Tan contact and universal high momentum behavior of the fermion propagator in the BCS-BEC crossover

We derive the universal high momentum factorization of the fermion self-energy in the BCS-BEC crossover of ultracold atoms using nonperturbative quantum field theoretical methods. This property is then employed to compute the Tan contact as a function of interaction strength, temperature, and chemical potential or density. We clarify the mechanism of the factorization from an analysis of the self-consistent Schwinger–Dyson equation for the fermion propagator and compute the perturbative contact on the BCS and Bose–Einstein condensate (BEC) sides within this framework. A functional renormalization group approach is then put forward, which makes it possible to determine the contact over the whole crossover and, in particular, for the unitary Fermi gas. We present numerical results from an implementation of the renormalization group equations within a basic truncation scheme.

Figure 1

Schwinger–Dyson equation for the inverse fermion propagator. A single line corresponds to a classical propagator and a double line denotes a full propagator. Fermions and bosons are represented by solid and dashed lines, respectively. In the loop integral we have one fully dressed Feshbach coupling and a microscopic one. The latter is momentum independent.

Figure 2

RG-scale dependence of the flowing contact $C_k$ at unitarity $a^{-1}=0$. We have $k=\Lambda e^t$ such that $t=0$ corresponds to the ultraviolet and $t\to -\infty$ to the infrared. We observe that the contact is unaffected by fluctuations of ultraviolet modes and it starts to build up on the many-body scales of the system, which are set here by the chemical potential and temperature corresponding to $t_{\mu }=\ln ({\mu }^{1/2}/\Lambda )=-6.9$ and $t_T\simeq t_{\mu }$. Obviously, all curves saturate at a certain value of $t$ and we can read off the physical value at $k=0$.

Figure 3

Zero temperature contact on the BEC side of the crossover. The many-body chemical potential is defined as ${\mu }_{\mathrm{mb}}=\mu -{\varepsilon }_{\mathrm{B}}/2$ and thus a positive quantity. [See, for instance, Eq. (E5) in the context of a weakly interacting Bose gas.] The FRG treatment captures the Lee–Huang–Yang (LHY) correction; see Eq. (E9). Mean-field (MF) theory is shown by the dashed curve.

Figure 4

The contact close to resonance as a function of the scattering length for both $T=0$ and $T=T_{\mathrm{c}}(a,\mu )$. The labels of the axes are analogous to Fig. 3. At unitarity we obtain $C(T=0)/{\mu }^2=0.34$ and $C(T=T_{\mathrm{c}})/{\mu }^2=0.24$. Far on the BCS side our present truncation becomes inappropriate, as is discussed in Appendix . For better visibility we show the asymptotic BCS value up to ${\left(\sqrt{\mu }a\right)}^{-1}=-0.5$, which is already beyond the applicability of BCS theory.

Figure 5

The contact of the unitary Fermi gas normalized by $k_{\mathrm{F}}^4$ with Fermi momentum $k_{\mathrm{F}}={\left(3{\pi }^{2}n\right)}^{1/3}$. We compare predictions from functional renormalization group (FRG, this work, $T_{\mathrm{c}}/T_{\mathrm{F}}=0.276$), non-self-consistent $T$-matrix theory ($G_0{G}_0$, [31], $T_{\mathrm{c}}/T_{\mathrm{F}}=0.242$), Gaussian pair fluctuations and Nozières–Schmitt-Rink theory (GPF/NSR, [33], $T_{\mathrm{c}}/T_{\mathrm{F}}=0.235$), self-consistent $T$-matrix theory ($GG$, [32], $T_{\mathrm{c}}/T_{\mathrm{F}}=0.15$), and quantum Monte Carlo calculations (QMC, [29], $T_{\mathrm{c}}/T_{\mathrm{F}}=0.15$) for lattice sizes $N_x=12,14$. In this list, the brackets indicate the label in the plot, the corresponding reference and the chosen value for the critical temperature. For the experimental data of Ref. [25] we employed $T_{\mathrm{c}}/T_{\mathrm{F}}=0.16$, which suffices here to obtain a qualitative comparison of the data.

Figure 6

The blue solid line shows the temperature dependence of the contact normalized by the chemical potential for the unitary Fermi gas. We observe a decrease of $C/{\mu }^2$ as we approach the critical temperature from below, resulting in a sharp dip at $T_{\mathrm{c}}$. The function is monotonic for $T/T_{\mathrm{c}}\gtrsim 1.75$. The green, red, and orange curves correspond to different contributions to the contact and are explained in Eqs. (45, 46, 47).

Figure 7

The asymptotic approach of the contact for the unitary Fermi gas at the critical temperature. We plot the real part of $P_{\psi ,\mathrm{cl}}(-P){\Sigma }_{\psi }\left(P\right)$ with $P_{\psi ,\mathrm{cl}}(-P)=-i{p}_0+{\vec{p}}^2-\mu$ and $p_0=\pi T$ (red dots). In accordance with formula (5), the factorization at large momenta leads to the approach of the constant value $4C$ (blue dashed line). For very low momenta our perturbative treatment of the fermion propagator becomes quantitatively less accurate, but still the deviations do not exceed 20$\%$ even for $p\to 0$. We restrict the self-energy here to the diagram in Fig. 11 where the external momentum $P$ appears in the fermion line and which is responsible for the contact term. This effectively adds the constant $\delta \mu$ to ${\Sigma }_{\psi }$ for large momenta.

Figure 8

The fermion self-energy ${\Sigma }_{\psi ,k=0}\left(P\right)$ at $a^{-1}=0$ and $T=T_{\mathrm{c}}$ computed from the first diagram in Fig. 11. We evaluate the function for $p_0=\pi T$ and find the large momentum behavior to be a reasonable approximation for both the real and imaginary parts even at low momenta. The asymptotic form (5) is shown by a dashed line. The self-energy vanishes for large momenta since we effectively added the constant $\delta \mu$ for $p\to \infty$ by neglecting the second diagram in Fig. 11.

Figure 9

Schwinger–Dyson equation for the inverse boson propagator. The notation is chosen as in Fig. 1.

Figure 10

Schwinger-Dyson equation for the inverse fermion propagator in a purely fermionic theory. Again, a single or double line denotes a classical or full propagator. The open (solid) dot corresponds to a classical (full) four-fermion vertex. We can combine the last two terms such that the correction is only given by the tadpole diagram but with a momentum-dependent effective vertex.

Figure 11

Flow equation for the fermion self-energy. The notation is chosen as in Fig. 1. The crossed circle indicates an insertion of ${\dot{R}}_k$ in the loop integral.