Universal four-component Fermi gas in one dimension

A four-component Fermi gas in one dimension with a short-range four-body interaction is shown to exhibit a one-dimensional analog of the BCS-BEC crossover. Its low-energy physics is governed by a Tomonaga-Luttinger liquid with three spin gaps. The spin gaps are exponentially small in the weak coupling (BCS) limit where they arise from the charge-density-wave instability, and become large in the strong coupling (BEC) limit because of the formation of tightly bound tetramers. We investigate the ground-state energy, the sound velocity, and the gap spectrum in the BCS-BEC crossover and discuss exact relationships valid in our system. We also show that a one-dimensional analog of the Efimov effect occurs for five bosons while it is absent for fermions. Our work opens up a very rich field of universal few-body and many-body physics in one dimension.

Figure 1

Feynman diagrams describing the four-body scattering in vacuum. The dot represents the bare vertex ${\mathit{ic}}_0$ and the dashed line represents the scattering amplitude $i\mathcal{A}(E,p)$.

Figure 2

Feynman diagrams to renormalize five-body composite operators. The dashed line is a resummed propagator of $\varphi$ field which is equal to $-i\mathcal{A}$. The shaded bulb represents the vertex function $z(p)$.

Figure 3

Contribution of the contact density $\mathcal{C}$ to the fermion propagator ${\mathit{iG}}_{\sigma }(k_0,k)$. The integration over $k_0$ leads to the momentum distribution in Eq. (55).