Holography for Ising spins on the hyperbolic plane

Motivated by the $\mathrm{AdS}/\mathrm{CFT}$ correspondence, we use Monte Carlo simulation to investigate the Ising model formulated on tessellations of the two-dimensional hyperbolic disk. We focus in particular on the behavior of boundary-boundary correlators, which exhibit power-law scaling both below and above the bulk critical temperature indicating scale invariance of the boundary theory at any temperature. This conclusion is strengthened by a finite-size scaling analysis of the boundary susceptibility which yields a scaling exponent consistent with the scaling dimension extracted from the boundary correlation function. This observation provides evidence that the connection between continuum boundary conformal symmetry and isometries of the bulk hyperbolic space survives for simple interacting field theories even when the bulk is approximated by a discrete tessellation.

Figure 1

A representation of hyperbolic space: the $\{3,7\}$ Poincaré disk. The edge length and area of the triangle is constant throughout the hyperbolic lattice.

Figure 2

Markov Chain Monte Carlo simulation is conducted for several lattices with a total number of lattice sites $N_0$. The bulk magnetic susceptibility showed here is computed using just the $N_{\mathrm{bulk}}=591$ spins in the $n_{\text{inner}}=10$ innermost layers in each case. Additional spins are added layer by layer and correspond to different $N_0$.

Figure 3

The peak in the bulk susceptibility vs $\ell =\log (N_{\mathrm{bulk}})$ in log-log coordinates. The scaling exponent ${\gamma }_{\mathrm{B}}/{\nu }_{\mathrm{B}}$ corresponds to the slope of the fitted line.

Figure 4

The boundary susceptibility plotted against bulk-temperature for 10- to 13-layered Poincaré disks with increasing numbers of boundary spins $N_{\partial }$.

Figure 5

Logarithm of the boundary susceptibility (${\chi }_{\partial }$) is plotted against logarithm of the total number of boundary points ($N_{\partial }$) for $T=$ (a) 2.7, (b) 3.0, (c) 3.4. Scaling exponents of the boundary susceptibility ($\gamma /\nu$) are computed from the linear fit of the data.

Figure 6

The boundary-boundary correlation function at $T=$ (a) 2.7, (b) 3.0, (c) 3.4, plotted against boundary distance squared $r^2\sim (1-\cos \theta )$. Results shown here are from the analysis of a 12 layered Poincaré disk with boundary length $N_{\partial }=1062$. ${\chi }^2$ per degree of freedom and the p-value of the fits are noted in the figures.

Figure 7

The scaling exponent of the boundary spin operator computed from fits to the boundary two-point correlator, denoted by $\mathrm{\Delta }$, and from the finite-size scaling of the boundary susceptibility, denoted by $(1-\gamma /\nu )/2$.

Figure 8

(a) Bulk magnetization, (b) internal energy, and (c) heat capacity are computed from $N_{\mathrm{bulk}}=591$ spins on 10 innermost layers of the tessellation as additional outer layers are added up to a maximum of $N_{\text{total}}=3495$.

Figure 9

The boundary-boundary correlation function of the dual spin variable ($C=⟨{\sigma }_0{\sigma }_r⟩$) at $\tilde{T}=$ (a) 1.1, (b) 1.2, and (c) 1.3 plotted against boundary distance squared $r^2\sim (1-\cos \theta )$. Results shown here are from the analysis of a six-layered $\{7,3\}$ Poincaré disk with boundary length $N_{\partial }=3647$.

Figure 10

Scaling exponent of the dual boundary-spin operator ($\tilde{\mathrm{\Delta }}$) computed from the fits of the boundary correlator.

Figure 11

(a) Scaling exponents at direct lattice $\mathrm{\Delta }$ vs. dual inverse temperature ($\tilde{\beta }$), and (b) scaling exponent at dual lattice ($\tilde{\beta }$) vs. inverse temperature at direct lattice ($\beta$). Linear fits at high temperature are shown with the extracted slope denoted in the figure.