Two-orbital physics of high-spin fermionic alkaline-earth atoms confined in a one-dimensional chain

We study the effect of the coupling between the electronic ground state of high-spin alkaline-earth fermionic atoms and their metastable optically excited state, when the system is confined in a one-dimensional chain, and show that the system provides a possible realization of a finite momentum pairing (Fulde-Ferrell-Larkin-Ovchinnikov-like) state without spin- or bare mass imbalance. We determine the $\beta$-functions of the renormalization group trajectories for general spin and analyze the structure of the possible gapped and gapless states in the hydrodynamic limit. Due to the SU($N$) symmetry in the spin space, complete mode separation can not be observed even in the fully gapless 2$N$-component Luttinger liquid state. Contrary, four velocities characterize the system. We solve the renormalization group equations for spin-9/2 strontium-87 isotope and analyze in detail its phase diagram. The fully gapless Luttinger liquid state does not stabilize in the two-orbital system of the ${}^{87}$Sr atoms, instead, different gapped non-Gaussian fixed points are identified either with dominant density or superconducting fluctuations. The superconducting states are stable in a nontrivial shaped region in the parameter space as a consequence of the coupling between the two electronic states.

Figure 1

The various bare vertices corresponding to the definition in Eq. (7). The solid lines refer to right moving particles, the dashed lines refer to left moving particles, and the wavy lines denote the interaction. With these definitions none of the interactions flips the spin, while the $\tilde{g}$ scatterings exchange the orbital states.

Figure 2

The fixed point structure of the ${}^{87}$Sr isotope on the plane $(g_{ge}^{+},g_e)$ settled by the value $g_{ge}^{-}/g_g=-3$. The different symbols related to different fixed point of the RG trajectories, the definition of the various symbols can be found in Table . Note that on both the horizontal and vertical axes, we used logarithmic scale.

Figure 3

The phase diagram of the ${}^{87}$Sr isotope on the plane $g_{ge}^{-}/g_g=-3$. The dark (blue) region shows the parameter regime where the density fluctuations dominates, while in the white regions the superconducting instabilities show slowest decay.

Figure 4

A zoom of the phase diagram in Fig. 3 to the moderated values of the interactions is presented here in order to get better visibility of the structure of the phase boundaries.

Figure 5

The vertex corrections up to one-loop order. The vertices with the same structure can be classified into two different channel: the upper line shows the contributions of the Cooper channel, the lower line shows that of the zero-sound channel. The solid lines refer to a right moving particle, the dashed lines refer to a left moving one, and the wavy lines denote the interaction. Note that in the short hand notation of the vertices, the interactions are denoted by a black dot.

Figure 6

The possible spin and orbital configurations for the first type of the Cooper channel vertices in Fig. 5.