Non-hypercubic lattices

3.9 Non-hypercubic lattices

Simulations on lattices with irregular link geometry (still with $T 4$ -topology) have been performed by Beirl et al. [31 , 36

]. They were obtained by adding a few vertices of low coordination number to otherwise regular lattices. One finds that the average curvature $⟨R⟩$ increases from negative to positive values, even for $k < kc$ , as a result of the formation of spikes. In [36

], the averages $2 ⟨l⟩$ were monitored separately at the regular and the inserted vertices.

The dependence of $ℛ$ on $k$ is rather interesting: One observes two “critical” points, a smaller one $k1$ , where the extra vertices develop spikes, and a second one $k2$ where the remaining vertices follow. $ℛ$ undergoes a small jump at $k 1$ , and a larger one at $k 2$ . There is also a transition point to a phase with collapsed simplices at large negative $k$ , with a jump to large negative $ℛ$ . Additional transition points at negative $k$ were also found in simulations of the “compactified” Regge action $∑ S = b(− k Ab sinδb + λVb)$ (this action was discussed in [104 ]; see also [71 , 140 ]) and a $ℤ2$ -version of Regge gravity [33 ]. Correlation functions at those points were computed in [34 ] for short distances, but no evidence for long-range correlations was found.

The same authors studied the inclusion of the higher-derivative term (15 ) in [36 ]. On the regular lattice, their findings for $ℛ (k,a )$ confirmed those by Hamber, apart from the fact that they found stable expectation values even for positive $ℛ$ . Inserting irregular vertices pushes $ℛ$ to larger values and leads again to the appearance of an additional transition point.