Non-scalar operators for the Potts model in arbitrary dimension

Romain Couvreur (1, 2) , Jesper Lykke Jacobsen (2, 1) , Romain Vasseur (3, 4)

1 LPTENS - Laboratoire de Physique Théorique de l'ENS [École Normale Supérieure]
2 IPHT - Institut de Physique Théorique - UMR CNRS 3681
3 Department of Physics [Berkeley]
4 Materials Science Division [LBNL Berkeley]

We investigate the operator content of the Q-state Potts model in arbitrary dimension, using the representation theory of the symmetric group. In particular we construct all possible tensors acting on N spins, corresponding to given symmetries under $S_Q$ and $S_N$, in terms of representations involving any Young diagram. These operators transform non-trivially under the group of spatial rotations, with a definite conformal spin. The two-point correlation functions are then computed, and their physical interpretation is given in terms of Fortuin-Kasteleyn clusters propagating between two neighbourhoods of each N spins. In two dimensions, we obtain analytically the critical exponent corresponding to each operator. In the simplest and physically most relevant cases, we confirm the values of the critical exponent and the conformal spin by numerical measurements, using both Monte Carlo simulations and transfer matrix diagonalisations. Our classification partially provides the structure of Jordan cells of the dilatation operator in arbitrary dimensions, which in turn gives rise to logarithmic correlation functions.

Domains: Statistical Mechanics [cond-mat.stat-mech]

Romain Couvreur, Jesper Lykke Jacobsen, Romain Vasseur. Non-scalar operators for the Potts model in arbitrary dimension. 2017. ⟨cea-01569421⟩