Hyperbolic lattice for scalar field theory in ${\mathrm{AdS}}_3$

We construct a tessellation of ${\mathrm{AdS}}_3$, by extending the equilateral triangulation of ${\mathrm{AdS}}_2$ on the Poincaré disk based on the (2,3,7) triangle group, suitable for studying strongly coupled phenomena and the $\mathrm{AdS}/\mathrm{CFT}$ 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 ${\varphi }^4$ theory using Monte Carlo simulations. Applications of this anti–de Sitter Hamiltonian formulation to real time evolution and quantum computing are discussed.

Figure 1

${\mathrm{AdS}}_3$ spacetime with our choice of coordinate system looks like a solid cylinder. At fixed time the space is the hyperbolic disk, which can be tessellated using equilateral hyperbolic triangles. Here, (2,3,7) triangles are used.

Figure 2

The full hyperbolic disk ${\mathbb{H}}^2$ and an enlarged view of the edge, tessellated with three layers ($L=3$) of (2,3,7) triangles. The dashed circle near the lattice edge is the effective lattice boundary $r_{\max }$ and the larger, solid circle the conformal boundary at $r=1$. The UV lattice cutoff is $\epsilon =1-r_{\max }$.

Figure 3

A general bulk geodesic between two points bends into the bulk due to the time dilation of the coefficient $g_{00}(\rho )$ in the metric (2.3) when moving toward the boundary of the ${\mathrm{AdS}}_3$ cylinder.

Figure 4

Checks in the noninteracting regime of the ${\mathrm{AdS}}_3$ lattice realization from a direct inversion of the massless Green’s function $G_{bb}(\sigma )$ for $L=4$. Top left: the propagator for all pairs of points with zero temporal separation (i.e., pairs of points in the same time slice). Top right: the propagator for all pairs of points with zero spatial separation. Bottom: all-to-all propagator.

Figure 5

Checks between the direct inversion and Monte Carlo simulation for propagators for the massless case with $L=4$. All of the fits shown are from the direct inversion data. Left: propagator from the center point to all other spatial points on the same time slice. Right: propagator along the center axis of the cylinder to all other temporal center points.

Figure 6

Evidence for a second-order phase transition for bulk ${\varphi }^4$ theory in the Euclidean ${\mathrm{AdS}}_3$ cylinder at ${\lambda }_0=1$. Left: the magnetic susceptibility $\chi$. Right: the fourth-order Binder cumulant $U_4$. We have scaled $\chi$ and $T$ by the lattice volume raised to an exponent, with the exponents chosen so that the data collapses onto a single curve.