Hyperbolic Lattice for Scalar Field Theory in AdS$_3$

Abstract

We construct a tessellation of AdS 3 , by extending the equilateral triangulation of AdS 2 on the Poincaré 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 φ 4 theory using Monte Carlo simulations. Applications of this AdS Hamiltonian formulation to real time evolution and quantum computing are discussed.

Breitenlohner-Freedman bound on hyperbolic tilings

“…Motivated by the goal to provide a new example of holographic duality, as well as possible realizations in tabletop experiments, in this Letter we report novel insights in this direction for discretized systems. A prime candidate is a scalar field defined on discretizations of AdS space via regular hyperbolic tilings [6,7], which have been recently investigated using methods from lattice gauge theory in [8][9][10][11]. These works consider discretization schemes for the scalar action, the Laplace operator, and lattice bulk propagators, finding good agreement of the scaling behaviour of correlation functions with analytic continuum results.…”