Uniform bound of the entanglement for the ground state of the one-dimensional quantum Ising model with non-homogeneous transverse field

We consider the ground state of the one-dimensional quantum Ising model with transverse field $h_x$ in one dimension depending on the site $x\in \mathbb{Z}$ in a finite volume Λ_{m}:=\{-m,-m+1,\ldots,m+L\}\. We make suitable assumptions on the regions where the field is small and prove that if the field is sufficiently large on the complementary set, then the entanglement of the interval ${\mathrm{\Lambda }}_0:=\{0,\mathrm{.}\mathrm{.},L\}$ relative to its complement ${\mathrm{\Lambda }}_m\mathrm{\backslash }{\mathrm{\Lambda }}_0$ is bounded uniformly in $m$ and $L$. The result applies in particular to periodic transverse fields. The bound is established by means of a suitable cluster expansion.