The Ising model coupled to 2d orders

Abstract

In this article we make first steps in coupling matter to causal set theory in the path integral. We explore the case of the Ising model coupled to the 2d discrete Einstein Hilbert action, restricted to the 2d orders. We probe the phase diagram in terms of the Wick rotation parameter β and the Ising coupling j and find that the matter and the causal sets together give rise to an interesting phase structure. The couplings give rise to five different phases. The causal sets take on random or crystalline characteristics as described in [1] and the Ising model can be correlated or uncorrelated on the random orders and correlated, uncorrelated or anti-correlated on the crystalline orders. We find that at least one new phase transition arises, in which the Ising spins push the causal set into the crystalline phase.Causal set theory is a theory of discrete quantum gravity, in which space-time is encoded as a discrete partial order, called a causal set because the partial order relations correspond to causal relations [2] (see [3] for a relatively recent review). The aim of the theory is to quantise gravity by taking the path integral over a suitable class of causal sets.One approach to quantise causal set theory is to sum over the class of all causal sets. However, it is unclear if this approach will work, since the class of all partial orders is dominated by the Kleitmann Rothschild (KR) orders [4,5]. These are a particular class of three layer orders and it has been proven that in the N → ∞ limit any random partial order is almost certainly of KR type. In practice, simulations exploring the state space over all causal sets give rise to the expectation that the KR orders will start to dominate the path integral for N > 100 [6]. This entropic dominance of the KR orders could possibly be broken by weighting causal sets with an action that strongly suppresses this type of orders. It is not clear whether the current causal set action of choice, the Benincasa-Dowker (BD) action, which was introduced in [7] for 2 and 4 dimensions and later generalised to any dimension [8,9], can achieve this. However, recent work shows that it does suppress another class of pathological orders, the so called bilayer orders [10]. In addition, summing over all causal sets does not fix the dimension of the spaces summed over, this leads to the question which dimension to chose for the BD-action. In an ideal world, the choice of action might influence the dimension of space-time, so using a 2d action could create a path integral dominated by 2d causal sets; however, reality is rarely ideal.The 2d orders are a class of causal sets that has proven very useful for study through computer simulations. While the ultimate goal will be to simulate a path integral over a wider class of causal sets, the 2d orders are an interesting model that gives us a fixed dimension and comes with embedding coordinates, which are useful to visualise the system. Another interesting property of the 2d orders is that they entropically favour states that correspond to 2d f...

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