Universal relations for disordered systems

Abstract The universal relations for spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> 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 δ -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . It is shown that the energy-pressure relation manifests a kind of scale anomaly in two and three dimensions.