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Article

Stochastic processes with $Z_N$ symmetry and complex Virasoro representations. The partition functions.

Journal of Physics A: Mathematical and Theoretical. 2014. Vol. 47. No. 46. P. 462001.
Francisco C. Alcaraz, Pavel Pyatov, Rittenberg V.

In a previous letter (Alcaraz F C et al 2014 J. Phys. A: Math. Theor. 47 212003) we have presented numerical evidence that a Hamiltonian expressed in terms of the generators of the periodic Temperley–Lieb algebra has, in the finite-size scaling limit, a spectrum given by representations of the Virasoro algebra with complex highest weights. This Hamiltonian defines a stochastic process with a ZN symmetry. We give here analytical expressions for the partition functions for this system which confirm the numerics. For N even, the Hamiltonian has a symmetry which makes the spectrum doubly degenerate leading to two independent stochastic processes. The existence of a complex spectrum leads to an oscillating approach to the stationary state. This phenomenon is illustrated by an example.