UC Berkeley

Magnetic impurities in a superconductor offer a promising platform for realizing topologicalsuperconductivity. The presence of both classical and quantum impurities in a superconduc- tor can induce a localized subgap state called the Yu-Shiba-Rusinov (YSR) state, which can mediate the effective interaction between magnetic impurities. In this thesis, we explore the effective interaction between classical and quantum impurity spins. We analytically obtain the second-order effective Hamiltonian for the classical impurity spins in the presence of microwave radiation, demonstrating that the effective Hamiltonian exhibits a long-range in- teraction. Additionally, we propose an experimental implementation that enables the control and readout of the impurity spin state ensemble embedded on a Josephson junction from the junction’s integrated conductivity. Extending on the previous result, we also study the effective interaction between quantum impurity spins using a well-controlled perturbation theory. We discover that at the third-order correction, the transverse magnetic field and the quantum spin fluctuation can induce a scalar chirality interaction, which is absent for classical spins. To this end, we investigate the effect of the scalar chirality interaction on the quantum impurity spin order on a two-dimensional lattice using large-scale Density Matrix Renormalization Group (DMRG) calculations. Finally, we study the effects of impurities and disorder in a different context: the bosonic quantum Hall effect. Using quantum Monte Carlo, we uncover a disorder-induced compressible bosonic integer quantum Hall state and characterize its properties.