Symmetry breaking in tensor models

In this paper we analyze a quartic tensor model with one interaction for a tensor of arbitrary rank. This model has a critical point where a continuous limit of infinitely refined random geometries is reached. We show that the critical point corresponds to a phase transition in the tensor model associated to a breaking of the unitary symmetry. We analyze the model in the two phases and prove that, in a double scaling limit, the symmetric phase corresponds to a theory of infinitely refined random surfaces, while the broken phase corresponds to a theory of infinitely refined random nodal surfaces. At leading order in the double scaling limit planar surfaces dominate in the symmetric phase, and planar nodal surfaces dominate in the broken phase.

Figure 1

An example of a graph generated by the effective action in Eq. (34). Here the two dashed edges represent two $+-$ vertices of the types $\mathrm{Tr}[M_{+}^2]\mathrm{Tr}[M_{-}^2]$ and $\mathrm{Tr}[M_{+}^2]\mathrm{Tr}[M_{-}]$.

Figure 2

Examples of graphs dominating the double scaling regime. Left: A graph dominating the symmetric phase. Right: A graph dominating the broken phase.