Free Probability Theory for Floquet and Quantum Many-Body Systems

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Suppose the eigenvalue distributions of two matrices $M_1$ and $M_2$ are known. What is the eigenvalue distribution of the sum $M_1+M_2$? This problem has a rich pure mathematics history dating back to H. Weyl (1912) with many applications in various fields. Free probability theory (FPT) answers this question under certain conditions, which often involves some degree of randomness (disorder). We will describe FPT and show examples of its powers for the qualitative understanding (often approximations) of physical quantities such as density of states, and gapped vs. gapless phases of some Floquet systems. These physical quantities are often hard to compute exactly. Nevertheless, using FPT and other ideas from random matrix theory excellent approximations can be obtained. Besides the applications presented, we believe the techniques will find new applications in new contexts.