Evidence for a second-order transition?

4.6 Evidence for a second-order transition?

It sometimes seems to be assumed that if one were to find a continuum theory at a second-order phase transition, it would have flat Minkowski space as its ground state (in spite of the $S4$ -topology), and gravitonic spin-2 excitations. An alternative scenario with a constant-curvature sphere-metric has been put forward by de Bakker and Smit [90

, 92

Agishtein and Migdal [2 ] initially reported a hysteresis in the average curvature $⟨R ⟩(κ2)$ , indicating a first-order transition. However, subsequent authors [11 , 191 , 192 , 75 , 76 ] found numerical data not incompatible with the existence of a second-order transition, and also Agishtein and Migdal [1 ] retracted their original claim as a result of a closer examination of the fixed point region.

One tries to discriminate between a first- and second- (or higher-)order transition by looking at the Binder parameter [191 , 192 , 7 ], or scaling exponents $α$ governing the scaling behaviour $∼ |κc2 − κ2|α$ of suitable observables [1 , 192 ], or the peak height of susceptibilities as a function of the volume $N4$ [16 , 75 , 76 , 14 , 48 , 87 ]. Other scaling relations are discussed in [90 , 91 , 96 , 95 ]. However, since critical parameters are hard to measure, and it is difficult to estimate finite-size effects, none of the data can claim to be conclusive.

Some doubts were cast on the conjectured continuous nature of the phase transition by Bialas et al. [48 ], who found an unexpected two-peak structure in the distribution of nodes near the fixed point. This was strengthened further by data taken at 64k by de Bakker [87 ], with an even more pronounced double peak (see also [51 ]). Most likely previous simulations were simply too small to detect the true nature of the phase transition. Both Bialas et al. and de Bakker observed that the finite size scaling exponents extracted from the node susceptibility $χ 0$ grow with volume and may well reach the value 1 expected for a first-order transition as $V → ∞$ .

The origin of this behaviour was further elucidated by Bialas et al. [49 , 47 ] and Bialas and Burda [46 ], who found a simple mean-field model that reproduces qualitatively the phase structure of 4d dynamically triangulated quantum gravity. With an appropriate choice of local weights, this model has a condensed and a fluid phase, with a first-order transition in between. A similar behaviour was found by Catterall et al. [77 ], who made a related mean-field ansatz, with the local weights depending on the local entropies around the vertices.