Exponential decay of correlations at high temperature in $H^{2\mathrm{\mid }2n}$ nonlinear sigma models

We consider a family of nonlinear sigma models on ${\mathbb{Z}}^d$ whose target space is the hyperbolic super manifold $H^{2\mathrm{\mid }2n}$, $n>1$, introduced by Crawford as an extension of Zirnbauer’s $H^{2\mathrm{\mid }2}$ model for disordered systems. We prove exponential decay of the two-point correlation function in the high-temperature regime $\beta \le C{n}^{-1}$, with $C>0$ a universal constant, for any $n>1$ and any dimension $d\ge 1$, with mass $\log {\beta }^{-1}$. We also consider models with long-range interaction and prove fast decay in the same high-temperature regime. The proof is based on the reduction to a marginal fermionic theory and combines a high-temperature cluster expansion, exact combinatorics and bounds derived via Grassmann norms.