Current response, structure factor and hydrodynamic quantities of a two- and three-dimensional Fermi gas from the operator-product expansion

We apply the operator-product expansion to determine the asymptotic form of the current response of a Fermi gas in two and three dimensions. The leading-order term away from the one-particle peak is proportional to a quantity known as the contact, the coefficient of which is determined exactly. We also calculate the dynamic structure factor and the high-frequency tails of the spectral viscosities as a function of the scattering length. Our results are used to derive certain sum rules for the viscosities.

Figure 1

(a) Feynman diagram for the two-particle scattering amplitude. (b) Current insertion on a fermion line.

Figure 2

Diagrams that contribute to the expectation value of $A_{ij}^{\sigma {\sigma }^{\prime }}(\mathbf{q},\omega )$ [Eq. (7)] between a one-particle state with momentum $\mathbf{p}$ and energy $p_0$.

Figure 3

Insertion of a bilinear operator on an internal line. The insertion on an external line matches the insertion of two current operators on the same line and need not be considered.

Figure 4

Diagrams that contribute to the matrix element of $A_{ij}^{\sigma {\sigma }^{\prime }}(\mathbf{q},\omega )$ [Eq. (7)] between the two-particle state.

Figure 5

3D: Transverse response function at (a) fixed energy and (b) fixed momentum as a function of the variable $x=q^2/2m\omega$ and $x^{-1}$, respectively. The continuous blue line denotes the spin-symmetric and the red dashed line the spin-antisymmetric response.

Figure 6

3D: Longitudinal response function at (a) fixed energy and (b) fixed momentum as a function of the variable $x=q^2/2m\omega$ and $x^{-1}$, respectively. The continuous blue line denotes the spin-symmetric and the red dashed line the spin-antisymmetric response. The spin-symmetric response is evaluated at unitarity.

Figure 7

3D: (a) Longitudinal response function as a function of $x$ and inverse scattering length $a_{3\mathrm{D}}^{-1}/\sqrt{m\omega }$. (b) Detail of (a) at unitarity. The peak at unitarity is due to the strong interaction. The thick blue line denotes the response at unitarity and corresponds to the thick blue line in Fig. 6a.

Figure 8

2D: Transverse response function at (a) fixed energy and (b) fixed momentum as a function of the variable $x=q^2/2m\omega$ and $x^{-1}$, respectively. The continuous blue line denotes the spin-symmetric and the red dashed line the spin-antisymmetric response.

Figure 9

2D: Longitudinal response function at (a) fixed energy and (b) fixed momentum as a function of the variable $x=q^2/2m\omega$ and $x^{-1}$, respectively. The continuous blue line denotes the spin-symmetric and the red dashed line the spin-antisymmetric response. The spin-symmetric response is evaluated at $a_{2\mathrm{D}}\sqrt{m\omega }=5$ and $a_{2\mathrm{D}}q=5$.

Figure 10

2D: Longitudinal response function as a function of $x$ and the 2D scattering length $a_{2\mathrm{D}}^{-1}/\sqrt{m\omega }$. There is a peak at $(a_{2\mathrm{D}}^{-1}=0,x=2)$ due to the enhanced interaction in this limit. The thick blue line denotes the response at $a_{2\mathrm{D}}^{-1}/\sqrt{m\omega }=0.2$ and corresponds to the thick blue line in Fig. 9a.