Hyperbolic Lattice for Scalar Field Theory in AdS$_3$

We construct a tessellation of AdS$_3$, by extending the equilateral triangulation of AdS$_2$ on the Poincar\'{e} disk based on the $(2,3,7)$ triangle group, suitable for studying strongly coupled phenomena and the AdS/CFT correspondence. A Hamiltonian form conducive to the study of dynamics and quantum computation is presented. We show agreement between lattice calculations and analytic results for the free scalar theory and find evidence of a second order critical transition for $\phi^4$ theory using Monte Carlo simulations. Applications of this AdS Hamiltonian formulation to real time evolution and quantum computing are discussed.Comment: 20 pages, 5 figure