Running scales in causal dynamical triangulations

The search for typical length scales, eventually diverging at a critical point, is a major goal for lattice approaches looking for a continuum theory of quantum gravity. Within the simplicial Monte Carlo approach known as causal dynamical triangulations, we study the spectrum of the Laplace operator to infer the geometrical properties of triangulations. In some phase of the theory a discrete set of length scales emerges, persisting in the infinite volume limit; such scales run as a function of the bare couplings, consistently with a critical behavior around a possible second order transition.

Figure 1

Average eigenvalues of the LB operator on spatial slices versus $1/V_S$ in phase $C_{ds}$ ($k_0=0.75$, $\mathrm{\Delta }=0.7$), mostly for $V_{S,\mathrm{tot}}=40\mathrm{K}$. We also report best fits to Eq. (2).

Figure 2

Average eigenvalues versus $1/V_S$ in phase $B$ ($k_0=1.0$, $\mathrm{\Delta }=-0.2$). Data points at $1/V_S=0$ have been extrapolated according to Eq. (3).

Figure 3

Scatter plot of ${\lambda }_n$ versus $n/V_S$ for slices with $V_S>200$ from sample configurations at $k_0=0.75$ and three values of $\mathrm{\Delta }$ in the $C_b$ phase.

Figure 4

$⟨{\lambda }_n⟩(V_S)$ for $dS$-like slices in the $C_b$ phase ($k_0=0.75$, $\mathrm{\Delta }=0.4$, $V_{S,\mathrm{tot}}=40\mathrm{K}$). We report also best fits to Eq. (2).

Figure 5

As in Fig. 4, for $B$-like slices. We report also some $V_S\to \infty$ extrapolations with $1/V_S$ corrections (see text).

Figure 6

Sketch of the phase diagram of CDT with $S^3$ slice topology. The isolated points correspond to the simulations performed (total volume not shown); The points with error bars are the quantitative estimates for the corresponding critical values of ${\mathrm{\Delta }}_c$ obtained in this study and reported in the text.

Figure 7

$⟨{\lambda }_n{⟩}_{\infty }$ in $B$-like slices as a function of $\mathrm{\Delta }$ for different values of $k_0$ and $n$, together with best fits to Eq. (4).