Duality analysis in symmetric group and its application to random tensor network model

The Ising model is the simplest to describe many-body effects in classical statistical mechanics. Duality analysis leads to a critical point under several assumptions. The Ising model itself has $Z(2)$ symmetry. The basis of the duality analysis is a nontrivial relationship between low and high-temperature expansions. However, the discrete Fourier transformation finds the hidden relationship automatically. The duality analysis can be naturally generalized into the case with the degrees of freedom with $Z(q)$ symmetry and random spin systems. We further obtain the duality in a series of permutation models in the present study by considering the symmetric group $S_q$ and its Fourier transformation. The permutation model in the symmetric group is closely related to the random quantum circuits and random tensor network model, often discussed in the context of quantum computing and the holographic principle, a property of string theories and quantum gravity. We provide a systematic way by our duality analysis to analyze the phase transition in these models.