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The Parity-Time $\mathcal{PT}$-symmetric non-Hermitian Sine-Gordon(nhSG)model derived from the nonequilibrium spin-boson model. We have derived theKeldysh rotation for spin operators, from which the SG model can be derived. Weperform renormalization group calculations in the Keldysh fields and comparethe fixed points and the flow of the effective couplings of nonequilibrium andnon-Hermitian models. Also, we explicitly find the self-energies and comparethe two methods to understand the regimes where $\mathcal{PT}$ symmetrypreserved regime, and nonequilibrium regimes persist.RG flow of couplings ofnonequilibrium model and non-hermitian model both capture the standardBerezinskii-Kosterlitz-Thouless(BKT) physics in a strong coupling regime.

Derivation of $\mathcal{PT}$-symmetric Sine-Gordon model and its relevance to non-equilibrium