Efimov effect from functional renormalization

We apply a field-theoretic functional renormalization-group technique to the few-body (vacuum) physics of nonrelativistic atoms near a Feshbach resonance. Three systems are considered: one-component bosons with a U(1) symmetry, two-component fermions with a $\text{U}\left(1\right)\times \text{SU}\left(2\right)$ symmetry, and three-component fermions with a $\text{U}\left(1\right)\times \text{SU}\left(3\right)$ symmetry. We focus on the scale-invariant unitarity limit of an infinite scattering length. The exact solution for the two-body sector is consistent with the unitary fixed-point behavior of the considered systems. Nevertheless, the numerical three-body solution in the $s$-wave sector develops a limit cycle scaling in case of U(1) bosons and SU(3) fermions. The Efimov parameter for the one-component bosons and the three-component fermions is found to be $s\approx 1.006$, consistent with the result of Efimov.

Figure 1

Schematic graphical representation of the flow for the inverse diatom propagator $P_{\phi }$. Diatoms are denoted by dashed lines, and atoms by solid lines.

Figure 2

Kinematics of the vertex ${\lambda }_3\left(Q_1^{\psi },Q_2^{\psi },E\right)$ in the center-of-mass frame. The atoms and diatoms have momenta $Q_1^{\psi }=\left(E_{\psi 1},{\mathbf{q}}_1\right)$, $Q_1^{\phi }=\left(-E_{\psi 1}+E,-{\mathbf{q}}_1\right)$ and $Q_2^{\psi }=\left(E_{\psi 2},{\mathbf{q}}_2\right)$, $Q_2^{\phi }=\left(-E_{\psi 2}+E,-{\mathbf{q}}_2\right)$.

Figure 3

Graphical representation of the flow equation for ${\lambda }_3$. Full lines denote atoms $\psi$ and dashed lines denote diatoms $\phi$. The shaded circle denotes ${\lambda }_3$.

Figure 4

(Color online) The RG evolution of the momentum-dependent modified vertex $f_t\left(t_1,t_2\right)=4\left(3+p\right)q_1{q}_2{\lambda }_3\left(q_1,q_2,E\right)$ for bosons (a1–a4) and SU(2) fermions (b1–b4). Spatial momenta $t_1$ and $t_2$ and the RG time $t$ are discretized to $N=50$ intervals with a step $\Delta t=0.4$. The cartoons for bosons and fermions correspond to the discretized steps 10, 25, 35, and 50.

Figure 5

(Color online) Numerical evolution in the RG time $t$ of $f_t\left(t_1=0,t_2=0\right)$ for (A) system I and (B) system II. For SU(2) fermions [panel (B)] the modified vertex approaches a fixed-point solution. In the case of bosons [panel (A)], a limit cycle behavior is developed with a period $T_{\text{temp}}\approx 3.1$.

Figure 6

$\beta$ function of RG equation (D1) for (A) $D>0$ and (B) $D<0$. The arrows show the direction of the RG flow toward the IR.