Crossover duality for the quantum long-range Ising chain: perturbation theory around the short-range endpoint and a DMRG test

We propose a weakly coupled description of the long-range transverse-field Ising chain near its short-range crossover at $s_*=7/4$, where the couplings decay as $r^{-1-s}$. The Ising conformal field theory is coupled to an instantaneous Gaussian field through $σχ$, extending the classical long-range Ising duality to quantum chains. The equal-time interaction generates a stress-tensor deformation that causes the velocity to run and renormalizes the dynamical critical exponent $z$. Conformal perturbation theory gives $z=1-0.720\,δ+O(δ^2)$, $1/ν=(1+z)/2+O(δ^2)$, and $ω=δ+O(δ^2)$, with $δ=s_*-s$. DMRG calculations on periodic chains of up to $128$ sites provide a consistency test using theoretical inputs for $1/ν$ and $ω$. After accounting for slow finite-size corrections, static and dynamical estimates of $z$ agree, supporting the crossover expansion at $s=1.5$ and a two-sided Padé interpolation at $s=1$, $1.25$, and $1.5$.