Analytic bound on the generalized HMWC denominator

Establish analytically the bound on the quantity D_N(k_L)=\langle\{C_{k_L},\{C_{k_L}^*,H\}\}\rangle used in the generalized Hohenberg–Mermin–Wagner–Coleman argument for one-dimensional noncompact classical fractons, and determine the range of kernels, densities, and temperatures for which the bound is valid.

Background

The paper’s proof that the translation-symmetry-breaking density order parameter vanishes depends on the equilibrium estimate D_N(k_L)\leq C k_B T N k_L^2, with an N-independent constant C. The authors support this estimate numerically through finite-size scaling for a particular exponentially decaying kernel, density, and temperature, but do not derive it analytically or establish its general domain of validity.

An analytic proof would determine whether the generalized Hohenberg–Mermin–Wagner–Coleman conclusion applies robustly across the noncompact classical-fracton parameter space rather than only in the numerically tested regime.