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Abstract
We reconsider the criticality of the Ising model on two-dimensional dynamical triangulations based on the N-by-N hermitian two-matrix model with the introduction of a loop-counting parameter and linear terms in the potential. We show that in the large-N limit even though the Ising model is classical, the critical temperature can reach absolute zero by tuning the loop-counting parameter, and the corresponding continuum theory turns out to be the quantised theory of neither pure gravity nor gravity coupled to conformal matter with central charge being 1/2.
Highlights
Two-dimensional dynamical triangulations (2d DT), first introduced in [1,2,3,4,5,6], are a quite powerful method to regularize two-dimensional Euclidean quantum gravity (Liouville quantum gravity) coupled to conformal matter with central charge less than or equal to one, or equivalently world sheets of noncritical string theories embedded in dimensions less than or equal to one
We have shown that the conventional nonzero critical temperature of the Ising model dressed by quantum gravity can reach absolute zero as θ → 0
We have identified the continuum theory defined around the critical end point of the critical line with c ≠ 0, which can be described by the nontrivial continuum two-matrix model (5.19)
Summary
Two-dimensional dynamical triangulations (2d DT), first introduced in [1,2,3,4,5,6], are a quite powerful method to regularize two-dimensional Euclidean quantum gravity (Liouville quantum gravity) coupled to conformal matter with central charge less than or equal to one, or equivalently world sheets of noncritical string theories embedded in dimensions less than or equal to one (see, e.g., [7,8]). Having the argument above in mind, it would be expected that if one defines the Ising model on 2d DT based on the N × N Hermitian two-matrix model with the introduction of the loop-counting parameter and linear terms, one can reduce the known conventional critical temperature of the Ising model on 2d DT to absolute zero by tuning the loop-counting parameter in the direction of reducing loops because the Ising model on connected tree graphs called branched polymers can be critical only at the zero temperature [30] Investigating this possibility is quite interesting since the critical point obtained in this manner might be related to a quantum critical point. We pursue this idea and as expected we find that in the large-N limit the critical temperature of the Ising model dressed by quantum gravity can reach absolute zero by tuning the loopcounting parameter; we identify the corresponding continuum theory around absolute zero, which is different from two-dimensional Euclidean quantum gravity coupled to conformal matter with central charge being 1=2.
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