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The Fishnet as Anti-ferromagnetic Phase of Worldsheet Ising Spins

We identify the strong coupling fishnet diagram with a certain Ising spin configuration in the lightcone worldsheet description of planar TrΦ^3 field theory. Then, using a mean field formalism, we take the remaining planar diagrams into account in an average way. The fishnet spin configuration requires two mean fields ϕ,ϕ' where the fishnet diagram is the case ϕ=1,ϕ'=0. For general values of these fields, the system is then approximated as a light-cone quantized string with a field dependent effective string tension T_{eff}(ϕ,ϕ'). We also calculate the worldsheet energy density E(ϕ,ϕ'), and find the field values that minimize it in the presence of a transverse space infra-red cutoff ε>0. The criterion for string formation is that the tension in this minimum energy state remains non-zero as εgoes to 0. In the most simple-minded implementation of the mean field method, we find that the tension vanishes for weak and moderate coupling, but for very large coupling stays non-zero. A more elaborate treatment, taking temporal correlations into account, removes this ``phase transition'' and the string tension of the minimum energy state vanishes for all values of the coupling when εgoes to 0. Our mean field analysis thus suggests that the ``fishnet phase'' of TrΦ^3 theory is unstable, and there is no string formation for any value of the coupling. This is probably a reasonable outcome given the instability of the underlying theory.