Exact Relations for a Strongly Interacting Fermi Gas from the Operator Product Expansion

The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like $1/k^4$ at large momentum $k$, as pointed out by Tan. He used novel methods to derive exact relations between the coefficient of the tail in the momentum distribution and various other properties of the system. We present simple derivations of these relations using the operator product expansion for quantum fields. We identify the coefficient as the integral over space of the expectation value of a local operator that measures the density of pairs.

Figure 1

Integral equation for the scattering of a pair of fermions. The blob represents the amplitude $i\mathcal{A}(E)$. The solid dot represents the interaction vertex $-ig(\Lambda )$.

Figure 2

Diagrams for matrix elements of the operators (a) ${\psi }_{\sigma }^{†}(-\frac{1}{2}\mathit{r}){\psi }_{\sigma }(+\frac{1}{2}\mathit{r})$, (b) ${\psi }_{\sigma }^{†}{\psi }_{\sigma }(0)$, (c) ${\psi }_1^{†}{\psi }_2^{†}{\psi }_1{\psi }_2^{(\Lambda )}(0)$, and (d) ${\psi }_1^{†}{\psi }_1(-\frac{1}{2}\mathit{r}){\psi }_2^{†}{\psi }_2(+\frac{1}{2}\mathit{r})$ between scattering states consisting of a pair of fermions. The open dots represent the operators. For each diagram, there are three other diagrams in which there is no scattering of the two incoming fermions or the two outgoing fermions or both.