The sinh-Gordon model beyond the self dual point and the freezing transition in disordered systems

The S-matrix of the well-studied sinh-Gordon model possesses a remarkable strong/weak coupling duality $b \to 1/b$. Since there is no understanding nor evidence for such a duality based on the quantum action of the model, it should be questioned whether the properties of the model for $b>1$ are simply obtained by analytic continuation of the weak coupling regime $0<b<1$. In this article we assert that the answer is no, and we develop a concrete and specific proposal for the properties when $b>1$. Namely, we propose that in this region one needs to introduce a background charge $Q_\infty = b + 1/b -2$ which differs from the Liouville background charge by the shift of $-2$. We propose that in this regime the model has non-trivial massless renormalization group flows between two different conformal field theories. This is in contrast to the weak coupling regime which is a theory of a single massive particle. Evidence for our proposal comes from higher order beta functions. We show how our proposal correctly reproduces the freezing transitions in the multi-fractal exponents of a Dirac fermion in $2+1$ dimensions in a random magnetic field, which provides a strong check since such transitions have several detailed features. We also point out a connection between a semi-classical version of this transition and the so-called Manning condensation phenomena in polyelectrolyte physics.