High-temperature expansion for interacting fermions

We present a general method for the high-temperature expansion of the self-energy of interacting particles. Although the method is valid for fermions and bosons, we illustrate it for spin-one-half fermions interacting via a zero range potential, in the Bose-Einstein-condensate–Bardeen-Cooper-Schrieffer (BEC-BCS) crossover. The small parameter of the expansion is the fugacity $z$. Our results include terms of order $z$ and $z^2$, which take into account, respectively, two- and three-body correlations. We give results for the high-temperature expansion of Tan's contact at order $z^3$ in the whole BEC-BCS crossover. We apply our method to calculate the spectral function at the unitary limit. We find structures that are different from those discussed in previous approaches, which included only two-body correlations. This shows that including three-body correlations can play an important role in the structures of the spectral function.

Figure 1

Lowest order diagram for the self-energy.

Figure 2

Spectral functions $A(k,\omega )$ versus $\omega +\mu$ at $T=T_F$. (a) $1/\left(k_{F}a\right)=1$ (BEC limit), (b) $1/\left(k_{F}a\right)=0$ (unitary limit), and (c) $1/\left(k_{F}a\right)=-1$ (BCS limit). In each panel, three values of the wave vectors are shown: $k/k_F=0$ (black solid line), $k/k_F=1$ (red dashed line), and $k/k_F=2$ (blue dash-dotted line).

Figure 3

The occupied spectral function ${\varepsilon }_F{A}_{-}(k,\omega )$ in a typical high wave vector $k$ limit. $T/T_F=1$ and $k/k_F=10$. (a) $1/\left(k_{F}a\right)=1$ (BEC regime), (b) $1/\left(k_{F}a\right)=0$ (unitary limit), (c) $1/\left(k_{F}a\right)=-1$ (BCS regime). The contributions with the self-energy of order $z$ is much smaller (blue continuous line) than the contribution of order $z^2$ (red dashed line).

Figure 4

Momentum distribution ${(k/k_F)}^4{n}_k$ for $T=T_F$. (a) $1/\left(k_{F}a\right)=1$ (red solid line), $1/\left(k_{F}a\right)=-1$ (blue dashed line). (b) $1/\left(k_{F}a\right)=0$. The asymptotic values, given by Tan's contact at order $z^2$ [see Eq. (17)] are shown as horizontal long dashed lines. For $1/\left(k_{F}a\right)=0$, the momentum distribution calculated with the term of order $z^2$ in the self-energy converges to Tan's contact (red full line). In contrast, without the term of order $z^2$ in the self-energy (blue dashed line), one does not recover Tan's contact.

Figure 5

Diagrammatic representation of $\mathrm{\Gamma }(\mathbf{P},{\beta }^{-})$.

Figure 6

The two-particle vertex $\mathrm{\Gamma }(\mathbf{P},{\beta }^{-})$ at lowest order in the fugacity $z$.

Figure 7

The two diagrams contributing to ${\mathrm{\Gamma }}^{\left(3\right)}(\mathbf{P},{\beta }^{-})$, and hence to $c_3$, the virial expansion term of Tan's contact coefficient of order $z^3$ [see Eq. (22)].

Figure 8

The coefficients $c_2$ (continuous line) and $c_3$ (dashed line) as defined in Eq. (22) for the virial expansion of Tan's contact, in the whole BEC-BCS crossover.

Figure 9

The Feynman diagram contributing to ${\mathrm{\Sigma }}^{\left(2\right)}$ containing one $G^{(0,2)}$ line (doubly slashed fermionic line).

Figure 10

The six diagrams contributing to ${\mathrm{\Sigma }}^{\left(2\right)}(k,{\tau }^{\prime \prime }-{\tau }^{\prime })$, including the three-body problem.

Figure 11

The imaginary part of the self-energy at order $z^3$ (see text) for $k{\mathrm{\Lambda }}_T=20\sqrt{\pi }$.

Figure 12

(a) Imaginary part of ${\mathrm{\Sigma }}^{(2,2,a)}(k,\omega )$ for $k{\mathrm{\Lambda }}_T=2\sqrt{\pi }$ and $z=0.3$. (b) Imaginary part of ${\mathrm{\Sigma }}^{\left(2\right)}(k,\omega )$, the second-order self-energy, for $k{\mathrm{\Lambda }}_T=2\sqrt{\pi }$ and $z=0.3$.

Figure 13

Spectral function at second order (blue solid line), compared to the lowest order ones (red dashed line). The fugacity is $z=0.3 \left(T/T_F=1.60\right)$. (a) $k{\mathrm{\Lambda }}_T=\sqrt{\pi }/5$, (b) $k{\mathrm{\Lambda }}_T=2\sqrt{\pi }$, and (c) $k{\mathrm{\Lambda }}_T=4\sqrt{\pi }$.

Figure 14

Unfolding trick. The slashed line $G^{(0,1)}({\tau }_2-{\tau }_1)$ can be drawn as the product of two unslashed $G^{(0,0)}$ lines. (a) ${\tau }_2>{\tau }_1$; (b) ${\tau }_2<{\tau }_1$.

Figure 15

The two diagrams of Figs. 7 and 7, drawn in the unfolded way.

Figure 16

Diagrams of Figs. 10–10 written in the unfolded way; (a) is for ${\mathrm{\Sigma }}^{2,2,d}$, (b) is for ${\mathrm{\Sigma }}^{2,2,e}$, and (c) is for ${\mathrm{\Sigma }}^{2,2,f}$. (d) Shows how they can be factorized.

Figure 17

The diagram considered in Appendix pp7 for ${\tau }^{\prime \prime }<{\tau }^{\prime }$.