Second- and first-order phase transitions in causal dynamical triangulations

Causal dynamical triangulations (CDT) is a proposal for a theory of quantum gravity, which implements a path-integral quantization of gravity as the continuum limit of a sum over piecewise flat spacetime geometries. We use Monte Carlo simulations to analyze the phase transition lines bordering the physically interesting de Sitter phase of the four-dimensional CDT model. Using a range of numerical criteria, we present strong evidence that the so-called A–C transition is first order, while the B–C transition is second order. The presence of a second-order transition may be related to an ultraviolet fixed point of quantum gravity and thus provide the key to probing physics at and possibly beyond the Planck scale.

Figure 1

Visualization of the phase diagram of CDT quantum gravity, based on actual measurements, illustrating the distinct volume profiles characterizing the three observed phases A, B and C.

Figure 2

Phase diagram of CDT quantum gravity [23]. Crosses represent actual measurements, dashed lines are extrapolations.

Figure 3

Monte Carlo time evolution (i) of $N_0$ at the A–C transition (top left) with associated histogram (bottom left), (ii) of $\mathrm{conj}(\Delta )$ at the B–C transition (top right) with associated histogram (bottom right).

Figure 4

Histograms of $N_0$ at the A–C transition for four different system sizes. The histograms have been normalized to obtain probability distributions $P$.

Figure 5

Measured A–C transition points ${\kappa }_0^c(N_4)$ at $\Delta =0.6$ for different system sizes $N_4$, together with a best fit to the inverse-volume expansion Eq. (8).

Figure 6

Dependence of the minimum $B_{N_0}^{\min }$ of the Binder cumulant $B_{N_0}$ on the system size $N_4$, at the A–C transition and for $\Delta =0.6$.

Figure 7

Histograms of $\mathrm{conj}(\Delta )$ at the B–C transition for three different system sizes. The histograms have been normalized to obtain probability distributions $P$.

Figure 8

Measured B–C transition points ${\Delta }^c(N_4)$ at ${\kappa }_0=2.2$ for different system sizes $N_4$, with a best fit to Eq. (10) to determine the shift exponent $\tilde{\nu }$.

Figure 9

Dependence of the minimum $B_{\mathrm{conj}(\Delta )}^{\min }$ of the Binder cumulant $B_{\mathrm{conj}(\Delta )}$ on the (inverse) system size $N_4^{-1}$, at the B–C transition and for ${\kappa }_0=2.2$. (Fit excludes the two points on the right).