Quantum simulators could provide an alternative to numerical simulations for understanding minimal models of condensed matter systems in a controlled way. Typically, cold atom systems are used to simulate e.g. Hubbard models. In this paper, we discuss a range of exotic interactions that can be formed when cold Rydberg atoms are loaded into optical lattices with unconventional geometries; such as long-range electron-phonon interactions and extended Coulomb like interactions. We show how these can lead to proposals for quantum simulators for complex condensed matter systems such as superconductors. Continuous time quantum Monte Carlo is used to compare the proposed schemes with the physics found in traditional condensed matter Hamiltonians for systems such as high temperature superconductors.
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