Solid State Quantum Simulation: From Hyperbolic Space to Emergent Spin-3/2 Fermions

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What are the basic truths we know about our world? For one, humankind believed for a long time that space is flat. And to a good approximation, it is: If you draw a triangle, the sum of the interior angles is 180 degrees. Later we discovered that our Earth is actually curved. If you were to draw a very large triangle on Earth's surface, its sum of interior angles would be larger than 180 degrees. Mathematicians call this a positive curvature. They also realized that a much more intriguing geometry exists, one that is called negatively curved, where the sum of interior angles in a triangle can be less than 180 degrees, even zero! This space, called hyperbolic space, has many features that evade our intuition, but Mathematicians love it because of its beauty, and Physicists realized over the last decades that many properties of our physical world may secretly be related to hyperbolic space. Another such truth, although you might not think about it often, is the existence of electrons --- those subatomic charged particles that are responsible for electronic current in solids and that populate the electronic shell of atoms. Electrons have an intrinsic property, called their spin quantum number, which takes two values, commonly called "up" or "down", or "+1/2" and "-1/2". (Because of the latter, we say that the electron has spin 1/2.) The properties of electrical currents and all chemical elements around us result from the fact that the electron can appear in these two states. Physicists have long searched for electron-like particles with spin 3/2, which have four states, called "+3/2", "+1/2", "-1/2", and "-3/2", but no fundamental particle with this property was found. This research project will investigate what would happen if the world would be negatively curved or if electrons had spin 3/2. We do this to satisfy our curiosity, of course, but the outcome of such an analysis will also affect our understanding of the fundamental Laws of Physics and could have technological applications. The last few years have seen two experimental breakthroughs that motivate this project. One is the realization of a hyperbolic lattice with superconducting resonators, which is a heptagonal tiling that only exists in negatively curved space. Particles moving on such a lattice think they are in hyperbolic space. The other is the synthesization of topological semimetal materials, where the charge carrying particles act like electrons with spin 3/2. Both settings are called "quantum simulations", because they create an artifical world in order to experimentally study its properties. Specifically, we will analyse how the physics of many particles is altered by hyperbolic space and we will theoretically help to create two more experimental platforms for their investigation: electrical circuits and ultracold quantum gases. We will investigate how the interactions between spin-3/2 particles alter the properties of solids and how this can be measured in experiment.