Shaken, but not stirred – Potts model coupled to quantum gravity

We investigate the critical behaviour of both matter and geometry of the threestate Potts model coupled to two-dimensional Lorentzian quantum gravity in the framework of causal dynamical triangulations. Contrary to what general arguments on the effects of disorder suggest, we find strong numerical evidence that the critical exponents of the matter are not changed under the influence of quantum fluctuations in the geometry, compared to their values on fixed, regular lattices. This lends further support to previous findings that quantum gravity models based on causal dynamical triangulations are in many ways better behaved than their Euclidean counterparts. 1 Matter and geometry in two dimensions A common difficulty for models of nonperturbative quantum gravity, which attempt to describe a Planckian regime of quantum-fluctuating and strongly coupled degrees of freedom, is to reproduce aspects of the classical theory of general relativity in a suitable large-scale limit. Assuming that one has a quantum model which is sufficiently complete to be considered a candidate for a quantum gravity theory, it will by construction not be given in terms of small metric fluctuations around some classical spacetime geometry. It is then a nontrivial step to show that classical gravity does indeed emerge on larger scales, and to elucidate the mechanism by which this happens. One way to probe the properties of (quantum) geometry is by coupling matter to the gravitational system and observing its behaviour as a function of scale. A necessary condition for the existence of a good classical limit is that on sufficiently large scales and for sufficiently weak gravity the matter should behave like on a fixed, classical background geometry. That this should happen is by no means obvious, if one starts from a Hilbert space or a path integral encoding nonperturbative Planck-scale excitations. The latter may be too numerous or violent to coalesce into a well-behaved macroscopic, four-dimensional spacetime. This kind of pathological behaviour is not just an abstract possibility, but has been found in Euclidean models of quantum gravity [1], and exhibits a certain genericity. The set of spacetime geometries underlying the approach of Causal Dynamical Triangulations (CDT) to quantum gravity seems to strike a balance between generating large quantum excitations on small scales [2], and managing to reproduce features of classical geometry on large scales [3, 4]. Something similar is apparently true for the two-dimensional quantum gravity theory derived from a CDT formulation [5, 6, 7, 8]. Although there is no classical limit per se for the purely gravitational degrees of freedom in this case – the Einstein-Hilbert action is trivial – one can still ask whether the dynamics of any additional matter fields is changed on a quantum-fluctuating “background geometry”, compared with the same matter on a fixed background. For the much-studied case of two-dimensional Euclidean (“Liouville”) quantum gravity, obtained from Euclidean dynamical triangulations (EDT) or their equivalent matrix models [9], one indeed finds that gravity alters the matter behaviour nontrivially. Namely, for matter with a central charge c, 0 < c ≤ 1, its critical exponents appear as “dressed” versions of their fixed-background counterparts. While these models present interesting examples of strongly interacting “gravity”-matter systems, their existence is restricted by the so-called c = 1-barrier for the central charge, beyond which no consistent unitary matter models have been found. A closer inspection of the geometric that is, the “histories” contributing to the “sum over histories”, a.k.a. the gravitational path integral