`Triviality' of universal relations for disordered systems

The universal relations for spin-$1\mathrm{/}2$ fermions with contact interaction in the presence of quenched disorder are discussed. The disorder is modeled by a random external potential with the Gaussian distribution and $\delta$-like two-point correlation function. Utilizing simple scaling arguments, the renormalizability of the theory, and the Hellmann-Feynman theorem we identified the large-momentum tail of particle distribution and analog of Tan’s energy relation for many-fermion systems with disorder in arbitrary dimension $d\ge 2$. It is shown that the energy-pressure relation manifests a kind of scale anomaly in two and three dimensions.