Abstract
We use a dispersion relation in conjunction with the operator product expansion (OPE) to derive model independent sum rules for the dynamic structure functions of systems with large scattering lengths. We present an explicit sum rule for the structure functions that control the density and spin response of the many-body ground state. Our methods are general, and apply to either fermions or bosons which interact through two-body contact interactions with large scattering lengths. By employing a Borel transform of the OPE, the relevant integrals are weighted towards infrared frequencies, thus allowing for greater overlap low energy data. Similar sum rules can be derived for other response functions. The sum rules can be used to extract the contact parameter introduced by Tan, including universality violating corrections at finite scattering lengths.
Highlights
The theory of non-relativistic spin-1/2 fermions near the unitarity limit, described by the Lagrangian
Refs. [6, 10] showed that the operator product expansion (OPE) predicts that the asymptotic large frequency behavior of dynamic structure functions is controlled by the Tan contact parameter C
We will use the OPE in conjunction with a set of weighted (“Borel transformed”) dispersion relations to derive sum rules that probe the infrared properties of many particle systems whose dynamics is described by Eq (1)
Summary
Taking ground state matrix elements on both sides of Eq (3) (or ensemble averages at finite temperature T ), one finds that the asymptotic short distance or time properties of two-point Green’s functions, for instance the response functions of the theory, are dominated by the condensates Oα of the first few operators of lowest dimension ∆α. Observables calculated using Eq (1) are accurate up to corrections parametrized by powers of r0 2mω, where r0 is a length scale characterizing the range of the two-body potential (e.g., the van der Waals interaction length scale lvDW ) This means that our results for the OPE are valid in the range of energies μ. Where bα,i are numerical coefficients proportional to Oα , and pα,i are integer
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