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

Abstract

The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like 1/k4 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.

Tan relations in one dimension

“…Its leading non-analytic contribution defines the contact and gives rise to Tan relations that are closely related to those in the 3D case [5,7,8] and also to those for 1D Bose gases [1]. In the following we will show that the contact also arises as a non-analytic contribution in the OPE of the two-particle density matrix which gives additional relations for the pair distribution function and the related static structure factor.…”

“…The contribution coming from these operators is analytic in x and is just equivalent to a Taylor expansion of the one-particle density matrix around x = 0. Note the slight difference compared to the matching performed in [8], because we have put the x-dependence only in the argument of the ψ σ -field. Thus, there are no analytic contributions involving derivatives of ψ † σ in our case.…”

“…, in complete analogy to the three-dimensional case [8]. Explicitly, the Wilson-OPE for the one-particle density matrix up to order x 3 reads…”

“…In analogy to the three-dimensional case there are, however, also non-analytic contributions in the short distance expansion of the one-particle density matrix. The leading term is, in fact, again connected with the operator ψ † ↑ ψ † ↓ ψ ↓ ψ ↑ that also appears in 3D [8]. To show this, we calculate the matrix element of the one-particle density matrix between the two-particle scattering states.…”

“…Except for this constraint, one may choose the simplest possible states one can think of. Our explicit calculations follow closely the method used by Braaten and Platter in the three-dimensional case [8,27]. For the coefficients of one-particle operators such as ψ † σ ψ σ , we will choose one-particle plane wave states p ′ | , |p containing one particle of the species σ with momentum p ′ or p respectively.…”