles in the system.By establishing an effective means of simulating the complex many-body physics of our target systems, we achieve predictive power over the resulting collective behaviour. Typically, we expect that this behaviour is very sensitive to the system parameters. Hence, the ability to effectively simulate the behaviour numerically provides a tremendous advantage in finding physically interesting regimes. We will use this ability in order to identify types of materials in which the desired quantum effects are most robust. We will also go beyond understanding what happens in a stationary regime, when the system is left to relax to its absolute ground state. We can learn even more about the intricate interplay of the constituent particles by exposing our system to a rapid change in parameters, and studying how it evolves afterwards. Think of taking a hammer and listening to the resulting sound of a bell when it is being struck in different places. Clearly, this is a lot more informative than just looking at the bell left to its own devices.