Looking for some fresh bathroom tiles? Why don't you try regular 7-gons this time, it looks really cool! Only requirement would be that you live in hyperbolic space of constant negative curvature. To get a feeling how that would be like, you should look into some recent breakthrough experiments in circuit quantum electrodynamics, where such tilings are realized with superconducting waveguide resonators and photons are tricked into believing that space is hyperbolic: I will present how these finite lattice geometries can be mapped onto quantum field theories in continuous negatively curved space. This provides a computational tool to determine observables of the discrete system analytically even for large lattices, where exact diagonalization fails due to exponential growth in system size. As an application we quantitatively reproduce ground state energy, spectral gap, and correlation functions of the noninteracting lattice system by means of analytic formulas on the Poincare disk, and show how conformal symmetry emerges for large lattices. This sets the stage for studying the effects of interactions and disorder on hyperbolic graphs, and to resolve fundamental open problems at the interface of interacting many-body systems, quantum field theory in curved space, and quantum gravity using tabletop experiments.
(pizza and drinks served 10 min. before talk)